Symmetric Distribution of Crossings and Nestings in Permutations of Type B
aa r X i v : . [ m a t h . C O ] J un SYMMETRIC DISTRIBUTION OF CROSSINGS AND NESTINGSIN PERMUTATIONS OF TYPE B ADEL HAMDIFaculty of Science of Gabes, Department of Mathematics,Cit´e Erriadh 6072, Zrig, Gabes, TunisiaandUniversit´e de Lyon, Universit´e Lyon 1, Institut Camille Jordan,CNRS UMR 5208, 43, boulevard du 11 novembre 191869622, Villeurbanne Cedex, France aadel [email protected]
Abstract.
This note contains two results on the distribution of crossing numbersand nesting numbers in permutations of type B. More precisely, we prove a B n -analogue of the symmetric distribution of crossings and nestings of permutationsdue to Corteel (Adv. in Appl. Math., 38(2)(2007), 149-163) as well as the sym-metric distribution of k -crossings and k -nestings of permutations due to Burrill etal. (DMTCS proc. AN, (2010), 461-468). Introduction
In the last years, many results on symmetric distributions of some statistics “crossing”and “nesting” have appeared in several combinatorial structures. At the heart of theseresults, on the set of matchings and partitions, there are Chen et al’s and Kasraouiet al’s theorems [2, 5] on the interchanging crossing (resp., 2-crossing, k -crossing)numbers and nesting (resp., 2-nesting, k -nesting) numbers. Then, some extensionsof type B and C have been given by Rubey and Stump [8], and Krattenthaler and deMier [6, 7] on the relation beteween increasing and decreasing chains in partitions andlink partitions, and fillings of Ferrers shapes. On the set of permutations, Corteel [3]has introduced the notion of crossings and nestings of permutations and proved thatfor any fixed number of weak exceedances, the distribution of crossing numbers andnestings numbers of permutations is symmetric. Recently, Burrill et al [1] have proveda similar result for k -crossings and k -nestings of permutations. The purpose of thispaper is to extend the last two results to their analogue of type B .2. Definitions and main results
For a positive integer n , let [ n ] := { , , . . . , n } . A type B permutation of rank n is an integer sequence σ := ( σ (1) , σ (2) , . . . , σ ( n )) such that {| σ (1) | , ..., | σ ( n ) |} = [ n ]. In this paper, we shall identify σ with a permutation of [ − n , n ] := {− n, . . . , − , − , , . . . , n } by σ ( − i ) = − σ ( i ) for each i ∈ [ n ]. Let neg( σ ) be the number of negativenumbers in { σ (1) , . . . , σ ( n ) } , and B n the set of type B permutations of rank n .In the sequel, we use the natural order of integers in Z .As in [3], it is convenient to represent a permutation σ ∈ B n by a permutationdiagram G = ( V, E ), where V = [ − n, n ] is the vertex set, and E is the set of edges( i, σ ( i )) for i ∈ [ − n, n ] such that the vertices − n , . . . , − −
1, 1, 2, . . . , n arearranged from left to right on a straight line. We draw an arc from i to σ ( i ) above(resp. under) the line if i σ ( i ) (resp. otherwise) such that two arcs cross at mostonce. A permutation diagram is given in Fig. 1. -3-4 -2 -1 1 2 3 4 5-5 6-6 rr r rr r r r rr rr Fig. 1. The permutation diagram of σ = (4 , − , , , , − We call the set of arcs that are above (resp. under) the line the upper (resp. under)permutation diagram and denoted
Upp ( σ ) (resp. Und ( σ )).We start with an easy lemma that follows immediately from the definition of thepermutation diagram since there is an easy bijective between upper and under dia-garms. Lemma 2.1.
Let σ ∈ B n . The diagram of σ is completely determined by the Upp( σ ). Note that there are five geometric patterns for two arcs above the line as illustratedin Fig. 2 by rrr r (i) rr r (ii) r r rr (iii) r r r (iv) r r rr (v)
Fig. 2. Five patterns between two arcs above the line.
These patterns are called: (i) a proper crossing, (ii) a skew crossing, (iii) a propernesting, (iv) a skew nesting and (v) an alignment. In another sense, one can recoverthese geometric patterns as in the two following definitions.The first is the notion of crossings of type B given by Corteel et al. in [4] as follows. Definition 2.2.
Let σ ∈ B n . The number of weak exceedances of σ , denoted by wex B ( σ ) , is the cardinality of the set { j ∈ [ n ]; σ ( j ) ≥ j } . For two integers i and j in [ n ] , two arcs ( i, σ ( i )) and ( j, σ ( j )) form a crossing of σ if they satisfy either therelation i < j σ ( i ) < σ ( j ) (upper crosing), or − i < j σ ( − i ) < σ ( j ) (uppercrosing) or σ ( i ) < σ ( j ) < i < j (lower crosing). ROSSINGS AND NESTINGS IN PERMUTATIONS OF TYPE B Similarly, in the second, we can define the notion of nesting of type B . Definition 2.3.
Let σ ∈ B n . A pair of arcs ( i, σ ( i )) and ( j, σ ( j )) , with i and j in [ n ] , is a nesting of σ if they satisfy either the relation i < j σ ( j ) < σ ( i ) (uppedernesting), or − i < j σ ( − j ) < σ ( i ) (upper nesting) or σ ( j ) < σ ( i ) < i < j (lowernesting). The number of crossings ( resp., nestings ) of σ is denoted by cro B ( σ ) ( resp., nes B ( σ )) . Example 1.
Let σ = (4, −
6, 3, 5, 1, − ∈ B . Then the nestings in σ are { ( − σ ( − σ (1)) } , { (3, σ (3)), (1, σ (1)) } , { (3, σ (3)), ( − σ ( − } , { (4, σ (4)), ( − σ ( − } and { (6, σ (6)), (5, σ (5)) } . The crossings are { ( − σ ( − σ (1)) } , { (1, σ (1)), (4, σ (4)) } , { (5, σ (5)), (2, σ (2)) } and { (6, σ (6)), (2, σ (2)) } (see Fig. 1). Hence nes B ( σ ) = 5 and cro B ( σ ) = 4.The following is our B n -analogue of Corteel’s result for type A permutations [3,Proposition 4]. Theorem 2.4.
The number of permutations in B n with k weak exceedances, l minussigns, i crossings and j nestings is equal to the number of permutations in B n with k weak exceedances, l minus signs, i nestings and j crossings. In other words, we have X σ ∈ B n p nes B ( σ ) q cro B ( σ ) y wex B ( σ ) a neg ( σ ) = X σ ∈ B n p cro B ( σ ) q nes B ( σ ) y wex B ( σ ) a neg ( σ ) . (1)Note that the a = 0 case of Theorem 2.4 corresponds to Proposition 4 in [3].Now, we extend the definition of k -crossings and k -nestings for permutations of type A in [1] to permutations of type B . Definition 2.5.
Let σ ∈ B n . A set { a , a , . . . , a k } of k integers in [ n ] is a k -crossingof σ if they satisfy either the relation a < a < . . . < a k ≤ σ ( a ) < σ ( a ) < . . . <σ ( a k ) ( upper k -crossing ) , or − a < a < . . . < a k ≤ − σ ( a ) < σ ( a ) < . . . < σ ( a k )( upper k -crossing ) or σ ( a k ) < σ ( a k − ) < . . . < σ ( a ) < a k < a k − < . . . < a ( lower k -crossing ) . Definition 2.6.
Let σ ∈ B n . A set { a , a , . . . , a k } of k integers in [ n ] is a k -nestingof σ if they satisfy either the relation a < a < ... < a k ≤ σ ( a k ) < σ ( a k − ) < . . . <σ ( a ) ( upper k -nesting ) , or − a < a < . . . < a k ≤ − σ ( a k ) < σ ( a k − ) < . . . < σ ( a )( upper k -nesting ) or σ ( a k ) < σ ( a k − ) < . . . < σ ( a ) < a < a < . . . < a k (lower k -nesting). As in [1], the k-crossing number (resp., k-nesting number ) of a permutation σ oftype B , denoted by cro ∗ B ( σ ) (resp., nes ∗ B ( σ )) is the size of the largest k such that σ contains a k -crossing (rsep. k -nesting). Example 2.
Let σ = (4, 5, 6, 2, − − ∈ B . Then we have cro ∗ B ( σ ) = 4and nes ∗ B ( σ ) = 2 that are illustrated respectively, in Fig. 3, by { , , , } and {
4, 5 } or {
4, 6 } since − < < < − σ (5) < σ (1) < σ (2) < σ (3), σ (5) < σ (4) < < σ (6) < σ (4) < < ADEL HAMDI-3-4 -2 -1 1 2 3 4 5-5 6-6 rr r rr r r r rr rr
Fig. 3. The permutation diagram of σ = (4, 5, 6, 2, -3, -1). We also recall the following definition from [1]. Let σ ∈ B n . The degree sequence ofthe upper permutation diagram U pp ( σ ) is the sequence ( indegree σ ( i ) , outdegree σ ( i )) i ∈ [ − n,n ] ,where indegree σ ( i ) (resp. outdegree σ ( i )) is the left (resp. right) degree of the vertex i , i.e. , the number of arcs joining i to a vertex j with j < i (resp. j > i ). If an upperpermutation diagram U pp ( σ ) has d as its degree sequence (some other sources callthis left-right degree sequence), we say that U pp ( σ ) is a diagram on d .But there is a straightforward difference that we do not put a loop (an arc if σ ( i ) = i )on the isolated vertex i with negative index in U pp ( σ ), i.e. , we put (0,0) as a degree.By Lemma 2.1, we limit ourselves to study the upper permutation diagram of type B.The vertices with degree (0,1) (resp., (1,0), (1,1)) are called openers (resp., closers ,closer-opener or transient). For instance, if we let σ = (4, 5, 6, 2, -3, -1), then thedegree sequence of the upper permutation diagram of σ is d := d ( σ ) = (0 , , , , , , , , , , , , . Let B dn be the set of the permutations in B n that has the degree sequence d.The following is our B n -analogue of [1, Theorem 1]. Theorem 2.7.
Let
N C B dn ( i, j, m ) be the number of permutations in B n with i-crossings,j-nestings, m minus signs and degree sequence specified by d . Then N C B dn ( i, j, m ) = N C B dn ( j, i, m ) . (2) In other words, we have X σ ∈ B dn x nes ∗ B ( σ ) y cro ∗ B ( σ ) z neg ( σ ) = X σ ∈ B dn p cro ∗ B ( σ ) q nes ∗ B ( σ ) z neg ( σ ) . (3)Note that when z = 0 in (3), we recover Theorem 1 of [1].Now, we sketch the opener, closer and transient vertices of the upper (resp., under)permutation diagram with degree (0,1), (1,0) and (1,1) (resp., (1,0), (0,1), (1,1))respectively. Then a vertex is said to be: r r (resp., )(i) an opener if it is illustrated by r r (resp., )(ii) a closer if it is illustrated by r r (resp., ).(iii) a transient if it is illustrated byWe shall prove Theorem 2.4 in Section 3 by constructing an explicit involution on B n that interchanges the number of crossings and number of nestings. In fact, it is ROSSINGS AND NESTINGS IN PERMUTATIONS OF TYPE B an extension of the involution defined in [5]. To prove the Theorem 2.7 in Section 4,we shall adopt the map defined by de Mier in [7] to B n .3. Proof of Theorem 2.4
First, for each σ ∈ B n , the number of crossings of σ is equal, in U pp ( σ ), to the numberof proper crossings plus the number of transient vertices i with i in [ n ]. Similarly,the number of nestings of σ is equal, in U pp ( σ ), to the number of proper nestingsplus the number of arcs ( i , k ) with i and k in [ − n, n ] such that there exist two fixedvertices j and − j satisfy i < | j | < k . For instance, in the left diagram of Fig. 4, wecan count the 4 crossings and the 5 nestings of σ given in Fig. 1.Now, we introduce some notations. For a positive integer n , let Λ n be the setof the subsets of [ n ] and B n the set of U pp ( σ ) for each σ ∈ B n . Let also F and T be two maps defined by: for each σ ∈ B n , F ( σ ) := { j ∈ [ n ]; σ ( j ) = j } and T ( σ ) := { j ∈ [ n ]; j is an upper transient vertex of σ } . We notice that, in [8],Rubey and Stump have studied the symmetry distribution of the number of crossingsand number of nestings in a kind of set partitions of type B. Then, we study a simileresult in B n where we count the transient vertex (resp. an arc covers a fixed vertex)as a crossing (resp. nesting) and our arrangement of the set [ − n, n ] is different oftheir. Proof of Theorem 2.4 . There are two steps.
First step.
Let σ ∈ B n . We define a map ψ by: let ( U pp ( σ ), F ( σ ), T ( σ )) ∈ B n × Λ n .Then ψ transforms each element i of F ( σ ) to an arc ( i , i ′ ) (see (a)) and each element j of T ( σ ) to a proper crossing (see (b)), which in the two cases, we have i < i ′ (resp. j < j ′ ) and no vertex between i and i ′ (resp. j and j ′ ). We adopte from [1] thefollowing graphs j r −→ ψ j j ′ r r (b) r −→ ψ i i i ′ rr (a)For instance, the permutation diagram of σ in Example 1 has a fixed vertex indexedby 3 and a transient vertex indexed by 4, see the left diagram of Fig. 4. Then, itsinverse ψ − reduces each two vertices i and i ′ introduced by ψ into a one vertex i . -3-4 -2 -1 1 2 3 4 5-5 6-6 rr r rr r r r rr rr -3-4 -2 -1 1 2 33’ 44’ 5-5 6-6 rr r rr r r r r r rr rr ✲ ψ Fig. 4. The left digram is the upper permutation diagram of σ = (4, -6, 3, 5, 1, -2)and the right is its image by ψ . Second step.
We give an outline of the involution ϕ of [5]. Let π be a partitionof type A and G be its partition diagram defined as the upper permutation diagram.For each two vertices k and j of G, we adopt that j is a vacant vertex for the k thposition if j < k and its corresponding closer vertex l satisfies l > k . Then for each ADEL HAMDI arc (i, j) of G, we denote by δ ( i, j ) (resp. γ ( i, j )) the number of vacant vertex k suchthat k < i (resp. k > i ) for the position j . The algorithm describing the involution ϕ is to construct a partition diagram G ′ from G, vertex by vertex and from left toright in the following paragraph.For each vertex k of G from 1 to the rank of π , if k is a fixed (resp. opener) vertexthen we conserve its form; fixed (resp. opener) vertex, at the position k in G ′ and if k is a closer (resp. transient) vertex, we also conserve its form; closer (resp. transient)vertex, at the same position in G ′ , but we exchange the arc ( s , k ) where s is thecorresponding opener of k in G into an arc ( t , k ) in G ′ with t is the γ ( s, k )th vacantvertex, from left to right, for the position k . So, ϕ is a proper crossings and nestingsinterchanging map. -3-4 -2 -1 1 2 33’ 44’ 5-5 6-6 rr r rr r r r r r rr rr ✲ ϕ Fig. 5. The left diagram is the diagram of ψ ( σ )and the right is its image by ϕ . -3-4 -2 -1 1 2 33’ 44’ 5-5 6-6 rr r rr r r r r r rr rr It remains to prove that the number of the minus signs is unchanged. Since, foreach σ ∈ B n , the number of minus signs is equal to the number of arcs, in the upperpermutation diagram U pp ( σ ), joining a vertex with negative index and a vertex withpositive index. Thus, according to the construction of the openers and closers verticesof ϕ ( U pp ( σ )), the last number is invariant.Finally, for instance, by the map ψ − ◦ ϕ ◦ ψ and Lemma 2. 1, we illustrate, inthe following figure, the corresponding permutation σ ′ = (2, -5, 4, 6, 1, -3) of thepermutation σ = (4, −
6, 3, 5, 1, −
2) in Example 1. -3-4 -2 -1 1 2 3 4 5-5 6-6 rr r rr r r r rr rr
Fig. 6. The permutation diagram of σ ’ = (2, -5, 4, 6, 1, -3). (cid:3) Proof of Theorem 2.7
Our proof is based on an extension of de Mier’s bijection in [7, Section 4] to B n .First, we extend the basic tool, in [7], that is the construction of a bijection betweenlink partitions of type A and fillings of Young diagram into a bijection, denoted by ξ , between upper permutation diagrams and fillings of Young diagrams on B n . For ROSSINGS AND NESTINGS IN PERMUTATIONS OF TYPE B each σ in B n , let i , ..., i c be the closers vertices of U pp ( σ ) and j , ..., j o the openersones. Let p ( i ), for each closer vertex i , be the number of vertices j with j < i thatare openers such that for each transient we associate a closer before an opener. Weconsider a Young diagram T of shape ( p ( i c ), ..., p ( i )), and if there is an arc goingfrom the opener j s to the closer i r , we fill the cell in column s and row c − r + 1 with 1.For instance, in the following figure, we illustrate the upper permutation diagram of σ of Example 2 and its corresponding filling of Young diagram. -3-4 -2 -1 1 2 3 4 5-5 6-6 qq q qq q q q qq qq Fig. 7. The upper permutation diagram of σ = (4, 5, 6, 2, -3, -1) andits corresponding filling of Young diagram. ✲ ξ Thus, we see that the k − nesting (one can also say k +1-nonnesting) (resp., k − crossing)in the upper permutation diagram corresponds to a matrix identity I k (resp., antiiden-tity J k called also the antidiagonal of I k ) in a largest rectangle in the correspondingYoung diagram.We can now state an important result due to de Mier [7]. Lemma 4.1. [7, Theorem 3.5]
For all diagrams T with prescribed row and colunmsums, the number of fillings T that avoid I k equals the number of fillings of T thatavoid J k . Moreover, de Mier proves that there is a map Ψ preserving the left-right degreesequence of a link partition of type A and it interchanges the k -noncrossings and k -nonnestings for the proper crossings and nestings and conversely. We extend thisinvolution on B n . In fact, it divides into two maps. The first is ϕ . For each σ ∈ B n ,we see the largest k such that the filling of Young diagram corresponding to U pp ( σ )contains a largest rectangle which contains a matrix antiidentity J k . If there aremany matrices antiidentities of rank k , we choose the one more to the right and thetopmost. So, the 1’s of J k , from left and bottom to right and top, correspond to( l , c ), ( l , c ), . . . , ( l k , c k ) cells in the diagram, i.e. , ( l i , c i ) is the intersection cell ofthe l i th line and c i th colunm, for each 1 i k . Thus ϕ changes the places of the1’s of J k in the diagram to new places define by: ( l , c ), ( l , c ), . . . , ( l k , c k − ) and( l , c k ) and we obtain in the first time the following matrix ϕ ( J k ) = (cid:18) J k −
00 1 (cid:19) where J k − is the matrix antiidentity of rank k − ϕ to J k − , J k − , . . . , until we get I k . ADEL HAMDI
The second is φ (the inverse of ϕ ). Let k be the largest integer such that the Youngdiagram contains a matrix I k (if there are many matrices identities of rank k , wechoose the one more to the right and the topmost). The 1’s of I k have ( a , b ),( a , b ), . . . , ( a k , b k ) as the cells in the diagram from top to bottom and left to rightwith ( a i , b i ) is the intersection cell between the a i th line and the b i colunm. So, theimage of I k by φ is ( a , b ), ( a , b ), ( a , b ), . . . , ( a k , b k ), i.e. , φ ( I k ) = I k − where I k − is the matrix identity of rank k −
2. The image of φ ( I k ) by φ is ( a , b ),( a , b ), ( a , b ), ( a , b ), . . . , ( a k , b k ). So on, we apply φ to φ i ( I k ), for i from 0 to k −
1, until we get J k .Then, by the two processes, Ψ interchanges I k and J k .It remains to prove that the number of minus signs is unchanged by the abovetransformations ξ and Ψ. We know that this number in such a permutation in B n isequal to the number of the arcs ( α i , β i ) in its upper permutation diagram such that α i (resp., β i ) is a negative (resp., positive) index. Let σ in B n . Suppose that neg ( σ )= m. Then there exist only m arcs ( α , β ), . . . , ( α m , β m ) in U pp ( σ ) such that, foreach 1 ≤ i ≤ m , α i and β i are, respectively, a negative and positive indices. Let ( i , j )be an arc in U pp ( σ ). We say that i (resp., j ) is an outpoint (resp., endponit ) and wedenote by N O (resp.,
N E ) the number of outpoints (resp., endpoints) with negativeindex. We have that m = ( N O - N E ). On the other hand, the degree sequence of
U pp ( σ ) is kept by the involution ϑ where ϑ = ξ − ◦ Ψ ◦ ξ . Then there exist only ( N O - N E ) arcs in ϑ ( U pp ( σ )) such that each arc has an outpoint with negative index andan endpoint with positive index. This establishes the desired equality.For instance, the following diagram is the permutation diagram of ϑ ( U pp ( σ )) ofthe permutation σ in Example 2. It is easy to see that cro ∗ ( ϑ ( U pp ( σ ))) = 2 and nes ∗ ( ϑ ( U pp ( σ ))) = 4. -3-4 -2 -1 1 2 3 4 5-5 6-6 qq q qq q q q qq qq Fig. 8. The upper permutation diagram ϑ ( U pp ( σ )). (cid:3) ✲ ϑ -3-4 -2 -1 1 2 3 4 5-5 6-6 qq q qq q q q qq qq We now come to conclude this paper with a straightforward enumerative result onthe maximum crossing (resp. maximum nesting) chains. A maximum crossing , insuch a permutation in B n , is the n -crossing. Then we can compute the number ofpermutations with n -crossing, denoted by C B n , in the following corollary. ROSSINGS AND NESTINGS IN PERMUTATIONS OF TYPE B Corollary 4.2.
Let n be a positive integer. Then the number of permutations in B n with n-crossings C B n satisfies C B n = (cid:26) if n ≤ , otherwise. (4)In fact, for n ≤
2, it is easy to see that. For n ≥
3, the only element σ in B n thathas n-crossing is the permutation σ defined by σ ( i ) = − ( n + 1) + i for 1 ≤ i ≤ n , i.e. , the only permutation σ that satisfies σ (1) < σ (2) < . . . < σ ( n ) < Acknowledgements
The author would like to thank Jiang Zeng for his advice and careful reading of thispaper.
References [1] S. Burrill, M. Mishna and J. Post, On k -crossings and kk