CCatalan Pairs and Fishburn Triples
V´ıt Jel´ınek ∗ July 16, 2018
Abstract
Disanto, Ferrari, Pinzani and Rinaldi have introduced the concept of
Catalan pair , which is a pair ofpartial orders (
S, R ) satisfying certain axioms. They have shown that Catalan pairs provide a naturaldescription of objects belonging to several classes enumerated by Catalan numbers.In this paper, we first introduce another axiomatic structure (
T, R ), which we call the
Catalan pairof type 2 , which describes certain Catalan objects that do not seem to have an easy interpretation interms of the original Catalan pairs.We then introduce
Fishburn triples , which are relational structures obtained as a direct commongeneralization of the two types of Catalan pairs. Fishburn triples encode, in a natural way, the structureof objects enumerated by the Fishburn numbers, such as interval orders or Fishburn matrices. Thisconnection between Catalan objects and Fishburn objects allows us to associate known statistics onCatalan objects with analogous statistics of Fishburn objects. As our main result, we then show thatseveral known equidistribution results on Catalan statistics can be generalized to analogous results forFishburn statistics.
The Catalan numbers C n = n +1 (cid:0) nn (cid:1) are one of the most ubiquitous number sequences in enumerativecombinatorics. As of May 2013, Stanley’s Catalan Addendum [35] includes over 200 examples of classes ofcombinatorial objects enumerated by Catalan numbers, and more examples are constantly being discovered.The Fishburn numbers F n (sequence A022493 in OEIS [30]) are another example of a counting sequencethat arises in several seemingly unrelated contexts. The first widely studied combinatorial class enumeratedby Fishburn numbers is the class of interval orders, also known as ( + )-free posets. Their study waspioneered by Fishburn [16, 17, 18]. Later, more objects counted by Fishburn numbers were identified,including non-neighbor-nesting matchings [36], ascent sequences [3], Fishburn matrices [12, 14], severalclasses of pattern-avoiding permutations [3, 31] or pattern-avoiding matchings [26].It has been observed by several authors that various Fishburn classes contain subclasses enumerated byCatalan numbers [4, 7, 15, 24]. For instance, the Fishburn class of ( + )-free posets contains the subclassesof ( + , + )-free posets and ( + , N )-free posets, which are both enumerated by Catalan numbers. Theaim of this paper is to describe a close relationship between Catalan and Fishburn classes, of which theabove-mentioned inclusions are a direct consequence.At first sight, Catalan numbers and Fishburn numbers do not seem to have much in common. TheCatalan numbers can be expressed a simple formula C n = n +1 (cid:0) nn (cid:1) , they have a simply exponential asymp-totic growth C n = Θ(4 n n − / ), and admit an algebraic generating function, namely (1 − √ − x ) / F n is known, their growth is superexponential( F n = Θ( n !(6 /π ) n √ n ) as shown by Zagier [38]), and their generating function (cid:80) n ≥ (cid:81) nk =1 (1 − (1 − x ) k ),derived by Zagier [38], is not even D-finite [3]. ∗ Computer Science Institute, Faculty of Mathematics and Physics, Charles University, Malostransk´e n´amˇest´ı 25, 118 00,Prague, [email protected] . Supported by project CE-ITI (GBP202/12/G061) of the Czech Science Foundation. a r X i v : . [ m a t h . C O ] J a n evertheless, we will show that there is a close combinatorial relationship between families of objectscounted by Catalan numbers and those counted by Fishburn numbers. To describe this relationship, it isconvenient to represent Catalan and Fishburn objects by relational structures satisfying certain axioms.An example of such a structure are the so-called Catalan pairs, introduced by Disanto, Ferrari, Pinzaniand Rinaldi [7, 11] (see also [2]). In this paper, we first introduce another Catalan-enumerated family ofrelational structures which we call Catalan pairs of type 2. Next, we introduce a common generalization ofthe two types of Catalan pairs, which we call Fishburn triples.This interpretation of Catalan objects and Fishburn objects by means of relational structures allows usto detect a correspondence between known combinatorial statistics on Catalan objects and their Fishburncounterparts. This allows us to discover new equidistribution results for Fishburn statistics, inspired byanalogous previously known results for statistics of Catalan objects.In this paper, we need a lot of preparation before we can state and prove the main results. After recallingsome basic notions related to posets and relational structures (Subsection 1.1), we introduce interval ordersand Fishburn matrices, and characterize N -free and ( + )-free interval orders in terms of their Fishburnmatrices (Subsection 1.2). In Section 2, we define Catalan pairs of type 1 and 2, which are closely related to N -free and ( + )-free interval orders, respectively. We then observe that several familiar Catalan statisticshave a natural interpretation in terms of these pairs. Finally, in Section 3, we introduce Fishburn triples,which are a direct generalization of Catalan pairs. We then state and prove our main results, which,informally speaking, show that certain equidistribution results on Catalan statistics can be generalized toFishburn statistics. A relation (or, more properly, a binary relation ) on a set X is an arbitrary subset of the Cartesian product X × X . For a relation R ⊆ X × X and a pair of elements x, y ∈ X , we write xRy as a shorthand for( x, y ) ∈ R . The inverse of R , denoted by R − , is the relation that satisfies xR − y if and only if yRx . Twoelements x, y ∈ X are comparable by R (or R -comparable ) if at least one of the pairs ( x, y ) and ( y, x ) belongsto R .A relation R on X is said to be irreflexive if no element of X is comparable to itself. A relation R is transitive if xRy and yRz implies xRz for each x, y, z ∈ X . An irreflexive transitive relation is a partialorder . A set X together with a partial order relation R on X form a poset .An element x ∈ X is minimal in a relation R (or R -minimal for short) if there is no y ∈ X suchthat yRx . Maximal elements are defined analogously. This definition agrees with the standard notion ofminimal elements in partial orders, but note that we will use this notion even when R is not a partial order.We let Min( R ) denote the set of minimal elements of R , and min( R ) its cardinality. Similarly, Max( R ) isthe set of R -maximal elements and max( R ) its cardinality.A relational structure on a set X is an ordered tuple R = ( R , R , . . . , R k ) where each R i is a relationon X . The relations R i are referred to as the components of R , and the cardinality of X is the order of R .Given R as above, and given another relational structure S = ( S , . . . , S m ) on a set Y , we say that R contains S (or S is a substructure of R ) if k = m and there is an injection φ : Y → X such that for every i = 1 , . . . , k and every x, y ∈ Y , we have xS i y if and only if φ ( x ) R i φ ( y ). In such case we call the mapping φ an embedding of S into R . If R does not contain S , we say that R avoids S , or that R is S -free . We saythat R and S are isomorphic if there is an embedding of S into R which maps Y bijectively onto X .By a slight abuse of terminology, we often identify a relation R of X with a single-component relationalstructure R = ( R ). Therefore, e.g., saying that a relation R avoids a relation S means that the relationalstructure ( R ) avoids the relational structure ( S ).Throughout this paper, whenever we deal with the enumeration of relational structures, we treat themas unlabeled objects, that is, we consider an entire isomorphism class as a single object.We assume the reader is familiar with the concept of Hasse diagram of a partial order. We shall bemainly interested in partial orders that avoid some of the three orders + , + , and N , depicted onFigure 1. These classes of partial orders have been studied before. The ( + )-free posets are known as interval orders , and they are the prototypical example of a class of combinatorial structures enumerated by2 + 2 3 + 1 N Figure 1: The posets + (left), + (center), and N (right), which will frequently play the role of forbiddenpatterns in this paper.Fishburn numbers. The N -free posets ∗ are also known as series-parallel posets . They are exactly the posetsthat can be constructed from a single-element poset by a repeated application of direct sums and disjointunions. The ( + )-free posets do not seem to admit such a simple characterization, and their structuraldescription is the topic of ongoing research [20, 27, 33, 34].The ( + , + )-free posets, i.e., the posets avoiding both + and + , are also known as semiorders ,and have been introduced by Luce [28] in the 1950s. They are enumerated by Catalan numbers [24]. The( + , N )-free posets are enumerated by Catalan numbers as well, as shown by Disanto et al. [7, 10]. Let us briefly summarize several known facts about interval orders and their representations. A detailedtreatment of this topic appears, e.g., in Fishburn’s book [18].Let R = ( X, ≺ ) be a poset. An interval representation of R is a mapping I that associates to everyelement x ∈ X a closed interval I ( x ) = [ l x , r x ] in such a way that for any two elements x, y ∈ X , we have x ≺ y if and only if r x < l y . Note that we allow the intervals I ( x ) to be degenerate, i.e., to consist of asingle point.A poset has an interval representation if and only if it is ( + )-free, i.e., if it is an interval order. Aninterval representation I is minimal , if it satisfies these conditions: • for every x ∈ X , the endpoints of I ( x ) are positive integers • there is a positive integer m ∈ N such that for every k ∈ { , , . . . , m } there is an interval I ( x ) whoseright endpoint is k , as well as an interval I ( y ) whose left endpoint is k .Each interval order R has a unique minimal interval representation. The integer m , which correspondsto the number of distinct endpoints in the minimal representation, is known as the magnitude of R . Notethat x ∈ X is a minimal element of R if and only if the left endpoint of I ( x ) is equal to 1, and x is a maximalelement if and only if the right endpoint of I ( x ) is equal to m .Two elements x, y of a poset R = ( X, ≺ ) are indistinguishable if for every element z ∈ X we have theequivalences x ≺ z ⇐⇒ y ≺ z and z ≺ x ⇐⇒ z ≺ y . In an interval order R with a minimal intervalrepresentation I , two elements x, y are indistinguishable if and only if I ( x ) = I ( y ). An interval order is primitive if it has no two indistinguishable elements. Every interval order R can be uniquely constructedfrom a primitive interval order R (cid:48) by replacing each element x ∈ R (cid:48) by a group of indistinguishable elementscontaining x .Given a matrix M , we use the term cell ( i, j ) of M to refer to the entry in the i -th row and j -th columnof M , and we let M i,j denote its value. We assume that the rows of a matrix are numbered top to bottom,that is, the top row has number 1. The weight of a matrix M is the sum of its entries † . Similarly, theweight of a row (or a column, or a diagonal) of a matrix is the sum of the entries in this row (or column ordiagonal). ∗ Beware that some authors (e.g. Khamis [23]) give the term ‘ N -free poset’ a meaning subtly different from ours. † Some earlier papers use the term size of M instead of weight of M . However, in other contexts the term ‘size’ often refersto the number of rows of a matrix. We therefore prefer to use the less ambiguous term ‘weight’ in this paper. bc d e fg a b c d ef g Figure 2: Three Fishburn structures: an interval order, its minimal interval representation, and its Fishburnmatrix. In this paper, we use the convention that cells of value 0 in a matrix are depicted as empty boxes.A
Fishburn matrix is an upper-triangular square matrix of nonnegative integers whose every row andevery column has nonzero weight. In other words, an m × m matrix M of nonnegative integers is a Fishburnmatrix if it satisfies these conditions: • M i,j = 0 whenever i > j , • for every i ∈ { , . . . , m } there is a j such that M i,j >
0, and • for every j ∈ { , . . . , m } there is an i such that M i,j > R = ( X, ≺ ) of magnitude m with minimal representation I , we represent it by an m × m matrix M such that M i,j is equal to |{ x ∈ X ; I ( x ) = [ i, j ] }| . We then say that an element x ∈ X is represented by the cell ( i, j ) of M , if I ( x ) = [ i, j ]. Thus, every element of R is represented by a unique cellof M , and M i,j is the number of elements of R represented by cell ( i, j ) of M . This correspondence yieldsa bijection between Fishburn matrices and interval orders. For an element x ∈ X , we let c x denote the cellof M representing x . Conversely, for a cell c of M , we let X c be the set of elements of X represented by c .Note that the value in cell c is precisely the cardinality of X c .In the rest of this paper, we will use Fishburn matrices as our main family of Fishburn-enumeratedobject. We now show how the basic features of interval orders translate into matrix terminology. Observation 1.1.
Let R = ( X, ≺ ) be an interval order, and let M be the corresponding Fishburn matrix.Let x and x (cid:48) be two elements of R , represented respectively by cells ( i, j ) and ( i (cid:48) , j (cid:48) ) of M . The followingholds: • The size of R is equal to the weight of M . • The number of minimal elements of R is equal to the weight of the first row of M , and the number ofmaximal elements of R is equal to the weight of the last column of M . • The elements x and x (cid:48) are indistinguishable in R if and only if they are represented by the same cellof M , i.e., i = i (cid:48) and j = j (cid:48) . • We have x (cid:48) ≺ x if and only if j (cid:48) < i . Let M be a Fishburn matrix, and let c = ( i, j ) and c (cid:48) = ( i (cid:48) , j (cid:48) ) be two cells of M such that i ≤ j and i (cid:48) ≤ j (cid:48) , i.e., the two cells are on or above the main diagonal. We shall frequently use the followingterminology (see Figure 3): • Cell c is greater than cell c (cid:48) (and c (cid:48) is smaller than c ), if j (cid:48) < i . The cells c and c (cid:48) are incomparable ifneither of them is greater than the other. These terms are motivated by the last part of Observation 1.1.Note that two cells c and c (cid:48) of M are comparable if and only if the smallest rectangle containing both c and c (cid:48) also contains at least one cell strictly below the main diagonal of M .4 c j ji i cc j ji ic cj j ii ( i ) ( ii ) ( iii ) Figure 3: Mutual positions of two cells c and c (cid:48) in a Fishburn matrix: (i) c is greater than c (cid:48) , (ii) c is strictlySouth-West of c (cid:48) , and (iii) c is strictly South-East of c (cid:48) . • Cell c is South of c (cid:48) , if i > i (cid:48) and j = j (cid:48) . North, West and East are defined analogously. Note that inall these cases, the two cells c and c (cid:48) are incomparable. • Cell c is strictly South-West (or strictly SW) from c (cid:48) (and c (cid:48) is strictly NE from c ) if i > i (cid:48) and j < j (cid:48) .This again implies that the two cells are incomparable. • Cell c is strictly South-East (or strictly SE) from c (cid:48) (and c (cid:48) is strictly NW from c ) if c and c (cid:48) areincomparable, and moreover i > i (cid:48) and j > j (cid:48) . • Cell c is weakly SW from c (cid:48) is c is South, West of strictly SW of c (cid:48) . Weakly NE, weakly NW andweakly SE are defined analogously.With this terminology, ( + )-avoidance and N -avoidance of interval orders may be characterized interms of Fishburn matrices, as shown by the next lemma, whose first part essentially already appears in thework of Dukes et al. [12]. Refer to Figure 4. Lemma 1.2.
Let R = ( X, ≺ ) be an interval order represented by a Fishburn matrix M . Let x and y be twoelements of X , represented by cells c x and c y of M .1. The cell c x is strictly SW of c y if and only if X contains two elements u and v which together with x and y induce a copy of + in R such that u ≺ x ≺ v , and y is incomparable to each of u, x, v .2. The cell c x is strictly NW of c y if and only if X contains two elements u and v which together with x and y induce a copy of N in R such that u ≺ y , u ≺ v and x ≺ v , and the remaining pairs among u, v, x, y are incomparable.In particular, R is ( + )-free if and only if the matrix M has no two distinct nonzero cells in a strictlySW position, and R is N -free if and only if M has no two distinct nonzero cells in a strictly NW position.Proof. We will only prove the second part of the lemma, dealing with N -avoidance. The first part can beproved by a similar argument, as shown by Dukes at al. [12, proof of Proposition 16].Suppose first that R contains a copy of N induced by four elements u , v , x and y , which form exactlythree comparable pairs u ≺ y , u ≺ v and x ≺ v . Let c u = ( i u , j u ), c v = ( i v , j v ), c x = ( i x , j x ) and c y = ( i y , j y )be the cells of M representing these four elements. We claim that c x is strictly NW from c y . To see this,note that c x is smaller than c v while c y is not smaller than c v , hence j x < i v ≤ j y . Similarly, c y is greaterthan c u while c x is not, implying that i x ≤ j u < i y . Since c x and c y are incomparable, it follows that c x isstrictly NW from c y .Conversely, suppose that M has two nonzero cells c x = ( i x , j x ) and c y = ( i y , j y ), with c x being strictlyNW from c y . We thus have the inequalities i x < i y ≤ j x < j y , where the inequality i y ≤ j x follows fromthe fact that c x and c y are incomparable. Let c u be any nonzero cell in column i x . This choice guaranteesthat c u is incomparable to c x and smaller than c y . Similarly, let c v be a nonzero cell in row j y . Then c v isincomparable to c y and greater than both c x and c u . Any four elements x , y , u and v of R represented bythe four cells c x , c y , c u and c v induce a copy of N . 5 x c y c v c u uxv y c x c y c v c u uxv y Figure 4: The two parts of Lemma 1.2. Left: if c x is strictly SW of c y , we get an occurrence of + . Right:if c x is strictly NW of c y , we get an an occurrence of N . We begin by defining the concept of Catalan pairs of type 1, originally introduced by Disanto et al. [7], whocalled them simply ‘Catalan pairs’.
Definition 2.1. A Catalan pair of type 1 (or
C1-pair for short) is a relational structure (
S, R ) on a finiteset X satisfying the following axioms.C1a) S and R are both partial orders on X .C1b) Any two distinct elements of X are comparable by exactly one of the orders S and R .C1c) For any three distinct elements x, y, z satisfying xSy and yRz , we have xRz .We let C In denote the set of unlabeled C1-pairs on n vertices.Note that axiom C1c might be replaced by the seemingly weaker condition that X does not contain threedistinct elements x , y and z satisfying xSy , yRz and xSz (see the left part of Figure 5). Indeed, if xSy and yRz holds for some x , y and z , then both zRx and zSx would contradict axiom C1a or C1b, so therole of C1c is merely to exclude the possibility xSz , leaving xRz as the only option. This also shows thatthe definition of C1-pairs would not be affected if we replaced C1c by the following axiom, which we denoteC1c*: “For any three distinct elements x, y, z satisfying xSy and zRy , we have zRx .”We remark that it is easy to check that in a C1-pair ( S, R ) on a vertex set X , the relation S ∪ R is alinear order on X .As shown by Disanto et al. [7], if ( S, R ) is a C1-pair, then R is a ( + , N )-free poset, and conversely,for any ( + , N )-free poset R , there is, up to isomorphism, a unique C1-pair ( S, R ). Since there are,up to isomorphism, C n distinct ( + , N )-free posets on an n -elements set, this implies that C1-pairs areenumerated by Catalan numbers. In fact, the next lemma shows that the first component of a C1-pair ( S, R )can be easily reconstructed from the Fishburn matrix representing the second component.
Lemma 2.2.
Let ( S, R ) be a C1-pair, and let M be the Fishburn matrix representing the poset R . Then S has the following properties:(a) If c is a nonzero cell of M and X c the set of elements represented by c , then X c is a chain in S , thatis, the restriction of S to X c is a linear order.(b) If c and d are distinct nonzero cells of M , representing sets of elements X c and X d respectively, andif c is weakly SW from d , then for any x ∈ X c and y ∈ X d we have xSy .(c) Apart from the situations described in parts (a) and (b), no other pair ( x, y ) ∈ X is S -comparable. y zS SR xy zT TR xy zT TR Figure 5: Left: the substructure forbidden in C1-pairs by axiom C1c. Right: the two substructures forbiddenin C2-pairs by axiom C2c.
Proof.
Let (
S, R ) be a C1-pair. Property (a) of the Lemma follows directly from the fact that the elements of X c form an antichain in R . To prove property (b), fix distinct nonzero cells c = ( i c , j c ) and d = ( i d , j d ) suchthat c is weakly SW from d , and choose x ∈ X c and y ∈ X d arbitrarily. Since x and y are R -incomparableby Observation 1.1, we must have either xSy or ySx . Suppose for contradiction that ySx holds. As c isweakly SW from d , we know that i c > i d or j c < j d . Suppose that i c > i d , as the case j c < j d is analogous.Let e be any nonzero cell in column i d of M , and let z ∈ X be an element represented by e . Note that e is smaller than c and incomparable to d . It follows that y and z are comparable by S . However, if zSy holds, then we get a contradiction with the transitivity of S , due to ySx and zRx . On the other hand, if ySz holds, we see that x , y and z induce the substructure forbidden by axiom C1c.To see that property (c) holds, note that by Lemma 1.2, the matrix M has no two nonzero cells instrictly NW position. Thus, if x and y are distinct elements of X represented by cells c x and c y , then either c x = c y and x, y are S -comparable by property (a), or c x and c y are in a weakly SW position and x, y are S -comparable by (b), or one of the two cells is smaller than the other, which means that the two elementsare R -comparable.Various Catalan-enumerated objects, such as noncrossing matchings, Dyck paths, or 132-avoiding per-mutations, can be encoded in a natural way as C1-pairs (see [7]). However, there are also examples ofCatalan objects, such as nonnesting matchings or ( + , + )-free posets, possessing a natural underlyingstructure that satisfies a different set of axioms. This motivates our next definition. Definition 2.3. A Catalan pair of type 2 (or
C2-pair for short) is a relational structure (
T, R ) on a finiteset X with the following properties.C2a) R and T ∪ R are both partial orders on X .C2b) Any two distinct elements of X are comparable by exactly one of the relations T and R .C2c) There are no three distinct elements x, y, z ∈ X satisfying xT y , xT z and yRz , and also no threeelements x, y, z ∈ X satisfying yT x , zT x and yRz .We let C IIn denote the set of unlabeled C2-pairs on n vertices. Lemma 2.4. If ( T, R ) is a C2-pair on a vertex set X , then R is a ( + , + )-free poset.Proof. Let (
T, R ) be a C2-pair. Suppose for contradiction that R contains a copy of + induced by fourelements { x, y, u, v } , with xRy and uRv . Any two R -incomparable elements must be comparable in T ,so we may assume, without loss of generality, that xT u holds. Then xT v holds by transitivity of T ∪ R ,contradicting axiom C2c. Similar reasoning shows that R is ( + )-free.As in the case of C1-pairs, a C2-pair ( T, R ) is uniquely determined by its second component, and its firstcomponent can be easily reconstructed from the Fishburn matrix representing the second component.
Lemma 2.5.
Let R be ( + , + )-free poset represented by a Fishburn matrix M . Let T be a relationsatisfying these properties: a) If c is a nonzero cell of M and X c the set of elements represented by c , then X c is a chain in T , thatis, the restriction of T to X c is a linear order.(b) If c and d are distinct nonzero cells of M , representing sets of elements X c and X d respectively, andif c is weakly NW from d , then for any x ∈ X c and y ∈ X d we have xT y .(c) Apart from the situations described in parts (a) and (b), no other pair ( x, y ) ∈ X is T -comparable.Then ( T, R ) is a C2-pair. Moreover, if ( T (cid:48) , R ) is another C2-pair, then ( T, R ) and ( T (cid:48) , R ) are isomorphicrelational structures. It follows that on an n -element vertex set X , there are C n isomorphism types ofCatalan pairs, where C n is the n -th Catalan number.Proof. Let R be a ( + , + )-free partial order on a ground set X , represented by a Fishburn matrix M .Let T be a relation satisfying properties (a), (b) and (c) of the lemma.We claim that ( T, R ) is a C2-pair. It is easy to see that (
T, R ) satisfies axiom C2a. Let us verify that C2bholds as well, i.e., any two distinct elements x, y ∈ X are comparable by exactly one of T , R . Since clearlyno two elements can be simultaneously T -comparable and R -comparable, it is enough to show that any two R -incomparable elements x, y ∈ X are T -comparable. If x and y are also R -indistinguishable, then they arerepresented by the same cell c of M , and they are T -comparable by (a). If x and y are R -distinguishable,then they are represented by two distinct cells, say c x and c y . By Lemma 1.2, M has no pair of nonzerocells in strictly SW position, and hence c x and c y must be in weakly NW position. Hence x and y are T -comparable by (b).To verify axiom C2c, assume first, for contradiction, that there are three elements x, y, z ∈ X satisfying xT y , xT z and yRz . Let c x = ( i x , j x ), c y = ( i y , j y ) and c z = ( i z , j z ) be the cells of M representing x , y and z , respectively. Since c y is smaller than c z , we see that j y < i z . Since ( x, y ) belongs to T , we see that either c x = c y or c x is weakly NW from c y . In any case, c x is smaller than c z , contradicting xT z . An analogousargument shows that there can be no three elements x, y, z satisfying yT x , zT x and yRz . We conclude that( T, R ) is a C2-pair.It remains to argue that for the ( + , + )-free poset R , there is, up to isomorphism, at most onerelation T forming a C2-pair with R . Fix a relation T (cid:48) such that ( T (cid:48) , R ) is a C2-pair. We will first showthat T (cid:48) satisfies the properties (a), (b) and (c) of the lemma. Property (a) follows directly from axioms C2aand C2b.To verify property (b), fix elements x, y ∈ X represented by distinct cells c x = ( i x , j x ) and c y = ( i y , j y )such that c x is weakly NW from c y . We claim that ( x, y ) ∈ T (cid:48) . Suppose for contradiction that this is notthe case. Since x and y are R -incomparable, this means that ( y, x ) ∈ T (cid:48) by axiom C2b. Since c x is weaklyNW from c y , we know that i x < i y or j x < j y . Suppose that i x < i y (the case j x < j y is analogous). Let c z be any nonzero cell in column i x and z ∈ X an element represented by c z . This choice of z guaranteesthat ( z, y ) ∈ R while z and x are R -incomparable. From zRy and yT (cid:48) x we deduce, by the transitivity of R ∪ T (cid:48) and the R -incomparability of x and z , that ( z, x ) belongs to T (cid:48) , which contradicts axiom C2c. Thefact that T (cid:48) satisfies (c) follows from axiom C2b and the fact that no two nonzero cells of M are in strictlySW position.We conclude that when ( T, R ) and ( T (cid:48) , R ) are C2-pairs, the relations T and T (cid:48) may only differ byprescribing different linear orders on the classes of R -indistinguishable elements. This easily implies that( T, R ) and ( T (cid:48) , R ) are isomorphic relational structures. We have seen that Catalan pairs naturally encode the structure of Fishburn matrices representing ( + , N )-free and ( + , + )-free posets. However, to exploit known combinatorial properties of Catalan-enumeratedobjects, it is convenient to relate Catalan pairs to more familiar Catalan objects. Our Catalan objects ofchoice are the Dyck paths. Definition 2.6. A Dyck path of order n is a lattice path P joining the point (0 ,
0) with the point ( n, n ),consisting of n up-steps and n right-steps , where an up-step joins a point ( i, j ) to ( i, j + 1) and a right-step8 , t s , Figure 6: A Dyck path (solid) with a tunnel t (dashed). The two steps associated to t are highlighted inbold. The unit square s , is below the path while s , is above it.joins ( i, j ) to ( i + 1 , j ), and moreover, every point ( i, j ) ∈ P satisfies i ≤ j . We let D n denote the set ofDyck paths of order n .For a Dyck path P , we say that a step s of P precedes a step s (cid:48) of P if s appears on P before s (cid:48) whenwe follow P from (0 ,
0) to ( n, n ).Given a Dyck path P , a tunnel of P is a segment t parallel to the diagonal line of equation y = x , suchthat the bottom-left endpoint of t is in the middle of an up-step of P , the top-right endpoint is in the middleof a right-step of P , and all the internal points of t are strictly below the path P . See Figure 6. We refer tothe up-step and the right-step that contain the two endpoints of t as the steps associated to t .Note that a Dyck path of order n has exactly n tunnels, and every step of the path is associated to aunique tunnel.Let t and t be two distinct tunnels of a Dyck path P . Let u i and r i be the up-step and right-stepassociated to t i , respectively. Suppose that u precedes u on the path P . If r precedes r on P , we saythat t is nested within t . If r precedes r , then we easily see that r also precedes u on P . In such case,we say that t precedes t .The following construction, due to Disanto et al. [7], shows how to represent a Dyck path by a C1-pair. Fact 2.7 ([7]) . Let P be a Dyck path and let X be the set of tunnels of P . Define a relational structure C I ( P ) = ( S, R ) on X as follows: for two distinct tunnels t and t , put ( t , t ) ∈ S if and only if t isnested within t , and ( t , t ) ∈ R if and only if t precedes t . Then C I ( P ) = ( S, R ) is a C1-pair, and thisconstruction yields a bijection between Dyck paths of order n and isomorphism types of C1-pairs of order n . There is also a simple way to encode a Dyck path by a C2-pair, which we now describe. Let P be aDyck path of order n and let ( i, j ) ∈ { , . . . , n } be a lattice point. Let s i,j be the axis-aligned unit squarewhose top-right corner is the point ( i, j ) (see Figure 6). We say that s i,j is above the Dyck path P if theinterior of s i,j is above P (with the boundary of s i,j possibly overlapping with P ). Notice that s i,j is above P if and only if the i -th right-step of P is preceded by at most j up-steps. If s i,j is not above P , we saythat it is below P ; see Figure 6. Lemma 2.8.
Let P be a Dyck path of order n , and let X be the set { , , . . . , n } . Define a relationalstructure C II ( P ) = ( T, R ) on X as follows: for any i, j ∈ X , put ( i, j ) ∈ T if and only if i < j and s i,j isbelow P , and put ( i, j ) ∈ R if and only if i < j and s i,j is above P . Then C II ( P ) is a C2-pair, and thisconstruction is a bijection between Dyck paths of order n and isomorphism types of C2-pairs.Proof. It is routine to verify that C II ( P ) satisfies the axioms of C2-pairs, and that distinct Dyck paths mapto distinct C2-pairs. Since both Dyck paths of order n and C2-pairs of order n are enumerated by Catalannumbers, the mapping is indeed a bijection. 9e will now focus on combinatorial statistics on Catalan objects. For our purposes, a statistic on a set A is any nonnegative integer function f : A → N . For an integer k and a statistic f , we use the notation A [ f = k ] as shorthand for { x ∈ A : f ( x ) = k } . This notation extends naturally to two or more statistics,e.g., if g is another statistic on A , then A [ f = k, g = (cid:96) ] denotes the set { x ∈ A : f ( x ) = k, g ( x ) = (cid:96) } .Two statistics f and g are equidistributed (or have the same distribution ) on A , if A [ f = k ] and A [ g = k ]have the same cardinality for every k . The statistics f and g have symmetric joint distribution on A , if A [ f = k, g = (cid:96) ] and A [ f = (cid:96), g = k ] have the same cardinality for every k and (cid:96) . Clearly, if f and g havesymmetric joint distribution, they are equidistributed.Recall that for a binary relation R on a set X , an element x ∈ X is minimal, if there is no y ∈ X \ { x } such that ( y, x ) is in R . Recall also that Min( R ) is the set of minimal elements of R , and min( R ) is thecardinality of Min( R ). Similarly, Max( R ) is the set of maximal elements of R , and max( R ) its cardinality.Let P be a Dyck path of order n , and let C I ( P ) = ( S, R ) be the corresponding C1-pair, as defined inFact 2.7. We will be interested in the four statistics min( S ), max( S ), min( R ) and max( R ). It turns outthat these statistics correspond to well known statistics of Dyck paths. To describe the correspondence, weneed more terminology.For a Dyck path P , the initial ascent is the maximal sequence of up-steps preceding the first right-step,and the final descent is the maximal sequence of right-steps following the last up-step. Let asc( P ) anddes( P ) denote the length of the initial ascent and final descent of P , respectively. A return of P is a right-step whose right endpoint touches the diagonal line y = x . Let ret( P ) denote the number of returns of P .Finally, a peak of P is an up-step of P that is immediately followed by a right-step, and pea( P ) denotes thenumber of peaks of P . Observation 2.9.
Let P be a Dyck path and let ( S, R ) = C I ( P ) be the corresponding C1-pair. • A tunnel t of P is minimal in R if and only if the up-step associated to t precedes the first right-stepof P . In particular min( R ) = asc( P ) . Symmetrically, t is maximal in R if and only if its associatedright-step succeeds the last up-step of P . Hence, max( R ) = des( P ) . • A tunnel t of P is maximal in S if and only if its associated right-step is a return of P . Hence, max( S ) = ret( P ) . • A tunnel t of P is minimal in S if and only if its associated up-step is immediately succeeded by itsassociated right-step. Hence, min( S ) = pea( P ) . Suppose now that (
T, R ) = C II ( P ) is a C2-pair representing a Dyck path P by the bijection of Lemma 2.8.We again focus on the statistics min( T ), max( T ), min( R ) and max( R ). It turns out that they againcorrespond to the Dyck path statistics introduced above. Observation 2.10.
Let P be a Dyck path of order n and let ( T, R ) = C II ( P ) be the corresponding C2-pair. • An element i ∈ { , , . . . , n } is minimal in R if and only if the unit square s ,i is below P , whichhappens if and only if asc( P ) ≥ i . In particular, asc( P ) = min( R ) . Symmetrically, des( P ) = max( R ) . • An element i ∈ { , , . . . , n } is minimal in T if and only if for every j < i , the unit square s j,i isabove P , which is if and only if the point ( i − , i − belongs to P . It follows that min( T ) = ret( P ) .Symmetrically, i ∈ { , , . . . , n } is maximal in T if and only if for every j > i the square s i,j is above P , which is if and only if the point ( i, i ) belongs to P . It follows that max( T ) = ret( P ) . The statistics asc, des, ret and pea are all very well studied. We now collect some known facts aboutthem.
Fact 2.11.
Let D n be the set of Dyck paths of order n . • The sets D n [asc = k ] , D n [des = k ] and D n [ret = k ] all have cardinality k n − k (cid:0) n − kn (cid:1) . See [30, A033184].Among the three statistics asc , des and ret , any two have symmetric joint distribution on D n . The set D n [pea = k ] has cardinality N ( n, k ) = k (cid:0) n − k − (cid:1)(cid:0) nk − (cid:1) . The numbers N ( n, k ) are known asNarayana numbers [30, A001263]. The Narayana numbers satisfy N ( n, k ) = N ( n, n − k + 1) , andhence D n [pea = k ] has the same cardinality as D n [pea = n − k + 1] . • In fact, for any n , k , (cid:96) , and m , the set D n [asc = k, ret = (cid:96), pea = m ] has the same cardinality asthe set D n [asc = (cid:96), ret = k, pea = n − m + 1] . This follows, e.g., from an involution on Dyck pathsconstructed by Deutsch [6]. For future reference, we rephrase some of these facts in the terminology of Catalan pairs. Recall that C In and C IIn denote respectively the set of C1-pairs and the set of C2-pairs of order n . Proposition 2.12.
For any n ≥ , there is a bijection ψ : C In → C IIn with the following properties. Let ( S, R ) ∈ C In be a C1-pair, and let ( T (cid:48) , R (cid:48) ) ∈ C IIn be its image under ψ . Then • max( S ) = max( T (cid:48) ) , and • max( R ) = max( R (cid:48) ) .Proof. Given a C1-pair (
S, R ) ∈ C In , fix the Dyck path P ∈ D n satisfying ( S, R ) = C I ( P ), and define theC2-pair ( T (cid:48) , R (cid:48) ) = C II ( P ). The mapping ( S, R ) (cid:55)→ ( T (cid:48) , R (cid:48) ) is the bijection ψ . By Observations 2.9 and2.10, we have max( S ) = ret( P ) = max( T (cid:48) ), and max( R ) = des( P ) = max( R (cid:48) ). Proposition 2.13.
For each n ≥ , the two statistics max( S ) and max( R ) have symmetric joint distributionon C1-pairs ( S, R ) ∈ C In .Proof. This follows from Observation 2.9 and from the first part of Fact 2.11.
In this section, we will show that objects from certain Fishburn-enumerated families can be representedby a triple (
T, S, R ) of relations, satisfying axioms which generalize the axioms of Catalan pairs. This willallow us to extend the statistics asc, des, ret and pea to Fishburn objects, where they admit a naturalcombinatorial interpretation. We will then show, as the main results of this paper, that some of the classicalequidistribution results listed in Fact 2.11 can be extended to Fishburn objects.
Definition 3.1. A Fishburn triple (or
F-triple for short) is a relational structure (
T, S, R ) on a set X satisfying the following axioms:Fa) S , R , and T ∪ R are partial orders on X .Fb) Any two distinct elements of X are comparable by exactly one of T , S , or R .C1c) For any three distinct elements x, y, z satisfying xSy and yRz we have xRz .C1c*) For any three distinct elements x, y, z satisfying xSy and zRy we have zRx .C2c) There are no three distinct elements x, y, z ∈ X satisfying xT y , xT z and yRz , and also no threeelements x, y, z ∈ X satisfying yT x , zT x and zRy .As with Catalan pairs, the Fishburn triples may equivalently be described as structures avoiding certainsubstructures of size at most three; see Figure 7.Observe that a relational structure ( S, R ) is a C1-pair if and only if ( ∅ , S, R ) is an F-triple, and a relationalstructure ( T, R ) is a C2-pair if and only if ( T, ∅ , R ) is an F-triple. In this way, F-triples may be seen as acommon generalization of the two types of Catalan pairs.Note also that ( T, S, R ) is an F-triple if and only if ( T − , S, R − ) is an F-triple. We will refer tothe mapping ( T, S, R ) (cid:55)→ ( T − , S, R − ) as the trivial involution on F-triples. We may restrict the trivialinvolution to C1-pairs and C2-pairs, with a C1-pair ( S, R ) being mapped to (
S, R − ) and a C2-pair ( T, R )being mapped to ( T − , R − ). When representing Catalan pairs of either type as Dyck paths, as we did inSubsection 2.1, the trivial involution acts on Dyck paths of order n as a mirror reflection whose axis is theline x + y = n . 11 y zS SR xy zT TR xy zT TRxy zS TRxy zS TR Figure 7: Left: the three minimal structures satisfying axioms Fa and Fb, but not C1c or C1c*. Right: thetwo structures excluded by axiom C2c.
Lemma 3.2. If ( T, S, R ) is an F-triple on a vertex set X , then R is an interval order. Let M be theFishburn matrix of R . Let x and y two elements of X , represented by cells c x and c y of M .1. If c x is strictly SW of c y , then ( x, y ) ∈ S .2. If c x is strictly NW of c y , then ( x, y ) ∈ T .3. If ( x, y ) ∈ S , then c x is weakly SW of c y .4. If ( x, y ) ∈ T , then c x is weakly NW of c y .Proof. Let (
T, S, R ) be an F-triple on a set X . Let us prove that R is ( + )-free. For contradiction,assume X contains four distinct elements x, x (cid:48) , y, y (cid:48) , such that ( x, x (cid:48) ) and ( y, y (cid:48) ) belong to R , while all otherpairs among these four elements are R -incomparable. By axiom Fb, x and y are either S -comparable or T -comparable. However, if xSy holds, then C1c implies that xRy (cid:48) holds as well, which is impossible. If xT y holds, then transitivity of T ∪ R implies xT y (cid:48) or xRy (cid:48) . However, xT y (cid:48) is excluded by axiom C2c and xRy (cid:48) contradicts the choice of x, x (cid:48) , y, y (cid:48) . This shows that R is ( + )-free.We now prove the four numbered claims of the lemma. Let M be the Fishburn matrix of R , and let x and y be two elements of X represented by cells c x = ( i x , j x ) and c y = ( i y , j y ).To prove the first claim, suppose that c x is strictly SW of c y . By Lemma 1.2, there are two elements u, v ∈ X such that u, v, x, y induce a copy of + , where ( u, x ), ( x, v ), and ( u, v ) belong to R , and y is R -incomparable to x , u , and v . Now ySx would imply yRv by C1c, which is impossible, since y and v are R -incomparable. Next, yT x would imply yT v by transitivity of T ∪ R , contradicting C2c. Similarly, xT y implies uT y , again contradicting C c . This leaves xSy as the only option.To prove the second claim, suppose that c x is strictly NW of c y . By Lemma 1.2, there are elements u and v such that u , v , x and y induce a copy of N in R , with precisely the three pairs ( x, v ), ( u, v ) and( u, y ) belonging to R . We want to prove that ( x, y ) is in T . Consider the alternatives: yT x forces yT v bytransitivity of T ∪ R , contradicting C2c; on the other hand xSy forces xRu by C1c*, while ySx forces yRv by C1c, which both contradict the choice of u and v . We conclude that yT x is the only possibility.For the third claim, proceed by contradiction, and assume that xSy holds, but c x is not weakly SWof c y . This means that i x < i y or j x > j y . Suppose that i x < i y , the other case being analogous. Let c z bea nonzero cell of M in column i x , and let z be an element represented by c z . By the choice of z , we knowthat zRy holds and that x and z are R -incomparable. This contradicts axiom C1c*.Finally, suppose that xT y holds and c x is not weakly NW from c y . Since x and y are R -incomparable,this means that i x > i y or j x > j y . Suppose that i x > i y . Let z be an element represented by a cellin column i y , so that zRx holds, while y and z are R -incomparable. Transitivity of T ∪ R implies zT y ,contradicting C2c.Suppose we are given an interval order R on a set X , with a corresponding Fishburn matrix M , and wewould like to extend R into an F-triple ( T, S, R ). The four conditions in Lemma 3.2 put certain constraintson T and S , but in general they do not determine T and S uniquely. In particular, if x and y are two12lements represented by cells c x and c y that belong to the same row or column of M , then Lemma 3.2 doesnot say whether x and y should be T -comparable or S -comparable.To obtain a unique Fishburn triple for a given R , we need to impose additional restrictions to disam-biguate the relations between elements represented by cells in the same row or column of M . We willconsider two ways of imposing such restrictions. In the first way, all ambiguous pairs end up S -comparable,while in the second way they will be T -comparable. Definition 3.3.
Let R be an interval order on a set X , and let M be its Fishburn matrix. The Fishburntriple of type 1 of R (or F1-triple of R ) is the relational structure ( T , S , R ) on the set X , in which T and S are determined by these rules: • For any two elements x, y ∈ X , represented by distinct cells c x and c y in M , we have xT y if and onlyif c x is strictly NW from c y , and we have xS y if and only if c x is weakly SW from c y . • If c is a cell of M and X c ⊆ X the set of elements represented by c , then S induces a chain in X c and T induces an antichain in X c .Similarly, for R and M as above, a Fishburn triple of type 2 of R (or F2-triple of R ) is the relationalstructure ( T , S , R ) on X , with T and S defined as follows: • For any two elements x, y ∈ X , represented by distinct cells c x and c y in M , we have xT y if and onlyif c x is weakly NW from c y , and we have xS y if and only if c x is strictly SW from c y . • If c is a cell of M and X c ⊆ X the set of elements represented by c , then S induces an antichain in X c and T induces a chain in X c .Note that the F1-triple and the F2-triple are up to isomorphism uniquely determined by the intervalorder R . Lemma 3.4.
For an interval order R , its F1-triple ( T , S , R ) and its F2-triple ( T , S , R ) are Fishburntriples.Proof. Consider the F1-triple ( T , S , R ). Clearly, it satisfies the axioms Fa and Fb of F-triples. To checkaxiom C1c, pick three elements x, y, z ∈ X , with xS y and yRz , and let c x = ( i x , j x ), c y = ( i y , j y ) and c z = ( i z , j z ) be the corresponding cells of M . Then xS y implies j x ≤ j y , and yRz implies j y < i z . Togetherthis proves j x < i z , and consequently xRz , and axiom C1c holds. Axiom C1c* can be proved by an analogousargument.To prove axiom C2c, consider again three elements x, y, z ∈ X represented by the cells c x = ( i x , j x ), c y = ( i y , j y ) and c z = ( i z , j z ). To prove the first part of the axiom, assume for contradiction that xT y , xT z and yRz holds. Since c x is strictly NW from c y , we have j x < j y . From yRz we get j y < i z , hence j x < i z implying xRz , which is a contradiction. The second part of axiom C2c is proved analogously.An analogous reasoning applies to ( T , S , R ) as well.Note that the trivial involution on F-triples maps F1-triples to F1-triples and F2-triples to F2-triples.From Lemma 1.2, we deduce that an interval order R is N -free if and only if its F1-triple has the form( ∅ , S, R ), which means that ( S, R ) is a C1-pair, while R is ( + )-free if and only if its F2-triple has theform ( T, ∅ , R ), implying that ( T, R ) is a C2-pair. Thus, F1-triples are a generalization of C1-pairs, whileF2-triples generalize C2-pairs.Our main goal is to use F1-triples and F2-triples to identify combinatorial statistics on Fishburn objectsthat satisfy nontrivial equidistribution properties. Inspired by the Catalan statistics explored in Subsec-tion 2.1, we focus on the statistics that can be expressed as the number of minimal or maximal elements ofa component of an F1-triple or an F2-triple.For an interval order R , let ( T , S , R ) be its F1-triple and ( T , S , R ) its F2-triple. We may then considerthe number of minimal and maximal elements in each of the five relations T , S , T , S , and R , for a totalof ten possible statistics. 13n fact, we have no nontrivial result for the two statistics min( S ) and min( S ). Furthermore, the trivialinvolution on F-triples maps the minimal elements of T to maximal elements of T and vice versa, and thesame is true for T and R as well. Therefore we will not treat the statistics min( T ), min( T ) and min( R )separately, and we focus on the five statistics max( S ), max( S ), max( T ), max( T ), and max( R ).To gain an intuition for these five statistics, let us describe them in terms of Fishburn matrices. Let( T , S , R ) and ( T , S , R ) be as above, and let M be the Fishburn matrix representing the interval order R .Recall from Observation 1.1 that max( R ) equals the weight of the last column of M . To get a similardescription for the remaining four statistics of interest, we need some terminology. Let us say that a cell c of the matrix M is strong-NE extreme cell (or sNE-cell for short), if c is a nonzero cell of M , and anyother cell strongly NE from c is a zero cell. Similarly, c is a weak-NE extreme cell (or wNE-cell ) if it is anonzero cell and any other cell weakly NE from c is a zero cell. Note that every weak-NE extreme cell isalso a strong-NE extreme cell; in particular, being strong-NE extreme is actually a weaker property thanbeing weak-NE extreme. In an obvious analogy, we will also refer to sSE-cells, wSE-cells, etc. Observation 3.5.
Let R be an interval order represented by a Fishburn matrix M , let ( T , S , R ) be itsF1-triple and ( T , S , R ) be its F2-triple. Then • max( S ) is equal to the number of wNE-cells of M , • max( S ) is equal to the total weight of the sNE-cells of M , • max( T ) is equal to the total weight of the sSE-cells of M , and • max( T ) is equal to the number of wSE-cells of M . We are finally ready to state our main results. Let F n denote the set of interval orders on n elements. Theorem 3.6.
Fix n ≥ . There is an involution φ : F n → F n with these properties. Suppose R ∈ F n isan interval order with F1-triple ( T , S , R ) and F2-triple ( T , S , R ) . Let R (cid:48) = φ ( R ) be its image under φ ,with F1-triple ( T (cid:48) , S (cid:48) , R (cid:48) ) and F2-triple ( T (cid:48) , S (cid:48) , R (cid:48) ) . Then the following holds: • max( S ) = max( T (cid:48) ) , and hence max( T ) = max( S (cid:48) ) , since φ is an involution, • max( S ) = max( T (cid:48) ) , and hence max( T ) = max( S (cid:48) ) , and • max( R ) = max( R (cid:48) ) .In other words, the pair of statistics (max( S ) , max( T )) and the pair of statistics (max( S ) , max( T )) bothhave symmetric joint distribution on F n , and the symmetry of both these pairs is witnessed by the sameinvolution φ which additionally preserves the value of max( R ) . Theorem 3.7.
Let ( T , S , R ) be the F1-triple of an interval order R . For any n ≥ , the pair of statistics (max( S ) , max( R )) has a symmetric joint distribution over F n . In matrix terminology, Theorem 3.7 states that the statistics ‘number of wNE-cells’ and ‘weight of thelast column’ have symmetric joint distribution over Fishburn matrices of weight n ; see Figure 8.Note that by combining Theorems 3.6 and 3.7, we may additionally deduce that max( R ) and max( T )also have symmetric joint distribution over F n .Before we present the proofs of the two theorems, let us point out how they relate to the results on Catalanstatistics discussed previously. Theorem 3.6 is a generalization of Proposition 2.12. To see this, consider thesituation when R is ( + , N )-free. With the notation of Theorem 3.6, this implies that T = ∅ and ( S , R ) isa C1-pair. Consider then the F2-triple ( T (cid:48) , S (cid:48) , R (cid:48) ). Theorem 3.6 states that max( T ) = max( S (cid:48) ), but since T = ∅ , it follows that S (cid:48) = ∅ as well, since ∅ is the only relation with n maximal elements. Consequently, R (cid:48) has an F2-triple of the form ( T (cid:48) , ∅ , R (cid:48) ), hence ( T (cid:48) , R (cid:48) ) is a C2-pair and R (cid:48) is ( + , + )-free. We concludethat by restricting the mapping φ from Theorem 3.6 to ( + , N )-free posets R , we get a bijection fromC1-pairs to C2-pairs with the same statistic-preserving properties as in Proposition 2.12.Theorem 3.7 is inspired by Proposition 2.13, and can be seen as extending the statement of this propo-sition from C1-pairs to F1-triples. 14 1 1 11 1 1 1 11 1 1 1 11212 21 2 21 21 21 2111 1 11 1 1 12 3Figure 8: Illustration of the symmetric joint distribution of the number of wNE-cells and the weight of lastcolumn over Fishburn matrices. Left: the Fishburn matrices of weight 5, with 3 wNE-cells, and last columnof weight 2. Right: the Fishburn matrices of weight 5, with 2 wNE-cells, and last column of weight 3. ThewNE-cells are shaded. To prove Theorems 3.6 and 3.7, it is more convenient to work with Fishburn matrices rather than relationalstructures, and to interpret the relevant statistics using Observation 3.5.Recall that an interval order is primitive if it has no two indistinguishable elements. Primitive intervalorders correspond to Fishburn matrices whose entries are equal to 0 or 1; we call such matrices primitiveFishburn matrices .An inflation of a primitive Fishburn matrix M is an operation which replaces the value of each 1-cellof M (i.e., a cell of value 1) by a positive integer, while the 0-cells are left unchanged. Clearly, by inflatinga primitive Fishburn matrix we again obtain a Fishburn matrix, and any Fishburn matrix can be uniquelyobtained by inflating a primitive Fishburn matrix.Another useful operation on primitive Fishburn matrices is the extension . Informally speaking, it createsa primitive Fishburn matrix P (cid:48) with k + 1 columns from a primitive Fishburn matrix P with k -columns,by splitting the last column of P into two new columns. Formally, suppose that P = ( P i,j ) ki,j =1 is a k -by- k primitive Fishburn matrix. We say that a ( k + 1)-by-( k + 1) matrix P (cid:48) = ( P (cid:48) i,j ) k +1 i,j =1 is an extension of P , if P (cid:48) has the following properties (see Figure 9): • The last row of P (cid:48) consists of k j ≤ k we have P (cid:48) k +1 ,j = 0,while P k +1 ,k +1 = 1. • For every j < k and for every i ≤ j , we have P i,j = P (cid:48) i,j . That is, the first k − P (cid:48) areidentical to the first k − P , except for an extra 0-cell in the last row. • If P i,k = 0 for some i , then P (cid:48) i,k = P (cid:48) i,k +1 = 0. That is, each 0-cell in the last column of P gives riseto two 0-cells in the same row and in the last two columns of P (cid:48) . • If P i,k = 1, then there are three options for the values of P (cid:48) i,k and P (cid:48) i,k +1 :1. P (cid:48) i,k = P (cid:48) i,k +1 = 1. In such case we say that the 1-cell P i,k is duplicated into P (cid:48) i,k and P (cid:48) i,k +1 .2. P (cid:48) i,k = 0 and P (cid:48) i,k +1 = 1. We then say that P i,k is shifted into P (cid:48) i,k +1 .3. P (cid:48) i,k = 1 and P (cid:48) i,k +1 = 0. We then say that P i,k is ignored by the extension.We say that an extension of P into P (cid:48) is valid , if there is at least one 1-cell in the penultimate column of P (cid:48) , or equivalently, if at least one 1-cell in the last column of P has been duplicated or ignored. It is easy15
11 11 11 11 11 1 11
DIS
Figure 9: Example of an extension of a primitive Fishburn matrix. The word DIS is the code of the extension.to see that if P (cid:48) is a valid extension of a primitive Fishburn matrix P , then P (cid:48) is itself a primitive Fishburnmatrix, and conversely, any primitive Fishburn matrix P (cid:48) with at least two columns is a valid extension ofa unique primitive Fishburn matrix P .Note that a primitive Fishburn matrix P whose last column has weight m has exactly 3 m extensions;one of them is invalid and 3 m − P (cid:48) of P by aword w = w · · · w m of length m over the alphabet { D, S, I } , defined as follows: suppose that c , c , . . . , c m are the 1-cells in the last column of P , listed in top-to-bottom order. Then w i is equal to D (or S or I ),if the cell c i is duplicated (or shifted, or ignored, respectively) in the extension P (cid:48) . We will call w the code of the extension from P to P (cid:48) . Notice that the 1-cell in the bottom-right corner of P (cid:48) is not represented byany symbol of w .Given a word w = w w · · · w m of length m , the reverse of w , denoted by w , is the word w m w m − · · · w . Observation 3.8.
Let P be a k -by- k primitive Fishburn matrix with m P (cid:48) be its valid extension, with code w = w · · · w m . Let c be a 1-cell in the j -th column of P (cid:48) . • Suppose that j < k , which implies, in particular, that c is also a 1-cell in P . Then c is an sNE-cell of P (cid:48) (or wNE-cell of P (cid:48) , or sSE-cell of P (cid:48) , or wNE-cell of P (cid:48) ) if and only if it is a sNE-cell of P (orwNE-cell of P , or sSE-cell of P , or wNE-cell of P , respectively). • Suppose that j = k , which means that c is also a 1-cell in P , and this 1-cell was duplicated or ignoredby the extension from P to P (cid:48) . Suppose c is the i -th 1-cell in the last column of P , counted from thetop (i.e., there are i − c and m − i c in the last column of P ). Then c isa wNE-cell of P (cid:48) if and only if i = 1 and w = I , while c is a wSE-cell of P (cid:48) if and only if i = m and w m = I . Furthermore, c is an sNE-cell of P (cid:48) if and only if all the 1-cells of P above it were ignored(i.e., w = w = · · · = w i − = I ), while c is an sSE-cell of P (cid:48) if and only if all the 1-cells of P belowit were ignored (i.e., w i +1 = w i +2 = · · · = w m = I ). • If j = k + 1 , i.e., c is in the last column of P (cid:48) , then c is an sNE-cell and also an sSE-cell. Moreover, c is a wNE-cell if and only if it is the topmost 1-cell of the last column of P (cid:48) , and it is a wSE-cell ifand only if it is the bottommost 1-cell of the last column of P (cid:48) . To prove Theorem 3.6, we first describe the involution φ , and then verify that it has the required properties.Let M be a k -by- k Fishburn matrix. As explained above, M can be constructed in a unique way fromthe 1-by-1 matrix 1 by a sequence of k − P , P , . . . , P k , M , where P = 1 , for i > P i is an extension of P i − ,and M is an inflation of P k .Define a new sequence P , . . . , P k as follows: • P = P = 1 . • For each i > P i is an extension of P i − , and the code of P i is the reverse of the code of P i .16bserve that for any 1 ≤ j ≤ i ≤ k , the j -th column of P i has the same weight as the j -th column of P i .Furthermore, from Observation 3.8, we immediately deduce that for every i and j such that 1 ≤ j ≤ i ≤ k ,the following relationships hold: • The number of sNE-cells in the j -th column of P i equals the number of sSE-cells in the j -th columnof P i . • The number of sSE-cells in the j -th column of P i equals the number of sNE-cells in the j -th columnof P i . • The number of wNE-cells in the j -th column of P i equals the number of wSE-cells in the j -th columnof P i . • The number of wSE-cells in the j -th column of P i equals the number of wNE-cells in the j -th columnof P i .As the last step in the definition of φ , we describe how to inflate P k into a matrix M . Fix a columnindex j ≤ k . Let m be the number of 1-cells in the j -th column of P k . Since M is an inflation of P k , it has m nonzero cells in its j -th column; let x , x , . . . , x m be the weights of these non-zero cells, ordered fromtop to bottom. As we know, P k also has m j -th column. We inflate these cells by using values x m , x m − , . . . , x , ordered from top to bottom. Doing this for each j , we obtain an inflation M of P k . Wethen define φ by φ ( M ) = M .Let us check that φ has all the required properties. Clearly, φ is an involution, and it preserves the weightof the last column (indeed, of any column). Moreover, the number of wNE-cells of M is equal to the numberof wSE-cells of M , since these numbers are not affected by inflations. It remains to see that the total weightof the sNE-cells of M equals the total weight of the sSE-cells of M . Fix a column j ≤ k , and suppose that M has exactly (cid:96) sNE-cells in its j -th column. These must be the (cid:96) topmost nonzero cells of the j -th columnof M , and in the notation of the previous paragraph, their total weight is x + x + · · · + x (cid:96) . It follows that P k also has (cid:96) sNE-cells in its j -th column, therefore P k has (cid:96) sSE-cells in its j -th column, and these are thebottommost (cid:96) j -th column of P k . In M , these (cid:96) cells have total weight x + · · · + x (cid:96) . We seethat the sNE-cells of M have the same weight as the sSE-cells of M , and Theorem 3.6 is proved.We remark that the involution φ actually witnesses more equidistribution results than those stated inTheorem 3.6. E.g., the total weight of wNE-cells of M equals the total weight of wSE-cells of φ ( M ), andthe number of sNE-cells of M equal the number of sSE-cells of φ ( M ). These facts do not seem to be easyto express in terms of F1-triples or F2-triples.Moreover, from the construction of φ it is clear that the conclusions of Theorem 3.6 remain valid evenwhen restricted to primitive Fishburn matrices, or to Fishburn matrices of a given number of rows. No suchrestriction is possible in Theorem 3.7, as seen from the examples in Figure 8. As with Theorem 3.6, our proof will be based on the concepts of extension and inflation. However, in thiscase we are not able to give an explicit bijection. Instead, we will proceed by deriving a formula for thecorresponding refined generating function.For a matrix M , let w ( M ) denote its weight, lc ( M ) the weight of its last column, pc ( M ) the total weightof the columns preceding the last one (so w ( M ) = lc ( M ) + pc ( M )), and ne ( M ) its number of wNE-cells.Our goal is to show that the statistics lc and ne have symmetric joint distribution over the set of Fishburnmatrices of a given weight.We will use the standard notation ( a ; q ) n for the product (1 − a )(1 − aq )(1 − aq ) · · · (1 − aq n − ).Let M be the set of nonempty Fishburn matrices. Our main object of interest will be the generatingfunction F ( x, y, z ) = (cid:88) M ∈M x w ( M ) y lc ( M ) z ne ( M ) . F ( x, y, z ) = F ( x, z, y ). Therefore, the theorem follows immediatelyfrom the following proposition. Proposition 3.9.
The generating function F ( x, y, z ) satisfies F ( x, y, z ) = xyz (cid:88) n ≥ (cid:0) (1 − xy )(1 − xz ); 1 − x (cid:1) n . Let us remark that the formula above is a refinement of a previously known formula for the generatingfunction G ( x, y ) = F ( x, y,
1) = (cid:88) M ∈M x w ( M ) y lc ( M ) , which also corresponds to refined enumeration of interval orders with respect to their number of maximalelements (see [30, sequence A175579]). Formulas for G ( x, y ) have been obtained in different contexts by sev-eral authors (namely Zagier [38], Kitaev and Remmel [25], Yan [37], Levande [26], Andrews and Jel´ınek [1]),and there are now three known expressions for this generating function: G ( x, y ) = (cid:88) n ≥ xy − xy (1 − x ; 1 − x ) n [25]= (cid:88) n ≥ (1 − xy ; 1 − x ) n [26, 37, 38]= − (cid:88) n ≥ pq n ( p ; q ) n ( q ; q ) n with p = 11 − xy , q = 11 − x . [1]The second of these three expressions can be deduced from Proposition 3.9 by the substitution z = 1 anda simple manipulation of the summands. Other authors have derived formulas for generating functions ofFishburn matrices (or equivalently, interval orders) refined with respect to various statistics [3, 13, 22], butto our knowledge, none of them has considered the statistic ne ( M ). Proof of Proposition 3.9.
Our proof is an adaptation of the approach from our previous paper [22]. We firstfocus on primitive Fishburn matrices. Let P k be the set of primitive k -by- k Fishburn matrices, and let P = (cid:83) k ≥ P k . Define auxiliary generating functions P k ( x, y, z ) = (cid:88) M ∈P k x pc ( M ) y lc ( M ) − z ne ( M ) , and P ( x, y, z ) = (cid:88) k ≥ P k ( x, y, z ) . Since lc ( M ) is positive for every M ∈ P , the exponent of the factor y lc ( M ) − in P k is nonnegative. The ideabehind subtracting 1 from the exponent is that y now only counts the 1-cells in the last column that arenot wNE-cells, while the unique wNE-cell of the last column is counted by the variable z only. Note thatthe wNE-cells of the previous columns contribute to x as well as to z .Consider a matrix M ∈ P k , with w ( M ) = n , lc ( M ) = m and ne ( M ) = (cid:96) . This matrix contributeswith the summand x n − m y m − z (cid:96) into P k ( x, y, z ); let us call x n − m y m − z (cid:96) the value of M . Let us determinethe total value of the matrices obtained by extending M . Let c be the topmost 1-cell in the last columnof M . If M (cid:48) is an extension of M , then ne ( M (cid:48) ) = ne ( M ) + 1 if c was ignored by the extension, and ne ( M (cid:48) ) = ne ( M ) otherwise. Any other 1-cell in the last column may be ignored, duplicated or shifted,contributing respectively a factor x , or xy , or y into the value of M (cid:48) . It follows that the total value of allthe matrices obtained by a valid extension of M is equal to xz ( x n − m ( x + y + xy ) m − z (cid:96) ) + xy ( x n − m ( x + y + xy ) m − z (cid:96) )+ y ( x n − m ( x + y + xy ) m − z (cid:96) ) − yx n − m y m − z (cid:96) , c has beenignored, duplicated and shifted, and the final subtracted term is the value of the matrix obtained by theinvalid extension. Note that an extension M (cid:48) also includes a 1-cell in the bottom-right corner, whichcontributes a factor of y into the value of M (cid:48) , but the effect of this extra factor is cancelled by the fact thatthe topmost 1-cell in the last column of M (cid:48) does not contribute any factor of y into the value of M (cid:48) .Summing over all M ∈ P k , and recalling that every matrix in P k +1 may be uniquely obtained as a validextension of a matrix in P k , we deduce that P k +1 ( x, y, z ) = ( xz + xy + y ) P k ( x, x + y + xy, z ) − yP k ( x, y, z ) . Summing this identity over all k ≥ P ( x, y, z ) = z , we see that P ( x, y, z ) − z = ( xz + xy + y ) P ( x, x + y + xy, z ) − yP ( x, y, z ) , or equivalently, P ( x, y, z ) = z y + xz + xy + y y P ( x, x + y + xy, z ) . From this functional equation, by a simple calculation (analogous to [22, proof of Theorem 2.1]), we obtainthe formula P ( x, y, z ) = (cid:88) n ≥ z (1 + x ) n (1 + y ) n − (cid:89) i =0 (1 + x ) i +1 (1 + y ) − − x + xz (1 + x ) i (1 + y )= z y (cid:88) n ≥ (cid:18) x − xz (1 + x )(1 + y ) ; 11 + x (cid:19) n . (1)Any Fishburn matrix may be uniquely obtained by inflating a primitive Fishburn matrix, which correspondsto the identity F ( x, y, z ) = xy − xy P (cid:18) x − x , xy − xy , z (cid:19) , where the factor xy − xy on the right-hand side corresponds to the contribution of the topmost 1-cell in thelast column. Substituting into (1) gives F ( x, y, z ) = xy − xy · z xy − xy (cid:88) n ≥ x − x − z x − x (cid:16) x − x (cid:17) (cid:16) xy − xy (cid:17) ; 11 + x − x n = xyz (cid:88) n ≥ (cid:0) (1 − xy )(1 − xz ); 1 − x (cid:1) n . This completes the proof of Proposition 3.9, and of Theorem 3.7.
Diagonal-free fillings of polyominoes.
We have seen in Lemma 1.2 that Fishburn matrices with no twopositive cells in strictly SW position correspond to ( + , + )-free posets, while those with no two positivecells in strictly SE position correspond to ( + , N )-free posets. Both classes are known to be enumeratedby Catalan numbers, and in particular, the two types of matrices are equinumerous. This can be seen as avery special case of symmetry between fillings of polyomino shapes avoiding increasing or decreasing chains.To be more specific, a k -by- k Fishburn matrix (or indeed, any upper-triangular k -by- k matrix) can alsobe represented as a right-justified array of boxes with k rows of lengths k, k − , k − , . . . ,
1, where eachbox is filled by a nonnegative integer. Such arrays of integers are a special case of the so-called fillings ofpolyominoes , which are a rich area of study. 19iven a Fishburn matrix M , we say that a m -tuple of nonzero cells c , . . . , c m forms an increasing chain (or decreasing chain ) if for each i < j , the cell c j is strictly NE from c i (strictly SE from c i , respectively).It follows from general results on polyomino fillings (e.g. from a theorem of Rubey [32, Theorem 5.3]) thatthere is a bijection which maps Fishburn matrices avoiding an increasing chain of length m to those thatavoid a decreasing chain of length m , and the bijection preserves the number of rows, as well as the weightof every row and column.We have seen that certain equidistribution results related to Fishburn matrices without increasing chainsof length 2 can be extended to general Fishburn matrices. It might be worthwhile to look at Fishburnmatrices avoiding chains of a given fixed length m >
2, and see if these matrices exhibit similar kinds ofequidistribution.
F-triples for other objects.
Disanto et al. [7] have shown that many Catalan-enumerated objects, such asDyck paths, 312-avoiding permutations, non-crossing partitions or non-crossing matchings, admit a naturalbijective encoding into C1-pairs. For other Catalan objects, e.g. non-nesting matchings, an encoding intoC2-pairs is easier to obtain.Since F1-triples and F2-triples are a direct generalization of C1-pairs and C2-pairs, it is natural to askwhich Fishburn-enumerated objects admit an easy encoding into such triples. We have seen that such anencoding exists for interval orders, as well as for Fishburn matrices. It is also easy to describe an F1-triplestructure on non-neighbor-nesting matchings [5, 36], since those are closely related to Fishburn matrices.A more challenging problem is to find an F-triple structure on ascent sequences. Unlike other Fishburnobjects, ascent sequences do not exhibit any obvious ‘trivial involution’ that would be analogous to dualityof interval orders or transposition of Fishburn matrices.
Problem . Find a natural way to encode an ascent sequence into an F-triple.There are several results showing that certain subsets of ascent sequences are enumerated by Catalannumbers (see e.g. Duncan and Steingr´ımsson [15], Callan et al. [4], or Mansour and Shattuck [29]). It mightbe a good idea to first try to encode those Catalan-enumerated classes into C1-pairs or C2-pairs.Another Fishburn-enumerated family for which we cannot find an F-triple encoding is the family of( )-avoiding permutations. We say that a permutation π = π · · · π n is ( )-avoiding, if there are no threeindices i, j, k such that i + 1 = j < k and π i + 1 = π k < π j . Parviainen [31] has shown that ( )-avoidingpermutations are enumerated by Fishburn numbers.Numerical evidence suggests that there is a close connection between the statistics of Fishburn matricesexpressible in terms of F-triples, and certain natural statistics on ( )-avoiding permutations. In a permu-tation π = π π · · · π n , an element π i is a left-to-right maximum (or LR-maximum ) if π i is larger than anyelement among π , . . . , π i − . Let LRmax( π ) denote the number of LR-maxima of π . Analogously, we defineLR-minima, RL-maxima and RL-minima of π , their number being denoted by LRmin( π ), RLmax( π ) andRLmin( π ), respectively. Let Av n ( ) be the set of ( )-avoiding permutations of size n . Observe that for a( )-avoiding permutation π , its composition inverse π − avoids ( ) as well. Let M n be the set of Fishburnmatrices of weight n . Conjecture 4.1.
For every n , there is a bijection φ : Av n ( ) → M n with these properties: • LRmax( π ) is the weight of the first row of φ ( π ) , • RLmin( π ) is the weight of the last column of φ ( π ) , • RLmax( π ) is the number of wNE-cells of φ ( π ) , • LRmin( π ) is the number of positive cells of φ ( π ) belonging to the main diagonal, and • φ ( π − ) is obtained from φ ( π ) by transpose along the North-East diagonal. Together with Theorem 3.7, Conjecture 4.1 implies the following weaker conjecture.
Conjecture 4.2.
For any n , LRmax and
RLmax have symmetric joint distribution on Av n ( ) . rimitive Fishburn matrices and Motzkin numbers. Recall that an interval order is primitive if ithas no two indistinguishable elements. Primitive interval orders are encoded by primitive Fishburn matriceswhich are Fishburn matrices whose every entry is equal to 0 or 1. Primitive interval orders (and thereforealso primitive Fishburn matrices) were enumerated by Dukes et al. [13], see also [30, sequence A138265]. Aswe pointed out before, every Fishburn matrix can be uniquely obtained from a primitive Fishburn matrixby inflation. Thus, knowing the enumeration of primitive objects we may obtain the enumeration of generalFishburn objects and vice versa.In particular, with the help of Lemma 1.2, we easily deduce that the number of primitive ( + , + )-free interval orders of size n , as well as the number of primitive ( + , N )-free interval orders of size n isthe ( n − Tamari-like lattices.
Disanto et al. [9] defined a lattice structure on the set of interval orders of a givensize, and showed that the restriction of this lattice to the set of ( + , N )-free posets yields the well-knownTamari lattice of Catalan objects [19, 21]. They also gave a simple description of the Tamari lattice on( + , N )-free posets [8].It might be worth exploring whether the lattice structure introduced by Disanto et al. for interval ordersadmits an easy interpretation in terms of Fishburn triples or Fishburn matrices. It might also be worthwhileto look at the restriction of this lattice to ( + , + )-free posets. The missing bijection.
For Theorem 3.7, we could provide a proof by a generating function argument,but not a direct bijective proof. By finding a bijective proof, or even better, a proof that generalizesDeutsch’s [6] involution on Catalan objects, we might, e.g., gain more insight into the statistic min( S ),which generalizes the Narayana-distributed statistic pea on Dyck paths. Problem . Prove Theorem 3.7 bijectively.
References [1] G. E. Andrews and V. Jel´ınek. On q -series identities related to interval orders. Eur. J. Combin. , 39:178–187,2014.[2] S. Bilotta, F. Disanto, R. Pinzani, and S. Rinaldi. Catalan structures and Catalan pairs.
Theoretical ComputerScience , 502:239–248, 2013.[3] M. Bousquet-M´elou, A. Claesson, M. Dukes, and S. Kitaev. (2+2)-free posets, ascent sequences and patternavoiding permutations.
J. Comb. Theory Ser. A , 117(7):884–909, 2010.[4] D. Callan, T. Mansour, and M. Shattuck. Restricted ascent sequences and Catalan numbers.
Applicable Analysisand Discrete Mathematics , 8:288–303, 2014.[5] A. Claesson and S. Linusson. n ! matchings, n ! posets. Proc. Am. Math. Soc. , 139:435–449, 2011.[6] E. Deutsch. An involution on Dyck paths and its consequences.
Discrete Math. , 204(1–3):163–166, 1999.[7] F. Disanto, L. Ferrari, R. Pinzani, and S. Rinaldi. Catalan pairs: A relational-theoretic approach to Catalannumbers.
Adv. Appl. Math. , 45(4):505–517, 2010.[8] F. Disanto, L. Ferrari, R. Pinzani, and S. Rinaldi. Catalan lattices on series parallel interval orders. InFolkert M¨uller-Hoissen, Jean Marcel Pallo, and Jim Stasheff, editors,
Associahedra, Tamari Lattices and RelatedStructures , volume 299 of
Progress in Mathematics , pages 323–338. Springer Basel, 2012.[9] F. Disanto, L. Ferrari, and S. Rinaldi. A partial order structure on interval orders.
ArXiv:1203.5948 , 2012. Toappear in Utilitas Mathematica.[10] F. Disanto, E. Pergola, R. Pinzani, and S. Rinaldi. Generation and enumeration of some classes of intervalorders.
Order , 30(2):663–676, 2013.[11] F. Disanto, S. Rinaldi, L. Ferrari, and R. Pinzani. Combinatorial properties of Catalan pairs.
Electronic Notesin Discrete Mathematics , 34(0):429–433, 2009.[12] M. Dukes, V. Jel´ınek, and M. Kubitzke. Composition matrices, (2 + 2)-free posets and their specializations.
Electronic J. Combin. , 18(1)(P44), 2011.
13] M. Dukes, S. Kitaev, J. Remmel, and E. Steingr´ımsson. Enumerating (2 + 2)-free posets by indistinguishableelements.
Journal of Combinatorics , 2(1):139–163, 2011.[14] M. Dukes and R. Parviainen. Ascent sequences and upper triangular matrices containing non-negative integers.
Electronic J. Combin. , 17(R53), 2010.[15] P. Duncan and E. Steingr´ımsson. Pattern avoidance in ascent sequences.
Electronic J. Combin. , 18(P226), 2011.[16] P. C. Fishburn. Intransitive indifference with unequal indifference intervals.
Journal of Mathematical Psychology ,7(1):144–149, 1970.[17] P. C. Fishburn. Interval graphs and interval orders.
Discrete Math. , 55(2):135–149, 1985.[18] P. C. Fishburn.
Interval orders and interval graphs: A study of partially ordered sets . John Wiley & Sons, 1985.[19] H. Friedman and D. Tamari. Probl`emes d’associativit´e: Une structure de treillis finis induite par une loidemi-associative.
J. Comb. Theory , 2(3):215–242, 1967.[20] M. Guay-Paquet, A. H. Morales, and E. Rowland. Structure and enumeration of (3+1)-free posets.
Ann. Comb. ,18(4):645–674, 2014.[21] S. Huang and D. Tamari. Problems of associativity: A simple proof for the lattice property of systems orderedby a semi-associative law.
J. Comb. Theory A , 13(1):7–13, 1972.[22] V. Jel´ınek. Counting general and self-dual interval orders.
J. Comb. Theory A , 119(3):599–614, 2012.[23] S. M. Khamis. Height counting of unlabeled interval and N-free posets.
Discrete Math. , 275(1-3):165–175, 2004.[24] K.H. Kim and F.W. Roush. Enumeration of isomorphism classes of semiorders.
Journal of Combinatorics,Information & System Sciences , 3:58–61, 1978.[25] S. Kitaev and J. B. Remmel. Enumerating (2+2)-free posets by the number of minimal elements and otherstatistics.
Disc. Appl. Math. , 159(17):2098–2108, 2011.[26] P. Levande. Fishburn diagrams, Fishburn numbers and their refined generating functions.
J. Comb. Theory A ,120(1):194–217, 2013.[27] J. B. Lewis and Y. X. Zhang. Enumeration of graded (3 + 1)-avoiding posets.
J. Comb. Theory A , 120(6):1305–1327, 2013.[28] R. D. Luce. Semiorders and a theory of utility discrimination.
Econometrica , 24(2):pp. 178–191, 1956.[29] T. Mansour and M. Shattuck. Some enumerative results related to ascent sequences.
Discrete Math. , 315-316:29–41, 2014.[30] OEIS Foundation, Inc. The On-Line Encyclopedia of Integer Sequences. http://oeis.org/, 2011.[31] R. Parviainen. Wilf classification of bi-vincular permutation patterns. arXiv:0910.5103 , 2009.[32] M. Rubey. Increasing and decreasing sequences in fillings of moon polyominoes.
Adv. Appl. Math. , 47:57–87,2011.[33] M. Skandera. A Characterization of (3+1)-Free Posets.
J. Comb. Theory A , 93(2):231–241, 2001.[34] M. Skandera and B. Reed. Total nonnegativity and (3 + 1)-free posets.
J. Comb. Theory A ∼ rstan/ec/catadd.pdf.[36] A. Stoimenow. Enumeration of chord diagrams and an upper bound for Vassiliev invariants. J. Knot TheoryRamifications , 7:93–114, 1998.[37] S. H. F. Yan. On a conjecture about enumerating (2+2)-free posets.
Eur. J. Combin. , 32(2):282–287, 2011.[38] D. Zagier. Vassiliev invariants and a strange identity related to the Dedekind eta-function.
Topology , 40(5):945–960, 2001., 40(5):945–960, 2001.