Enumeration of Graded (3+1)-Avoiding Posets
EEnumeration of Graded ( + )-Avoiding Posets Joel Brewster Lewis and Yan X ZhangMassachusetts Institute of TechnologyOctober 8, 2018
Abstract
The notion of ( + )-avoidance has shown up in many places in enumerative com-binatorics, but the natural goal of enumerating all ( + )-avoiding posets remainsopen. In this paper, we enumerate graded ( + )-avoiding posets for both reasonabledefinitions of the word “graded.” Our proof consists of a number of structural theoremsfollowed by some generating function computations. We also provide asymptotics forthe growth rate of the number of graded ( + )-avoiding posets. Keywords: posets, ( + )-avoidance, generating functions, asymptotic enumeration The notion of ( + )-avoiding posets appears in different different areas of combinatorics,such as in the Stanley-Stembridge conjecture about the e -positivity of certain chromaticsymmetric functions [SS93] and the characterization of interval semiorders [Fis70]. Graph-theoretically, ( + )-avoiding posets are exactly those posets whose comparability graphsare complements of claw-free graphs; as a result, they also are connected to a generalizationof the “birthday problem” [Fad].Despite these connections, the enumeration of ( + )-avoiding posets has remained elu-sive. This is particularly bothersome because the enumeration of posets that are both( + )- and ( + )-avoiding, the interval semiorders, is well-understood: the number ofunlabeled n -element interval semiorders is exactly the Catalan number C n [Fis70]. More-over, ( + )-avoiding posets have recently been enumerated, as well [BMCDK10]. Happily,there has been some progress: Skandera [Ska01] has given a characterization of all ( + )-avoiding posets involving the square of the antiadjacency matrix and Atkinson, Sagan andVatter [ASV12] have recently characterized and enumerated ( + )-avoiding permutations(i.e., permutations whose associated posets are ( + )-avoiding).In this paper, we consider a related problem and enumerate graded ( + )-avoidingposets (for both common meanings of the word graded) via structural theorems and gener-ating function computations. The property of gradedness is very natural and captures a lotof the complexity of the general case while making the problem much more tractable. Weremark that a substantially easier problem is to enumerate ( + )- and ( + )-avoidinggraded posets, and that the solution may be found in work of the second-named author1 a r X i v : . [ m a t h . C O ] M a r urrently in preparation; labeled ( + )- and ( + )-avoiding strongly graded posets arecounted by the generating function 1 + e x ( e x − e x − e x − e x − .In the rest of this introduction, we summarize our strategy and results. In Section 2, weoffer some definitions and notation that we will use throughout the paper. Then in Section 3,we give a useful local condition that is equivalent to ( + )-avoidance for graded posets.The main ideas of the paper are in Section 4, where we introduce several operations thatallow us to decompose strongly graded ( + )-avoiding posets into simpler objects. First, inSection 4.1 we reduce our problem of obtaining the generating function for all graded ( + )-avoiding posets to studying certain posets we will call trimmed which are slightly simpler butwhich capture most of the information of the original posets. Then, in Section 4.2, we showthat trimmed ( + )-avoiding posets arise from taking ordinal sums of sum-indecomposable ( + )-avoiding posets. Finally, in Section 4.3 we introduce two more operations, gluing and sticking . We show that sum-indecomposable ( + )-avoiding posets arise from gluingand sticking together basic units called quarks , which we enumerate in Section 5.This line of argument culminates in Section 6, in which we backtrack and use the resultsof the preceding sections and the transfer-matrix method to enumerate all strongly graded( + )-avoiding posets. We end with some extensions of these techniques. In Section 7 weuse similar generating functional arguments to enumerate strongly graded ( + )-avoidingposets by height. We use this modified enumeration in Section 8 to enumerate ( + )-avoiding weakly graded posets. Finally, in Section 9, we use the generating functions com-puted in Sections 6 and 8 to establish the asymptotic rate of growth of the number of graded( + )-avoiding posets.An extended abstract of this work appeared as [LZ12]. Note added in proof : In recent work [GPMR], Guay-Paquet, Morales and Rowland haveenumerated all ( + )-avoiding posets using similar techniques. A partially ordered set , or poset for short, is a set with an irreflexive, antisymmetricand transitive relation > . We say two elements a , b of a poset are comparable if a ≥ b or b ≤ a . In this paper, we concern ourselves only with posets of finite cardinality. We saythat an element w covers an element v , denoted v < · w , if v < w and there is no z suchthat v < z < w . Observe that the order relations of a finite poset follow by transitivity fromthe cover relations; this allows us to graphically represent posets by showing only the coverrelations. The resulting graph is called the Hasse diagram of the poset.A poset in which every pair of elements is comparable is called a chain , and a poset inwhich every pair of elements is incomparable is called an antichain .We say that four elements w , x , y , z in a poset P are a copy of 3 + if we have that x < y < z and w is incomparable to all of x , y , z . If P contains no copy of + , we saythat P avoids 3 + .Call a poset P weakly graded if there exists a rank function rk : P → N such that if a < · b then rk( b ) − rk( a ) = 1 and such that the minimal occurring rank in each connectedcomponent is 0. Call a weakly graded poset strongly graded if all minimal elements have2igure 1: Three posets: the first is strongly graded, the second is weakly graded but notstrongly graded, and the third is not weakly graded.Figure 2: All weakly graded ( + )-avoiding posets on four or fewer vertices. The doubledline separates the strongly graded posets (on the left) from the others.the same rank and all maximal elements have the same rank. (Equivalently, a poset isstrongly graded if all maximal chains in the poset have the same length; in this case therank function rk may be recovered by setting rk( v ) to be the length of a longest chain whosemaximal element is v .) Figure 1 gives examples of posets with these properties. The height of a weakly graded poset P is the number of vertices in a longest chain in P .A weakly graded poset P of height k + 1 has rank sets P (0), P (1), . . . , P ( k ), where P ( i ) = { v ∈ P | rk( v ) = i } . If P is strongly graded, all the minimal elements are in P (0)and all the maximal ones are in P ( k ).Figure 2 shows all unlabeled weakly graded ( + )-avoiding posets on four or fewervertices. Taking labelings into account, we see that for n = 1 , ,
3, and 4 the number ofweakly graded ( + )-avoiding posets on n vertices is 1, 3, 19, and 195, respectively. Ofthese, respectively 1, 3, 13 and 111 are strongly graded.In this paper, we avoid the use of the unmodified word “graded” because of an ambiguityin the literature: some sources (e.g., [Sta99]) use the word “graded” to mean “stronglygraded,” while many others (e.g., [Kla69]) use “graded” to mean “weakly graded.” In this section, we give a concise local condition which is equivalent to ( + )-avoidance forweakly graded posets.Given a weakly graded poset P , call a vertex v ∈ P of rank i up-seeing if every vertex3 Figure 3: In the (weakly graded) poset pictured, the vertices labeled 1, 8, 4, 2 and 5 areup-seeing and the vertices labeled 2, 3 and 5 are down-seeing. The vertices labeled 6 and 7are neither up- nor down-seeing.in P ( i + 1) covers v . Similarly, call v down-seeing if v covers every vertex in P ( i − i ) be the set of up-seeing vertices of rank i and let Λ( i ) be the set of all down-seeingvertices of rank i . (As a mnemonic, think of v at the point of the V or Λ, with lots of edgesgoing respectively up or down in the Hasse diagram of the poset.) These definitions areillustrated in Figure 3.Call a vertex that is both up- and down-seeing an all-seeing vertex. Also, call a weaklygraded poset P vigilant if every vertex of P is up-seeing, down-seeing, or all-seeing. One keyconsequence of our next result, Theorem 3.1, is that in our study of graded ( + )-avoidingposets it suffices to only consider vigilant posets. Theorem 3.1.
For a weakly graded poset P , the following are equivalent:I. P is ( + ) -avoiding;II. P is vigilant and every two vertices v , w such that rk( w ) − rk( v ) ≥ are comparable;III. P is vigilant and every two vertices v , w such that rk( w ) − rk( v ) = 2 are comparable.Proof. It is clear that II implies III , so we will show that
III implies II , that II implies I ,and that I implies II . III ⇒ II : Let P be a poset satisfying the conditions in III . We show that every twovertices whose ranks differ by 3 are comparable; the result follows by induction. Choosevertices v of rank i and w of rank i + 3. Since there is a vertex of rank i + 3, there must beat least one vertex z of rank i + 1, and by III we have w > z . Since P is graded, there issome vertex y of rank i + 2 such that w > y > z . But also y > v by III , so w > v , as desired. II ⇒ I : Let P be a poset satisfying the conditions in II ; we show P avoids + . Considerany 3-chain x < y < z in P and any other vertex w ∈ P . We claim that w is comparable toat least one of x , y , z . By the defining properties of P , if rk( w ) < rk( z ) − w < z whileif rk( w ) > rk( x ) + 1 then w > x , and in either case we have our result. The only remainingcase is rk( z ) − w ) = rk( x ) + 1. In this case, since w is either up- or down-seeing, weconclude that w is comparable to at least one of x and z . Thus, P avoids + , as desired. I ⇒ II : Let P be a weakly graded ( + )-avoiding poset. First, we show that twovertices whose ranks differ by 2 or more are comparable. Choose vertices u and w at ranks i and j respectively with j − i ≥
2. Since there are vertices at ranks at least i + 2, there mustbe a chain x < · y < · z with x at rank i . Because P avoids + , u must be comparable to atleast one of these vertices and so in particular u < z . Then there is a chain u < · v < · z in P ,4igure 4: The Hasse diagram for the vigilant poset at left will be displayed as the image atright: all-seeing vertices are represented as squares, other vertices as triangles.and by ( + )-avoidance we have that w is comparable to some member of this chain andso finally w > v as desired.Second, we show that P is vigilant. Suppose for contradiction that we have a vertex v of rank i that is neither up- nor down-seeing. This means v (cid:54)∈ Λ( i ) ∪ V( i ). Then there existvertices u , w such that rk( u ) = rk( v ) −
1, rk( w ) = rk( v ) + 1, and v is incomparable to both u and w . But by the preceding paragraph, u < w , and so there is some vertex v (cid:48) of rank i suchthat u < v (cid:48) < w . This chain together with v is a copy of + in P . This is a contradiction,so P is vigilant.We introduce the following convention for representing vigilant posets: vertices thatare all-seeing are represented by squares, vertices that are up-seeing are represented bydownwards-pointing triangles, and vertices that are down-seeing are represented by upwards-pointing triangles. (Thus, each vertex has horizontal edges on the sides on which it isconnected to all vertices.) This convention is illustrated in Figure 4. In this section, we introduce four operations that allow us to count vigilant posets by workinginstead with simpler objects. We show that ( + )-avoidance will be mostly compatible withthese simplifications, reducing the problem of enumerating graded ( + )-avoiding posetsbasically to studying vigilant posets of height 2. In Section 4.1 we work with weakly gradedposets, while in Sections 4.2 and 4.3 we restrict ourselves to strongly graded posets. (Wewill return to weakly graded posets in Section 8.) We call a vigilant poset P trimmed if it has the following properties: • every rank has at most one all-seeing vertex, • the all-seeing vertices are unlabeled, and • the other m vertices are labeled with [ m ].Given a weakly graded poset P , there is a naturally associated trimmed poset, denotedtrim( P ), that we get by removing the all-seeing vertices from P , adding a single unlabeled all-seeing vertex to every rank set from which we removed all-seeing vertices, and relabeling the5 trimming −→ Figure 5: A strongly graded ( + )-avoiding poset and the associated trimmed poset.other vertices so as to preserve the relative order of labels. Figure 5 provides one illustrationof this operation. Proposition 4.1.
The weakly graded vigilant poset P avoids + if and only if trim( P ) does.Proof. It is routine to check that neither of the conditions of Theorem 3.1(
III ) is affectedby the trimming map.Since we lose very little information when we replace the poset P by the trimmed posettrim( P ), Proposition 4.1 suggests that we can reduce the enumeration of labeled graded( + )-avoiding posets to the enumeration of trimmed ( + )-avoiding posets. The fol-lowing proposition makes this intuition precise. Let w n be the number of weakly graded( + )-avoiding posets on n vertices and let W ( x ) = (cid:88) n w n x n n !be the exponential generating function for labeled weakly graded ( + )-avoiding posets. Proposition 4.2.
The exponential generating function for labeled weakly graded ( + ) -avoiding posets is W ( x ) = (cid:88) n,r a n,r x n n ! ( e x − r . where a n,r is the number of trimmed ( + ) -avoiding posets with r all-seeing vertices and n other vertices.An analogous result holds if we restrict attention to the strongly graded posets.Proof. A weakly graded ( + )-avoiding poset P is uniquely determined by the associatedtrimmed poset T and the set of labels for the all-seeing vertices at each rank. Moreover,any trimmed ( + )-avoiding poset T with all-seeing vertices at r levels together with anappropriate tuple of r nonempty sets of labels yields a weakly graded ( + )-avoiding poset.Thus, by standard rules for generating functions (or equivalently from species-type consider-ations as in [JMM07, Section 4]), the generating function for weakly graded ( + )-avoidingposets with all-seeing vertices at exactly r ranks is ( e x − r · (cid:80) n a n,r x n n ! . Summing over r gives the result. 6 = Figure 6: An ordinal sum of two sum-indecomposable posets. (Labels are suppressed forreadability.)
Suppose we have two trimmed strongly graded posets P and P of heights a and b , respec-tively. We can take the ordinal sum of P and P by letting the lowest-ranked elementsin P cover all highest-ranked elements in P and relabeling in a way consistent with thelabelings of P and P . (Thus, there are many ways to take ordinal sums of P and P ; all theresulting posets are isomorphic under relabeling.) We denote one of the possible resultingposets of height a + b by P ⊕ P . See for example Figure 6. In the context of vigilantposets, the ordinal sum is an especially nice operation because a vertex in P or P which isup-seeing and/or down-seeing retains that property in P ⊕ P .Call a nonempty strongly graded poset P with height k sum-indecomposable if P istrimmed and there is no i < k − P ( i ) is up-seeing (equivalently,there is no i > P ( i ) is down-seeing). This word choice is motivatedby the existence of a decomposition of trimmed posets into sum-indecomposables. Proposition 4.3.
A trimmed strongly graded poset P can be written uniquely as P = P ⊕ P ⊕ · · · ⊕ P m , for a sequence ( P , P , . . . , P m ) of sum-indecomposable posets.Proof. Let P be a trimmed strongly graded poset and let k be the height of P . Takethe smallest rank i for which P ( i ) has all up-seeing vertices. If i = k , then P is sum-indecomposable. Otherwise, we can write P = P ⊕ P (cid:48) , where P has height i + 1 and issum-indecomposable by the minimality of i . Repeating this process gives us the desiredsequence, which is obviously unique. Proposition 4.4.
If a trimmed strongly graded poset P decomposes into sum-indecomposableposets as P = P ⊕ · · · ⊕ P m , then P avoids + if and only if all of the P i avoid + .Proof. One direction is trivial: if any of the P i contains a copy of + then certainly P does as well. For the other direction, suppose that P contains a copy of + with vertices u < v < w and x . If x ∈ P i , then u , v , and w must also all be in P i (since two vertices fromdifferent P j must be comparable), so P i itself does not avoid + .Propositions 4.3 and 4.4 simplify the problem of counting strongly graded ( + )-avoiding posets: it now suffices to count sum-indecomposable posets and then count the ways7igure 7: Three quarks. All three quarks are middle quarks; the left quark is also a bottomquark, while the center quark is also a top quark. The left and center quarks have isomorphicunderlying posets but are not isomorphic as quarks. The right quark has underlying poset ofheight 1 (with two vertices and no relations), while as a quark it has vertices at two differentrank sets.to combine them by ordinal sum. As we will see in Proposition 6.6, this is a simple task withgenerating functions. Thus, we now turn our attention to enumerating sum-indecomposable( + )-avoiding posets. In order to enumerate sum-indecomposable posets, we break them down into more manage-able pieces that we call quarks . We show that quarks can be combined to make posets usingtwo operations that we call sticking and gluing , that every sum-indecomposable poset can bewritten uniquely as a sticking and gluing of quarks, and that ( + )-avoidance is encodednicely in this decomposition.Observe that every poset of height 1 or 2 is weakly graded and so naturally has a rankfunction that assigns all minimal vertices to rank 0 and all other vertices to rank 1. A quark Q is a pair ( P, r ) of a poset P and function r : P → { , } , with the following restrictions: • P has height 1 or 2 (and so consequently is weakly graded); • P does not have both an up-seeing vertex at rank 0 and a down-seeing vertex at rank1; • if v ∈ P is not isolated then r ( v ) = rk( v ); • there exist vertices v and v in P such that r ( v ) = 0 and r ( v ) = 1.Equivalently, thinking in terms of Hasse diagrams, one may view a quark as a bipartite graphwith a designated bipartition of the vertices into nonempty lower and upper halves with therestriction that at most one part of the bipartition contains an all-seeing vertex.Given a vertex v ∈ Q , we say that r ( v ) is the rank of v . Note that two different quarksmay have the same underlying poset, and that the underlying poset of a quark may haveheight 1 even though the quark itself has two nonempty ranks.Given the close relationship between quarks and posets, we extend our poset terminologyto this new context in the natural way. Notably, the adjectives “vigilant” and “trimmed”have the same meaning for quarks as for posets, and since every quark is vigilant we usethe same convention for displaying their vertices as was introduced for vigilant posets inSection 3. 8 G ⊕ S = = Figure 8: Sticking and gluing a bottom quark and a top quark to build sum-indecomposableposets. (Labels are suppressed for readability.)There are three classes of quarks that will be of interest two us; see Figure 7 for examples.A bottom quark is a trimmed quark in which every isolated vertex (if any) is assigned torank 0 and there are no all-seeing vertices of rank 1. A middle quark is a quark with noall-seeing vertices of either rank. A top quark is a trimmed quark in which every isolatedvertex (if any) is assigned to rank 1 and there are no all-seeing vertices of rank 0. Observethat every trimmed quark belongs to at least one of these classes and that many quarksbelong to more than one of them.We now introduce the two operations that can be used to build every sum-indecomposableposet of height larger than 2 from quarks. We first describe the operations for height-3 posets,and afterwards the general case.Given a bottom quark Q and a top quark Q , we say that a poset P arises from sticking Q and Q if the following conditions hold: • The vertex set of P is the disjoint union of the vertex sets of Q and Q . • For i = 0 ,
1, if v, w ∈ Q i , then v < w in P if and only if v < w in Q i . • For j = 0 ,
1, if v ∈ Q and w ∈ Q have rank j in their respective quarks then v < w in P . • The only other order relations of P are those that follow by transitivity. • The labeling of vertices of P is consistent with the labelings of Q and Q .In an abuse of notation, we denote this relationship by P = Q ⊕ S Q . Similarly, we say that P arises from gluing Q and Q , and we write P = Q ⊕ G Q , if the following conditionshold: • P has a (not necessarily induced) subposet P (cid:48) = Q ⊕ S Q . • The rank set P (1) has an additional (unlabeled) all-seeing vertex, the additional orderrelations implied by the presence of this vertex and transitivity, and no other orderrelations.It is easy to check that posets of the form Q ⊕ S Q and Q ⊕ G Q are sum-indecomposableposets of height 3. Also observe that, as in the case of ordinal sums, a vertex in Q or Q that is up-seeing or down-seeing keeps this status after either gluing or sticking. Figure 8shows an example of the sticking and gluing of two quarks.Now we describe how to apply these operations to many quarks in order to create posetsof larger height. 9 S ⊕ G = Figure 9: An example of sticking and gluing quarks to build a sum-indecomposable poset.(Labels are suppressed for readability.)
Definition 4.5.
Suppose we are given a bottom quark Q , middle quarks Q , . . . , Q k − ,and a top quark Q k . For each choice ( α , . . . , α k ) ∈ { S, G } k , we say that a trimmed poset P is of the form Q ⊕ α Q ⊕ α · · · ⊕ α k Q k if the following conditions hold: • For i ∈ { , . . . , k + 1 } , the i th rank set P ( i ) consists of the disjoint union of Q i (0) and Q i − (1) and, if α i = G , an unlabeled all-seeing vertex. • For i ∈ { , . . . , k } , if v, w ∈ Q i then v < w in P if and only if v < · w in Q i . • For i ∈ { , . . . , k − } and j ∈ { , } , if v ∈ Q i ( j ) and w ∈ Q i +1 ( j ) then v < · w . • For i ∈ { , . . . , k − } , if v ∈ Q i (1) and w ∈ Q i +2 (0) then v < · w . • All other order relations of P follow by transitivity from those of the four precedingbullet points. • The labeling of vertices of P is consistent with the labelings of the Q i .As before, we denote this relation by P = Q ⊕ α · · · ⊕ α k Q k . An example of a poset ofheight 4 formed by sticking and gluing is shown in Figure 9.The quark decomposition is useful because it behaves nicely with resprect to up-seeingand down-seeing vertices. Proposition 4.6.
Suppose that Q is a bottom quark, Q , . . . , Q k − are middle quarks and Q k is a top quark, and P = Q ⊕ α Q ⊕ α · · · ⊕ α k Q k . A vertex v ∈ Q i is up-seeing (respectively, down-seeing) in Q i if and only if it is up-seeing(respectively, down-seeing) in P .Proof. Choose i ∈ { , . . . , k } and choose v ∈ Q i . If v is not down-seeing in Q i , then v ∈ Q i (1)and there is some w ∈ Q i (0) such that v (cid:54) > w . In this case, v ∈ P ( i + 1) is not larger than w ∈ P ( i ), and so v is not down-seeing in P . On the other hand, if v ∈ Q i is down-seeing in Q i then by construction v covers all vertices of one lower rank in P . Proposition 4.7.
Suppose that Q is a bottom quark, Q , . . . , Q k − are middle quarks and Q k is a top quark, and P = Q ⊕ α Q ⊕ α · · · ⊕ α k Q k . We have that P is sum-indecomposable. roof. It follows immediately from the construction that P is weakly graded, vigilant andtrimmed. Observe that the restrictions on the quarks guarantee that every vertex not ofmaximal rank is covered by something and that every vertex not of minimal rank coverssomething, so that P is strongly graded as well. Finally, for all i ∈ { , . . . , k } , Q i (0) containsa vertex that is not up-seeing. Thus, by Proposition 4.6, P ( i ) contains such a vertex, andthus P is sum-indecomposable.In fact, every sum-indecomposable poset may be written as a sticking and gluing ofquarks in a unique way, as the next result shows. Proposition 4.8.
For k ≥ , suppose that P is a sum-indecomposable poset of height k + 2 .There exists a unique bottom quark Q , top quark Q k , collection Q , . . . , Q k − of middlequarks, and choice ( α , . . . , α k ) ∈ { S, G } k such that P = Q ⊕ α Q ⊕ α · · · ⊕ α k Q k . Proof.
For some k ≥
1, choose a sum-indecomposable poset P of height k + 2. We firstdescribe the decomposition of P into quarks, then show that it is unique.For i ∈ { , . . . , k − } , define Q i as follows: the lower rank set Q i (0) consists of all verticesof P ( i ) that are not up-seeing, the upper rank set Q i (1) consists of all vertices of P ( i + 1)that are not down-seeing, and the vertices are labeled in accordance with the labeling of P .The top and bottom quark Q and Q k are defined similarly, except that Q (0) = P (0) and Q k (1) = P ( k + 1) (i.e., we remove the additional restriction in this case). For i ∈ { , . . . , k } ,we set α i = G if P ( i ) contains an all-seeing vertex, and α i = S otherwise.Since P is sum-indecomposable, both rank sets of every Q i are nonempty, and no Q i contains an all-seeing vertex except possibly in the bottom rank set of Q or the top rankset of Q k . Since P is strongly graded, every vertex in P (1) covers some vertex in P (0), so Q (1) has no isolated vertices (and likewise Q k (0) has no isolated vertices). Thus, Q is abottom quark, Q k a top quark, and Q , . . . , Q k − are middle quarks. Since P is trimmed,every vertex of P either belongs to exactly one of the Q i or is an all-seeing vertex not oftop or bottom rank. Finally, it’s easy to check that the cover relations of P and those of Q ⊕ α Q ⊕ α · · · ⊕ α k Q k are the same, as desired.The uniqueness of this decomposition is straightforward: the presence of all-seeing ver-tices indicates which of the α i are G , vertices of rank i that are not down-seeing can onlycome from Q i − (1), and vertices of rank i that are not up-seeing can only come from Q i (0).Thus, the partition of the underlying set of P into the underlying sets of the Q i is uniquelydetermined; the uniqueness of the Q i as quarks follows immediately.Now we can connect our characterization of sum-indecomposable posets as quarks thathave been glued or stuck together to our ultimate goal of studying ( + )-avoiding posets. Theorem 4.9.
A sum-indecomposable poset P is ( + ) -avoiding if and only if the decom-position P = Q ⊕ α Q ⊕ α · · · ⊕ α k Q k into quarks satisfies the following condition: forevery occurrence of Q i ⊕ S Q i +1 in the decomposition, either Q i has no isolated vertices onits bottom rank or Q i +1 has no isolated vertices on its top rank, or both. roof. The poset P avoids + if and only if the conditions of Theorem 3.1( III ) hold.The first condition holds for every sum-indecomposable poset by definition, so the desiredstatement reduces to the claim that P contains two incomparable vertices whose ranks differby 2 if and only if there is some i ≥ α i +1 = S , Q i has an isolated vertex on itslower rank set and Q i +1 has an isolated vertex on its upper rank set.The vertices u and w are comparable as elements of P if and only if there is some v ∈ P ( i + 1) such that u < v < w . If u (cid:54)∈ Q i (0) then we can take v to be any vertex in Q i (1), while if w (cid:54)∈ Q i +1 (1) then we can take v to be any vertex in Q i +1 (0), so in these cases u and w are always comparable. Now we consider the case that u ∈ Q i (0) and w ∈ Q i +1 (1).If α i +1 = G then we can take v to be the all-seeing vertex at rank i + 1, so in this case u and w are comparable. If α i +1 = S and u and w are not both isolated in their respective quarks,we may assume without loss of generality that there is some v ∈ Q i (1) such that u < v . Bythe sticking construction, v < w , and so u and w are comparable in this case as well. Finally,suppose that α i +1 = S and that u and w are both isolated in their respective quarks. We wishto show that u and w are incomparable. Since α i +1 = S , we have P ( i + 1) = Q i (1) ∪ Q i +1 (0).Thus, for any v ∈ P ( i + 1) we have that v is incomparable with u or with w .It follows that P contains two isolated vertices whose rank differs by 2, and so a copy of + , if and only if there is some i such that α i +1 = S , Q i has an isolated vertex of rank 0,and Q i +1 has an isolated vertex of rank 1, as desired.With this result in hand, we now turn to the task of counting quarks. Theorem 4.9 implies that studying sum-indecomposable ( + )-avoiding posets reduces tostudying quarks. In this section, we set out to enumerate quarks. Following the observationat the beginning of the previous section, this amounts to enumerating bipartite graphswith certain restrictions: a quark Q with m vertices in Q (0) and n vertices in Q (1) is,up to differences in the labeling scheme, just a particular kind of bipartite graph on thedisjoint union [ m ] (cid:93) [ n ]. We enumerate such graphs, keeping track of some simple structuralinformation about them.We define a family of sets A νµ ( m, n ), where µ and ν are subsets (possibly empty) of { (cid:3) , ◦ , (cid:2) , ⊗} , as follows: • A νµ ( m, n ) is the set of bipartite graphs on [ m ] (cid:93) [ n ] with some restrictions. The elementsof ν correspond to restrictions on the vertices in [ n ] and the elements of µ correspondto restrictions on the vertices of [ m ]. (Here the placement of indices is meant to suggestthat vertices in [ m ] form a bottom rank and the vertices in [ n ] a top rank.) An emptyset of symbols corresponds to no restrictions on the corresponding set. • A (cid:3) corresponds to the requirement that there be at least one all-seeing vertex; a (cid:2) corresponds to the requirement that there be no all-seeing vertex. • A ◦ corresponds to the requirement that there be an isolated vertex; a ⊗ correspondsto the requirement that there be no isolated vertex.12or example, A ( m, n ) is the set of all bipartite graphs on [ m ] (cid:93) [ n ] and A (cid:3)(cid:2) ( m, n ) is thesubset of A ( m, n ) containing those graphs with at least one all-seeing vertex in [ n ] but noall-seeing vertices in [ m ].Of these sets, we are particularly interested in those that contain quarks. The topquarks correspond to the graphs in A (cid:2) ⊗ , the bottom quarks correspond to the graphs in A (cid:2) ⊗ , and the middle quarks correspond to the graphs in A (cid:2)(cid:2) . In the next section, we willneed to consider a more refined count of middle quarks; thus, for ν, µ ⊂ { ◦ , ⊗} we define B νµ ( m, n ) = A { (cid:2) }∪ ν { (cid:2) }∪ µ ( m, n ). For example, B ⊗ ◦ ( m, n ) is the set of bipartite graphs on [ m ] (cid:93) [ n ]with no all-seeing vertices, no isolated vertices in [ n ], and at least one isolated vertex in [ m ].For each µ and ν , let F νµ ( x ) = (cid:88) m,n ≥ | B νµ ( m, n ) | x m + n m ! n ! (1)be the corresponding generating function, so for example the coefficient of x N N ! in F ⊗ ◦ ( x ) isthe number of middle quarks on N vertices with at least one isolated vertex of rank 0 butnone of rank 1. Finally, let B νµ be the union over m and n of all B νµ ( m, n ). Note that the setof middle quarks is a disjoint union B = B ◦◦ ∪ B ◦ ⊗ ∪ B ⊗ ◦ ∪ B ⊗⊗ , which manifests as a sum of formal power series F = F ◦◦ + F ◦ ⊗ + F ⊗ ◦ + F ⊗⊗ . Proposition 5.1.
Let Ψ( x ) = (cid:88) m,n ≥ mn x m + n m ! n ! and let F νµ be defined as in Equation (1) . We have F ◦◦ ( x ) = (1 − e − x ) Ψ( x ) ,F ◦ ⊗ ( x ) = F ⊗ ◦ ( x ) = (1 − e − x )((2 e − x − x ) − , and F ⊗⊗ ( x ) = (2 e − x − e − x − x ) − . Proof.
See Appendix A.
In this section, we use the F νµ as building blocks to obtain the generating function forsum-indecomposable ( + )-avoiding posets, and then proceed to enumerate all stronglygraded ( + )-avoiding posets. (Recall that by our definition, sum-indecomposable posetsare necessarily strongly graded.) We begin by encoding a sum-indecomposable poset in13erms of a word that keeps track of its quarks and how they are combined (i.e., gluing andsticking). Then we use the transfer-matrix method to enumerate words while keeping trackof the restrictions imposed by Theorem 4.9.Given any quark Q , we define its type as follows: if Q is a middle quark (i.e., an elementof B ), then the type of Q is the symbol B νµ corresponding to the unique subset among the four B νµ to which it belongs. (This is a slight abuse of notation that will never cause ambiguity incontext.) If Q is a top or bottom quark, first remove any all-seeing vertices from Q , leavinga middle quark Q (cid:48) ; then set the type of Q to be the type of Q (cid:48) . Define a word to be anymonomial in the noncommutative algebra R (cid:104)(cid:104) S, G, B ◦◦ , B ◦ ⊗ , B ⊗ ◦ , B ⊗⊗ (cid:105)(cid:105) . We now encode theproperties of sum-indecomposability and ( + )-avoidance into conditions on words. Definition 6.1.
We say that a word L is legal if for some k ≥ α i ∈ { S, G } and B i ∈ { B ◦◦ , B ◦ ⊗ , B ⊗ ◦ , B ⊗⊗ } such that L = α B α B α · · · B k − α k B k α k +1 , and none of thefollowing occur:1. α = S and B has a ◦ in the superscript;2. α k +1 = S and B k has a ◦ in the subscript;3. there is some i , 1 ≤ i ≤ k , such that B i − has a ◦ in the subscript, α i = S , and B i hasa ◦ in the superscript.We define a weight function wt : R (cid:104)(cid:104) S, G, B ◦◦ , B ◦ ⊗ , B ⊗ ◦ , B ⊗⊗ (cid:105)(cid:105) → R [[ x, z ]] as follows: weset wt( S ) = 1, wt( G ) = z , and wt( B νµ ) = F νµ and we extend by linearity and multiplication.Let I ≥ ( x, z ) be the generating function for sum-indecomposable ( + )-avoiding posetsof height at least 2, where the variable z counts all-seeing vertices, the variable x countsother vertices, and I ≥ ( x, z ) is exponential in x and ordinary in z . Theorem 6.2.
The generating function for sum-indecomposable ( + ) -avoiding posets ofheight at least is I ≥ ( x, z ) = (cid:88) L wt( L ) , where the sum is over all legal words L .Proof. First we handle the height-2 case. By considering the presence or absence of an all-seeing vertex of rank 0 or 1, it’s easy to see that the generating function for such posets isprecisely z · F + z · F ⊗ + z · F ⊗ + F ⊗⊗ and that this is equal to the sum of wt( L ) over all legal words L of length 3 (i.e., those ofthe form α B α satisfying certain restrictions). Now we handle the case of larger heights.Let P be a sum-indecomposable ( + )-avoiding poset of height k + 2 for some k ≥ P decomposes into quarks as P = Q ⊕ α · · · ⊕ α k Q k . Set W ( P ) to be the word α B α B α · · · B k − α k B k α k +1 defined as follows: • for 0 ≤ i ≤ k + 1, α i = G if P has an all-seeing vertex of rank i and α i = S otherwise; • for 0 ≤ i ≤ k , B i is the type of Q i . 14t is easy to check that the map W is well-defined and that the constraints imposed on the Q i and α i by Theorem 4.9 correspond precisely to the condition that W ( P ) is a legal word.Given a legal word L , we now show that the generating function for posets P such that W ( P ) = L is precisely wt( L ); our result follows immediately from summing over all legalwords L .Fix a word L = α B α · · · B k − α k B k α k +1 , and consider its preimage W − ( L ) = { P | W ( P ) = L } . Any P ∈ W − ( L ) can be written in the form P = Q ⊕ α · · · ⊕ α k Q k withthe types of the Q i determined by the B i . However, after we fix the type B i , any quarkof that type can be used as part of a sum-indecomposable ( + )-avoiding poset. Thus,the posets in the preimage of L contribute exactly F νµ for each occurrence of B i = B νµ .Furthermore, each occurrence of α i = G corresponds to a single all-seeing vertex, and socontributes z . Thus, by standard rules for generating functions, the generating function forposets in W − ( L ) is exactly wt( L ). It follows that I ≥ ( x, z ) is the result of summing wt( L )over the legal words L , as desired.Let I ( x, z ) be the generating function for nonempty sum-indecomposable ( + )-avoidingposets, where the variable z counts all-seeing vertices, the variable x counts other vertices,and I ( x, z ) is exponential in x and ordinary in z . Corollary 6.3.
The generating function for all nonempty sum-indecomposable ( + ) -avoiding posets is I ( x, z ) = z + (cid:88) L wt( L ) , where the sum is over all legal words L . The preceding results establish that to enumerate posets we may focus our energies onenumerating words. We accomplish this task with the transfer-matrix method. Let M W bethe matrix M W = G · B ◦◦ B ◦ ⊗ B ⊗ ◦ B ⊗⊗ B ◦◦ B ◦ ⊗ B ⊗ ◦ B ⊗⊗ B ◦◦ B ◦ ⊗ B ⊗ ◦ B ⊗⊗ B ◦◦ B ◦ ⊗ B ⊗ ◦ B ⊗⊗ + S · B ⊗ ◦ B ⊗⊗ B ◦◦ B ◦ ⊗ B ⊗ ◦ B ⊗⊗ B ⊗ ◦ B ⊗⊗ B ◦◦ B ◦ ⊗ B ⊗ ◦ B ⊗⊗ with entries in the noncommutative algebra R (cid:104)(cid:104) S, G, B ◦◦ , B ◦ ⊗ , B ⊗ ◦ , B ⊗⊗ (cid:105)(cid:105) of words. Proposition 6.4.
With M W as above, the sum of the legal words of length k + 3 is (cid:2) G · B ◦◦ G · B ◦ ⊗ ( S + G ) B ⊗ ◦ ( S + G ) B ⊗⊗ (cid:3) · ( M W ) k · GS + GGS + G . Proof.
Consider the graph G w with vertices {∗ , B ◦◦ , B ◦ ⊗ , B ⊗ ◦ , B ⊗⊗ } and the following directed,labeled edges: for every pair u, v of vertices (allowing u = v ), G w has a directed edge u G −→ v ,and G w has a directed edge u S −→ v unless u = B ν ◦ or u = ∗ and v = B ◦ µ or v = ∗ . The graph G w is illustrated in Figure 10. 15 ◦◦ ∗ B ⊗ ◦ B ⊗⊗ B ◦ ⊗ Figure 10: The S -labeled edges of the graph G w defined in the proof of Proposition 6.4.Each pair of vertices is also joined by directed edges labeled G (not shown).We identify each walk ∗ α −→ B α −→ · · · α k −→ B k α k +1 −−−→ ∗ with the word α B α · · · B k α k +1 . Observe that the first two conditions in Definition 6.1 correspond to the restrictions on edgesinvolving ∗ and the final condition corresponds to edges not involving ∗ . Thus the legal wordsare exactly the walks on this graph that start and end at ∗ , with no intermediate copies of ∗ . We enumerate these walks using the transfer-matrix method, as in [Sta97, Section 4.7].Let X = B ◦◦ B ◦ ⊗ B ⊗ ◦
00 0 0 B ⊗⊗ and Y = G · J + S · , where J is the 4 × G w and applying[Sta97, Theorem 4.7.1], we have that the sum of the words associated to the aforementionedwalks is (cid:2) G G S + G S + G (cid:3) XY XY · · · X GS + GGS + G , which is equivalent to the desired expression.16et M be the matrix M = z · F ◦◦ F ◦ ⊗ F ⊗ ◦ F ⊗⊗ F ◦◦ F ◦ ⊗ F ⊗ ◦ F ⊗⊗ F ◦◦ F ◦ ⊗ F ⊗ ◦ F ⊗⊗ F ◦◦ F ◦ ⊗ F ⊗ ◦ F ⊗⊗ + F ⊗ ◦ F ⊗⊗ F ◦◦ F ◦ ⊗ F ⊗ ◦ F ⊗⊗ F ⊗ ◦ F ⊗⊗ F ◦◦ F ◦ ⊗ F ⊗ ◦ F ⊗⊗ (whose entries are in R [[ x, z ]]). Corollary 6.5.
The generating function for all sum-indecomposable ( + ) -avoiding posetsof height at least is I ≥ ( x, z ) = (cid:2) zF ◦◦ zF ◦ ⊗ (1 + z ) F ⊗ ◦ (1 + z ) F ⊗⊗ (cid:3) · ( I − M ) − · z zz z , where I is the × identity matrix and M is as above.Proof. The result follows from Theorem 6.2, Proposition 6.4 and the fact that the weightmap wt is an algebra homomorphism between R (cid:104)(cid:104) S, G, B ◦◦ , B ◦ ⊗ , B ⊗ ◦ , B ⊗⊗ (cid:105)(cid:105) and R [[ x, z ]].Now that we have enumerated sum-indecomposable ( + )-avoiding posets, the onlyremaining step is to express the generating function for all ( + )-avoiding posets in termsof the generating function for sum-indecomposables. This turns out to be extremely simple. Proposition 6.6.
The generating function G T ( x, z ) for all trimmed strongly graded ( + ) -avoiding posets is given by G T ( x, z ) = (1 − I ( x, z )) − . (Recall that I ( x, z ) is the generating function for nonempty sum-indecomposable posets.)Proof. By Proposition 4.3 and Proposition 4.4 each trimmed strongly graded ( + )-avoidingposet P corresponds to a unique ordinal sum P ⊕ P ⊕· · ·⊕ P k of sum-indecomposable ( + )-avoiding posets, and all such sequences give a trimmed strongly graded ( + )-avoiding poset P . The result follows from the compositional formula for generating functions.The only thing remaining is arithmetic. Theorem 6.7.
The generating function for all strongly graded ( + ) -avoiding posets is e x (2 e x −
3) + e x ( e x − Ψ( x ) e x (2 e x + 1) + ( e x − e x − x ) . Proof.
This is just a calculation, combining Corollaries 6.3 and 6.5 with Propositions 4.2,5.1 and 6.6. For P = 0, 1, . . . , the resulting number of posets is 1, 1, 3, 13, 111, 1381,22383, . . . . 17 Strongly Graded Posets Counted by Height
In this section, we refine the generating function of the previous section to count stronglygraded ( + )-avoiding posets with n vertices of height k . (This refinement is a natural oneto ask for on its own terms; it will also be of use to us when we enumerate weakly graded( + )-avoiding posets in the next section.) The only change in our approach is that wekeep track of the height of the poset as we glue and stick quarks, and then again as we takethe ordinal sum of sum-indecomposables. To this end, let b n,k be the number of stronglygraded ( + )-avoiding posets on n vertices of height k and let H ( x, t ) = (cid:88) n,k b n,k x n n ! t k (2)be the generating function for these numbers. To compute H ( x, t ), we return to the ideasof Section 6. Using the same method as in the proof of Corollary 6.5 but keeping trackof height, we have that the generating function for sum-indecomposable ( + )-avoidingposets of height 2 or more is H I ( x, z, t ) = t (cid:2) zF ◦◦ zF ◦ ⊗ (1 + z ) F ⊗ ◦ (1 + z ) F ⊗⊗ (cid:3) · ( I − t · M ) − · z zz z . If we let H T ( x, z, t ) be the generating function for trimmed strongly graded ( + )-avoidingposets, then (as in Proposition 6.6) we have that H T ( x, z, t ) = (1 − tz − H I ( x, z, t )) − , while from the same reasoning as in the proof of Proposition 4.2 we have H ( x, t ) = H T ( x, e x − , t ). Working out the arithmetic gives the following result. Proposition 7.1.
Let H ( x, t ) be the generating function counting strongly graded ( + ) -avoiding posets by number of vertices and height (as in Equation (2) ). We have H ( x, t ) = e x ( e x + te x + t ( e x − ) + t ((1 − e x + e x ) + t ( e x − ( e x − x ) e x ( e x + te x + t ) + ((1 − e x + e x ) t + ( e x − t )Ψ( x ) . The resulting coefficients are shown in Table 1.
In this section, we expand our study to weakly graded posets. We seek to apply the samemethods that worked in the strongly graded case. The results of Section 3, the definitionof a trimmed poset, and the results of Section 4.1 carry over immediately to weakly gradedposets. We now seek to extend the rest of our work to this context.We begin by proving Proposition 8.1, which shows that weakly graded ( + )-avoidingposets mostly “look just like” strongly graded posets.18eight0 1 2 3 4 5 60 11 12 1 2 P + )-avoiding posets of six or fewer vertices, byheight. Proposition 8.1.
In a weakly graded ( + ) -avoiding poset of height k + 1 such that k ≥ ,all maximal elements are of rank k or k − and all minimal elements are of rank or .Proof. Let P be a weakly graded ( + )-avoiding poset of height at least 3. A maximalvertex in P is precisely the same as a vertex not comparable to vertices of any higher rank.However, by Theorem 3.1, every vertex of P is comparable to all vertices of rank two larger.Putting these two facts together, we immediately conclude that P has no vertices of ranktwo larger than any of its maximal vertices; this is the desired result. The case of minimalvertices is identical.With this result in hand, we can immediately extend the remaining results of Section 4to weakly graded posets. Corollary 8.2.
Every trimmed ( + ) -avoiding poset P can be decomposed uniquely asan ordinal sum of sum-indecomposable posets and each of the resulting sum-indecomposableposets can be decomposed uniquely by sticking and gluing quarks (including a top and bottomquark), where these objects and operations are defined as before, subject to the followingchanges: • when P is written as a maximal ordinal sum of nonempty posets, the topmost andbottommost summands may be weakly graded, not necessarily strongly graded; and • when P is written as an ordinal sum and the topmost summand is written as a stickingand gluing of quarks, the top quark may have isolated vertices on its lower rank set;similarly, the bottom quark of the bottommost summand may have isolated vertices onits upper rank set. (These isolated vertices are exactly the maximal vertices not ofmaximum rank and the minimal vertices not of minimum rank, respectively.) In otherwords, the topmost and bottommost quarks in these summands may be middle quarks. This result allows us to directly apply the methods of Section 6.Observe that any ( + )-avoiding poset with a chain containing 3 or more elements mustbe connected, while posets of height 2 can be somewhat more “wild.” This suggests that weshould consider separately posets of height 2 or less and posets of height 3 or more. We dothis in the following sections. 19 .1 Posets of height at most All posets of height at most 2 are weakly graded and avoid + . Of these, there is exactlyone with height 0 (the empty poset), and for each n ≥ n vertices of height 1 (the antichain on n vertices). The number of posets of height 2 on n vertices is precisely (cid:80) n − m =1 (cid:0) nm (cid:1) | A ⊗ ( n − m, m ) | : we choose m vertices to be at rank 1, andnone of these can be isolated. It follows from these three cases and from Appendix A thatthe generating function for weakly graded ( + )-avoiding posets of height at most 2 is1 + t ( e x −
1) + t ( e − x Ψ( x ) − e x ) . It follows from Corollary 8.2 that when we write a trimmed ( + )-avoiding poset as amaximal ordinal sum of smaller posets, the middle summands (if any) are all strongly graded,and so are sum-indecomposable under the definition of Section 4.2. The bottom summandsatisfies the same rules for gluing and sticking quarks as before, except that the bottommostquark may be a middle quark rather than a bottom quark, and similarly for the top summand.Equivalently, we may redefine legal word by removing conditions 1 and 2 in Definition 6.1 toallow words that begin with SB ◦◦ or SB ◦ ⊗ and end with B ◦◦ S or B ⊗ ◦ S . This corresponds toa straightforward change in the generating function computations of Section 6: the matrices M W and M that appear in Proposition 6.4 and Corollary 6.5 do not need to change at all,though the vectors by which we multiply on the left and right need to be adjusted. We mustonly take care in proving the analogue of Proposition 6.6 in this context.We now give a detailed plan of action. We handle separately those posets that can andcannot be a summand in a nontrivial ordinal sum. This gives us two cases: • posets of height k + 1, where k ≥
2, that consist of a single sum-indecomposable layerwith at least one minimal vertex of rank 1 and at least one maximal vertex of rank k −
1, and • posets of height k + 1, where k ≥
2, that do not fall into the previous class; theseposets have no minimal vertices above rank 0, or have no maximal vertices below rank k , or decompose as a nontrivial ordinal sum.We compute the generating functions in these cases following the transfer-matrix approachused previously. Note one important subtlety: in both cases, the transfer-matrix methodgenerates some posets of height 2 or less which we view as spurious. Thus, we use therefined version of the generating functions computed in Section 7 and make sure to eliminatethe height-0, 1 and 2 terms in the first two cases. (The reason for this approach is thatthe transfer-matrix method as applied here fundamentally works on quarks; thus, it countsposets with isolated vertices multiple times (once for every possible assignment of the isolatedvertices to rank 0 or rank 1). Strongly graded posets of height 2 or larger have no isolatedvertices and so this issue does not arise in the strongly graded case.)20 .2.1 Sum-indecomposable posets that cannot be used in an ordinal sum Some sum-indecomposable ( + )-avoiding posets of height k + 1 cannot be used in a non-trivial ordinal sum to make another ( + )-avoiding poset; these are exactly the ones withmaximal vertices of rank k − t · (cid:2) F ◦◦ F ◦ ⊗ (cid:3) · ( I − tM ) − · where M = z · F ◦◦ F ◦ ⊗ F ⊗ ◦ F ⊗⊗ F ◦◦ F ◦ ⊗ F ⊗ ◦ F ⊗⊗ F ◦◦ F ◦ ⊗ F ⊗ ◦ F ⊗⊗ F ◦◦ F ◦ ⊗ F ⊗ ◦ F ⊗⊗ + F ⊗ ◦ F ⊗⊗ F ◦◦ F ◦ ⊗ F ⊗ ◦ F ⊗⊗ F ⊗ ◦ F ⊗⊗ F ◦◦ F ◦ ⊗ F ⊗ ◦ F ⊗⊗ as before. Trimmed ( + )-avoiding posets not counted in the previous cases have height at least 3and can be written as (possibly trivial) ordinal sums of the following form: • they may or may not have a bottom layer (i.e., a sum-indecomposable ordinal sum-mand) with all maximal vertices of the same rank but with minimal vertices at ranks0 and 1; • they have some number (possibly 0) of “middle layers” that are strongly graded sum-indecomposable posets; and • they may or may not have a top layer with all minimal vertices at rank 0 but withmaximal vertices at the rank below maximum rank.The generating function for strongly graded sum-indecomposable ( + )-avoiding posetsis the function H I ( x, z, t ) defined in Section 7. We define top( x, z, t ) to be the generatingfunction for sum-indecomposable ( + )-avoiding posets with all minimal vertices of rank0 and with some maximal vertices of non-maximum rank, and analogously we define thegenerating function bot( x, z, t ). Then the generating function for posets in this class coincideswith (1 + top( x, z, t ))(1 − H I ( x, z, t )) − (1 + bot( x, z, t )) (3)for all powers of t greater than or equal to 3. Moreover, we havetop( x, z, t ) = t · (cid:2) zF ◦◦ zF ◦ ⊗ (1 + z ) F ⊗ ◦ (1 + z ) F ⊗⊗ (cid:3) · ( I − tM ) − · P + )-avoiding posets of six or fewer vertices, byheight.and bot( x, z, t ) = t · (cid:2) F ◦◦ F ◦ ⊗ (cid:3) · ( I − tM ) − · z zz z . ( + ) -avoiding posets Finally, we combine the work in the preceding subsections to enumerate weakly graded( + )-avoiding posets. Theorem 8.3.
The generating function for weakly graded ( + ) -avoiding posets countedby number of vertices and height is e x − t + ( e − x Ψ( x ) − e x ) t + t e x + e x t − e x (2 e x + (1 + 2 e x − e x ) t )Ψ( x ) − ((1 − e x + e x ) + ( e x − t )Ψ( x ) e x ( e x + te x + t ) + ((1 − e x + e x ) t + ( e x − t )Ψ( x ) . The generating function for weakly graded ( + ) -avoiding posets counted by number ofvertices is ( e − x − x ) + 2 e x + e x ( e x − x ) e x (2 e x + 1) + ( e x − e x − x ) . Proof.
The proof is a straightforward (albeit messy) computation: we add the generatingfunctions from Section 8.2.1 to the expression from Equation (3), kill the t , t and t terms,and add the result to the generating function from Section 8.1. Substituting t = 1 andrearranging slightly gives the second formula.The resulting coefficients are shown in Table 2. Disregarding height, the numbers ofweakly graded ( + )-avoiding posets on 0, 1, 2, . . . vertices are 1, 1, 3, 19, 195, 2551,41343, . . . . 22 Asymptotics
In this section, we compute asymptotics for the number of graded ( + )-avoiding posets.First, we give asymptotics for the coefficients of the series Ψ( x ); then we give the asymptoticsfor our posets in terms of the asymptotics of Ψ. Define ψ n = n (cid:88) i =0 i ( n − i ) i !( n − i )! so that Ψ( x ) =1 + (cid:80) n ≥ ψ n x n . In the next result, we give asymptotics for the coefficients ψ n . Proposition 9.1.
There exist constants C and C such that ψ k ∼ C · k ( k !) and ψ k +1 ∼ C · k ( k +1) k !( k + 1)! . Proof.
For n = 2 k even, we can write ψ k = k (cid:88) i =0 i (2 k − i ) i !(2 k − i )!= k (cid:88) i = − k k − i ( k + i )!( k − i )! ∼ k ( k !) ϑ (0 , / ϑ is a Jacobi theta function. Similarly, when n = 2 k + 1 is odd we find ψ k +1 ∼ / ϑ (0 , /
2) 2 k ( k +1) k !( k + 1)! , as needed.Let g n be the number of strongly graded ( + )-avoiding posets on n vertices and let w n be the number of weakly graded ( + )-avoiding posets on n vertices. Theorem 9.2.
We have g n ∼ w n ∼ n ! · ψ n . The proof relies on the following special case of a theorem of Bender [Ben75], which mayalso be found in [Odl95, Theorem 7.3]:
Theorem 9.3 ([Ben75, Theorem 1]) . Suppose A ( x ) = (cid:80) n ≥ a n x n , that F ( x, y ) is a formalpower series in x and y , and that B ( x ) = (cid:80) n ≥ b n x n = F ( x, A ( x )) . Let C = ∂∂y F (cid:12)(cid:12)(cid:12) (0 , .Suppose further that1. F ( x, y ) is analytic in a neighborhood of (0 , ,2. lim n →∞ a n − a n = 0 , and . n − (cid:88) k =1 | a k a n − k | = O ( a n − ) . Then b n = C · a n + O ( a n − ) and in particular b n ∼ Ca n . We now use this result to prove Theorem 9.2.
Proof.
Define F ( x, y ) = 1 + e x (2 e x −
3) + e x ( e x − ( y + 1) e x (2 e x + 1) + ( e x − e x − y + 1)and F ( x, y ) = ( e − x − y + 1) + 2 e x + e x ( e x − y + 1) e x (2 e x + 1) + ( e x − e x − y + 1)so that F ( x, Ψ( x ) −
1) = G ( x )is the exponential generating function for strongly graded ( + )-avoiding posets (compareTheorem 6.7) and F ( x, Ψ( x ) −
1) = W ( x )is the exponential generating function for weakly graded ( + )-avoiding posets (compareTheorem 8.3). (We compose with Ψ − G and W . To apply the theorem, we have three conditions to check. The first condition is that F and F are analytic in a neighborhood of (0 , n − (cid:88) k =1 ψ k ψ n − k = O ( ψ n − ) . We do that now.The proof of Proposition 9.1 not only gives asymptotics for ψ n but also shows that ψ n ≤ C n / (cid:98) n/ (cid:99) ! · (cid:100) n/ (cid:101) !for all n . In addition, by taking only one term of the sum we have ψ n − ≥ (cid:98) ( n − / (cid:99)·(cid:100) ( n − / (cid:101) (cid:98) ( n − / (cid:99) ! · (cid:100) ( n − / (cid:101) !for all n . Thus n − (cid:88) k =1 ψ k ψ n − k ψ n − ≤ C (cid:48) n − (cid:88) k =1 (cid:98) ( n − / (cid:99) ! · (cid:100) ( n − / (cid:101) ! · − ( k − n − k − / (cid:98) k/ (cid:99) ! · (cid:100) k/ (cid:101) ! · (cid:98) ( n − k ) / (cid:99) ! · (cid:100) ( n − k ) / (cid:101) !24or some constant C (cid:48) . This last summation is bounded by an absolute constant independentof n : the k = 1 and k = n − n while the k = 2 and k = n − O (cid:0) (cid:100) ( n − / (cid:101) − n/ (cid:1) = o (1)to the sum. Each of the remaining terms can be seen to be O ( n − ), so their total contributionis also o (1).We have shown that the conditions of Theorem 9.3 hold and we now apply it directly.By direct computation, ∂∂y F ( x, y ) (cid:12)(cid:12)(cid:12) (0 , = ∂∂y F ( x, y ) (cid:12)(cid:12)(cid:12) (0 , = 1. Thus, we have g n n ! = ψ n + O ( ψ n − ) ∼ ψ n and similarly for w n , as desired. Acknowledgments
We wish to thank Alejandro Morales and Richard Stanley for valuable conversations. Wewould also like to thank the two referees for many valuable comments, and in particularfor correcting a significant error in Section 4.3. YXZ was supported by an NSF graduateresearch fellowship. JBL was supported in part by NSF RTG grant NSF/DMS-1148634.
A Computing generating functions for quarks
In this appendix, we enumerate and compute generating functions counting those sets ofthe form A νµ (introduced in Section 5) that are of use to us. For bookkeeping purposes, wemake these generating functions bivariate in variables x and y , with each bipartite graph in A νµ ( m, n ) (i.e., each graph on the vertex set [ m ] (cid:93) [ n ] with appropriate restrictions) giving acontribution of x m y n m ! n ! .It is very convenient to introduce the generating functionΨ( x, y ) = (cid:88) m,n ≥ mn x m y n m ! n ! , as most of our generating functions are most easily expressed in terms of Ψ( x, y ).1. | A ( m, n ) | = 2 mn : we have no restrictions, so all of the mn edges may choose inde-pendently to be present or absent. Equivalently, we have (cid:80) m,n ≥ | A ( m, n ) | x m y n m ! n ! =Ψ( x, y ) − e x − e y + 1. (The extra terms at the end simply account for the fact that wesum here only over positive values of m , n .)2. | A (cid:2) ( m, n ) | = | A ⊗ ( m, n ) | = (2 n − m : we need every vertex on the m -side to be notall-seeing (respectively, isolated) and there are no other restrictions. It is not hard to25ompute the generating function (cid:88) m,n ≥ | A ⊗ ( m, n ) | x m y n m ! n ! = e − x Ψ( x, y ) − e y . It follows by symmetry that the generating function for A (cid:2) and A ⊗ is e − y Ψ( x, y ) − e x .3. | A (cid:2) ⊗ ( m, n ) | = (2 n − m : each vertex on the m -side can be connected to any subseton the n -side except the empty set or everything. The associated generating functionis e − x Ψ( x, y ) − e − x − e y + 1.4. | A (cid:2)(cid:2) ( m, n ) | = | B ( m, n ) | : First, we show that | B ( m, n ) | = m (cid:88) i =0 ( − i (cid:18) mi (cid:19) (2 m − i − n . (4)The proof is by inclusion-exclusion on the all-seeing vertices in [ n ]. For a subset S ⊆ [ n ],the number of graphs in which the vertices of S are all-seeing and no vertices of [ m ] areall-seeing is (2 n −| S | − m : each vertex in [ m ] may choose the union of S with any propersubset of [ n ] \ S to be its neighbors, and these choices may be made independently.Applying inclusion-exclusion immediately gives the result. (As an aside, this meansthat the summation expression on the right-hand side of Equation (4) is symmetric in m and n , a fact not immediately obvious from its formula.)Now, a routine calculation gives1 + (cid:88) m,n ≥ | B ( m, n ) | x m y n m ! n ! = (cid:88) m,n ≥ | B ( m, n ) | x m y n m ! n ! = e − x − y Ψ( x, y ) . | A (cid:2)(cid:2) ◦ ( m, n ) | = | B ◦ ( m, n ) | : From the definitions of the sets A νµ and the precedingcomputations we have | A (cid:2)(cid:2) ◦ | = | A (cid:2) ◦ | − | A (cid:3)(cid:2) ◦ | = | A (cid:2) ◦ | = (2 n − m − (2 n − m . The associated generating function is (1 − e − x )( e − x Ψ( x, y ) − | A (cid:2) ◦ (cid:2) ◦ ( m, n ) | = | B ◦◦ ( m, n ) | : We have, by similar computations, that | A (cid:2) ◦ (cid:2) ◦ | = | A | − | A ⊗ | − | A ⊗ | + | B | and so the associated generating function is (1 − e − x )(1 − e − y )Ψ( x, y ).Finally, we may use the work above to compute the generating functions we desire. We26ave F ◦◦ ( x ) = (cid:88) m,n ≥ | B ◦◦ ( m, n ) | x m + n m ! n != (1 − e − x ) Ψ( x, x ) ,F ⊗ ◦ ( x ) = (cid:88) m,n ≥ | B ⊗ ◦ ( m, n ) | x m + n m ! n != (cid:88) m,n ≥ (cid:0) | B ◦ ( m, n ) | − | B ◦◦ ( m, n ) | (cid:1) x m + n m ! n != (1 − e − x )((2 e − x − x, x ) − , and similarly F ◦ ⊗ ( x ) = (1 − e − x )((2 e − x − x, x ) − F ⊗⊗ ( x ) = (2 e − x − e − x − x, x ) − . References [ASV12] Mike D. Atkinson, Bruce E. Sagan, and Vincent Vatter. Counting (3 + 1)-avoiding permutations.
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