aa r X i v : . [ m a t h . C O ] N ov ENUMERATION OF BOUNDED LECTURE HALL TABLEAUX
SYLVIE CORTEEL AND JANG SOO KIM
To Christian Krattenthaler, our determinantal hero.
Abstract.
Recently the authors introduced lecture hall tableaux in their study of multivariatelittle q -Jacobi polynomials. In this paper, we enumerate bounded lecture hall tableaux. We showthat their enumeration is closely related to standard and semistandard Young tableaux. We alsoshow that the number of bounded lecture hall tableaux is the coefficient of the Schur expansionof s λ ( m + y , . . . , m + y n ). To prove this result, we use two main tools: non-intersecting latticepaths and bijections. In particular we use ideas developed by Krattenthaler to prove bijectivelythe hook content formula. Introduction
Recently the authors [9] introduced lecture hall tableaux in their study of multivariate little q -Jacobi polynomials P λ ( x ; a, b ; q ) with t = q . They showed that if we expand the Schur function s λ ( x ) in terms of P µ ( x ; a, b ; q ) and vice versa as s λ ( x ) = X µ M λ,µ P µ ( x ; a, b ; q ) , P λ ( x ; a, b ; q ) = X µ N λ,µ s µ ( x ) , then the coefficients M λ,µ and N λ,µ can be expressed as generating functions for lecture halltableaux of shape λ/µ .A lecture hall tableau is a certain filling of a skew shape λ/µ with nonnegative integers. Sincethe entries in a lecture hall tableau can be arbitrarily large, there are infinitely many lecture halltableaux of a given shape. If we give an upper bound on their entries we can consider the numberof lecture hall tableaux. The main goal of this paper is to enumerate such bounded lecture halltableaux.Bounded lecture hall objects were first enumerated by the first author, Lee and Savage in [10].They showed that the number of sequences λ = ( λ , . . . , λ n ) of integers such that m ≥ λ ≥ λ ≥ · · · ≥ λ n n ≥ λ = ( λ , . . . , λ n ) of integers such that m ≥ λ n ≥ λ n − ≥ · · · ≥ λ n ≥ . This number is equal to ( m +1) n . As remarked by Matt Beck [5], this is also the Ehrhart polynomialof the n -cube. This observation started a collection of very interesting papers connecting lecturehall partitions to geometric combinatorics and in particular polytopes. We cite for example [3, 4,16, 21]. An overview of the techniques and results is presented in the survey by Carla Savage [22].We will see that counting bounded lecture hall tableaux is naturally related to standard andsemistandard Young tableaux. To state our results we first give definitions of related objects.A partition is a weakly decreasing sequence λ = ( λ , . . . , λ k ) of positive integers. Each integer λ i is called a part of λ . The length ℓ ( λ ) of λ is the number of parts. We identify a partition λ = ( λ , . . . , λ k ) with its Young diagram , which is a left-justified array of squares, called cells ,with λ i cells in the i th row for 1 ≤ i ≤ k . In other words, we consider λ = ( λ , . . . , λ k ) as the set Date : November 7, 2019.2010
Mathematics Subject Classification.
Primary: 05A15; Secondary: 33D45, 33D50, 05A30.
Key words and phrases. lecture hall tableau, standard Young tableau, semistandard Young tableau, bijectiveproof, Schur function. of cells ( i, j ) such that 1 ≤ i ≤ k and 1 ≤ j ≤ λ i . For two partitions λ and µ we write µ ⊂ λ tomean that the Young diagram of µ is contained in that of λ as a set. In this case, a skew shape λ/µ is defined to be the set-theoretic difference λ \ µ of their Young diagrams. We denote by | λ/µ | the number of cells in λ/µ . A partition λ is also considered as a skew shape by λ = λ/ ∅ .A tableau of shape λ/µ is a filling of the cells in λ/µ with nonnegative integers. In other words,a tableau is a map T : λ/µ → N , where N is the set of nonnegative integers. A standard Youngtableau of shape λ/µ is a tableau of shape λ/µ such that every integer 1 ≤ i ≤ | λ/µ | appearsexactly once and the entries are decreasing in each row and in each column. Let SYT( λ/µ ) denotethe set of standard Young tableaux of shape λ/µ . We note that it is more common to definea standard Young tableau to have entries increasing in each row and column. However, for ourpurpose in this paper, it is more convenient to have entries decreasing.It is well known that the number of standard Young tableaux of shape λ is given by the hooklength formula due to Frame, Robinson, and Thrall [11]:(1) | SYT( λ ) | = | λ | ! Q ( i,j ) ∈ λ h ( i, j ) , where h ( i, j ) = λ i + λ ′ j − i − j + 1 and λ ′ j is the number of integers 1 ≤ r ≤ ℓ ( λ ) with λ r ≥ j . Thereare many proofs of the hook length formula, see the survey by Adin and Roichman [1]. Amongthese a remarkable bijective proof of (1) was found by Novelli, Pak, and Stoyanovskii [20] using a“jeu de taquin” sorting algorithm.A semistandard Young tableau of shape λ/µ is a tableau of λ/µ such that the entries are weaklydecreasing in each row and strictly decreasing in each column. We denote by SSYT( λ/µ ) theset of semistandard Young tableaux of shape λ/µ . We also denote by SSYT n ( λ/µ ) the set of T ∈ SSYT( λ/µ ) with max( T ) < n , i.e., the entries of T are taken from { , , . . . , n − } . Stanley[23] showed that the number of such bounded semistandard Young tableaux is given by the hook-content formula :(2) | SSYT n ( λ ) | = Y ( i,j ) ∈ λ n + c ( i, j ) h ( i, j ) , where c ( i, j ) = j − i is the content of the cell ( i, j ). There are also many proofs of the hookcontent formula. Krattenthaler [14] found a bijective proof of (2) that uses a modified jeu detaquin sorting algorithm. In this paper we will use Krattenthaler’s jeu de taquin to investigatelecture hall tableaux.An n -lecture hall tableau of shape λ/µ is a tableau L of shape λ/µ satisfying the followingconditions: L ( i, j ) n + c ( i, j ) ≥ L ( i, j + 1) n + c ( i, j + 1) , L ( i, j ) n + c ( i, j ) > L ( i + 1 , j ) n + c ( i + 1 , j ) . The set of n -lecture hall tableaux of shape λ/µ is denoted by LHT n ( λ/µ ). For L ∈ LHT n ( λ/µ ),let ⌊ L ⌋ be the tableau of shape λ/µ whose ( i, j )-entry is ⌊ L ( i, j ) / ( n − i + j ) ⌋ , see Figure 1 for anexample. The set of n -lecture hall tableaux L ∈ LHT n ( λ/µ ) with max( ⌊ L ⌋ ) < m is denoted byLHT n,m ( λ/µ ). Since the bounded lecture hall tableaux in LHT n, ( λ/µ ) play an important role inour paper, we give a special name for them. These objects have another description as follows.A semistandard n -content tableau of shape λ/µ is a semistandard Young tableau S of shape λ/µ with the additional condition that 0 ≤ S ( i, j ) < n − i + j for every ( i, j ) ∈ λ/µ . We denoteby SSCT n ( λ/µ ) the set of semistandard n -content tableaux of shape λ/µ . It is easy to see thatSSCT n ( λ/µ ) = LHT n, ( λ/µ ) , SSCT n ( λ ) = SSYT n ( λ ) . In this paper we prove the following formula for the number of bounded lecture hall tableaux.Given a partition µ , we use the convention that µ i = 0 for all integers i > ℓ ( µ ). NUMERATION OF BOUNDED LECTURE HALL TABLEAUX 3
25 25 2116 18 21 10 48 9 2 04 4 0
258 259 2110165 186 217 108 4983 94 25 0642 43 04
Figure 1.
On the left is a lecture hall tableau L ∈ LHT n ( λ/µ ) for n = 5, λ = (6 , , ,
3) and µ = (3 , L ( i, j ) / ( n + c ( i, j )) for each entry ( i, j ) ∈ λ/µ . The diagram on the right is thetableau ⌊ L ⌋ . Theorem 1.1.
For partitions λ and µ with µ ⊂ λ and ℓ = ℓ ( λ ) ≤ n , we have | LHT n,m ( λ/µ ) | = m | λ/µ | det (cid:18)(cid:18) λ i + n − iµ j + n − j (cid:19)(cid:19) ≤ i,j ≤ ℓ . Note that Theorem 1.1 implies that(3) | LHT n,m ( λ/µ ) | = m | λ/µ | | LHT n, ( λ/µ ) | = m | λ/µ | | SSCT n ( λ/µ ) | . The determinant in Theorem 1.1 has another description in terms of standard Young tableaux.
Proposition 1.2.
For partitions λ and µ with µ ⊂ λ and ℓ = ℓ ( λ ) ≤ n , we have | SSCT n ( λ/µ ) | = det (cid:18)(cid:18) λ i + n − iµ j + n − j (cid:19)(cid:19) ≤ i,j ≤ ℓ = | SYT( λ/µ ) || λ/µ | ! Y x ∈ λ/µ ( n + c ( x )) . Kirillov and Scrimshaw [13] recently conjectured that the number | SYT( λ/µ ) || λ/µ | ! Q x ∈ λ/µ ( n + c ( x ))on the right hand side of the identity in Proposition 1.2 is always an integer and proposed aproblem to find a combinatorial object for this number. Proposition 1.2 gives an affirmativeanswer to the problem. Theorem 1.1 and Proposition 1.2 together with (1) and (2) immediatelyimply the following corollary. Corollary 1.3.
For partitions λ and µ with µ ⊂ λ and ℓ ( λ ) ≤ n , we have | LHT n,m ( λ/µ ) | = m | λ/µ | | SYT( λ/µ ) || λ/µ | ! Y x ∈ λ/µ ( n + c ( x )) . In particular, the number of n -lecture hall tableaux of shape λ whose maximum entry is less than nm is | LHT n,m ( λ ) | = m | λ | | SSYT n ( λ ) | = m | λ | Y x ∈ λ n + c ( x ) h ( x ) . Using Naruse’s hook length formula for | SYT( λ/µ ) | in [19], we get another enumerative formula: Corollary 1.4.
For partitions λ and µ with µ ⊂ λ and ℓ ( λ ) ≤ n , the number of bounded lecturetableaux of shape λ/µ is | LHT n,m ( λ/µ ) | = m | λ/µ | Y x ∈ λ/µ ( n + c ( x )) X D Y x ∈ λ \ D h ( x ) , where the sum is over all excited diagrams D of λ/µ . See [18, 19] for details on excited diagrams. In this paper we also show that the number of bounded lecture hall tableaux occurs naturallyas the coefficient in the Schur expansion of s λ ( m + y , . . . , m + y n ). Recall that for a sequence ofvariables x = ( x , x , . . . ), the (skew) Schur function s λ/µ ( x ) is defined by s λ/µ ( x ) = X T ∈ SSYT( λ/µ ) x T , SYLVIE CORTEEL AND JANG SOO KIM where x T = Q ( i,j ) ∈ λ/µ x T ( i,j ) . Note that s λ ( x , x , . . . , x n − ) = X T ∈ SSYT n ( λ ) x T , and | SSYT n ( λ ) | = s λ (1 n ), where (1 n ) is the sequence (1 , , . . . ,
1) of n ones. Theorem 1.5.
For integers n, m ≥ , variables y , . . . , y n , and a partition λ with at most n parts,we have s λ ( m + y , . . . , m + y n ) = X µ ⊂ λ | LHT n,m ( λ/µ ) | s µ ( y , . . . , y n ) . If m = 1 in Theorem 1.5 we obtain the following formula due to Lascoux [15]:(4) s λ (1 + y , . . . , y n ) = X µ ⊂ λ det (cid:18)(cid:18) λ i + n − iµ j + n − j (cid:19)(cid:19) ≤ i,j ≤ ℓ ( λ ) s µ ( y , . . . , y n ) . Lascoux [15] used (4) to compute the Chern classes of the exterior square and symmetric squareof a vector bundle, see also [17, Chapter 1, §
3, Example 10]. We note Theorem 1.5 can also beobtained from (4) and Theorem 1.1.Our next theorem is a generalization of Theorem 1.5 to skew shapes. In order to state thetheorem we first need to introduce some definitions.For any tableau T of shape λ/µ , let x T = Y ( i,j ) ∈ λ/µ x T ( i,j ) . We define L nλ/µ ( x ) = X T ∈ LHT n ( λ/µ ) x ⌊ T ⌋ , and S nλ/µ ( x ) = X T ∈ SSCT n ( λ/µ ) x T . Note that S nλ ( x ) = s λ ( x , x , . . . , x n − ).The following theorem is the main theorem of this paper, which is a skew version of Theorem 1.5. Theorem 1.6.
Let λ and µ be partitions with µ ⊂ λ and ℓ ( λ ) ≤ n . For any sequences x =( x , x , . . . ) and y = ( y , y , . . . ) of variables, we have S nλ/µ ( | x | + y ) = X µ ⊂ ν ⊂ λ L nλ/ν ( x ) S nν/µ ( y ) , where | x | = x + x + · · · and | x | + y = ( | x | + y , | x | + y , . . . ) . In this paper we give two proofs of Theorem 1.6: one proof uses a Jacobi–Trudi type determinantidentity and the other proof is bijective. In particular the bijective proof of Theorem 1.6 uses avariation of jeu de taquin due to Krattenthaler [14].If µ = ∅ and x = (1 m ), in Theorem 1.6, we have S nλ ( m + y ) = X ν ⊂ λ L nλ/ν (1 m ) S nν ( y ) . Since S nν ( y ) = s ν ( y , y , . . . , y n − ) for any partition ν , we obtain Theorem 1.5.We can also deduce (3) from Theorem 1.6 as follows. If x = ( x , . . . , x m − ) and y = (0 , , . . . )in Theorem 1.6, we have(5) S nλ/µ ( | x | , | x | , . . . ) = L nλ/µ ( x ) . By definition we have L nλ/µ (1 m ) = | LHT n,m ( λ/µ ) | and(6) S nλ/µ ( | x | , | x | , . . . ) = | x | | λ/µ | S nλ/µ (1 , , . . . ) = | x | | λ/µ | | SSCT n ( λ/µ ) | . Then (3) follows from (5), (6) with x = (1 m ). NUMERATION OF BOUNDED LECTURE HALL TABLEAUX 5 ∞ ... ......012 · · ·· · · Figure 2.
The lecture hall graph G .The remainder of this paper is organized as follows. In Section 2 we give a simple proof ofTheorem 1.1 using a Jacobi–Trudi type determinant identity. We also prove Proposition 1.2. InSection 3 we prove Theorem 1.6 also using a Jacobi–Trudi identity. The main tool of Sections 2and 3 is to transform the tableaux into some system of non-intersecting paths on a planar graphand use the Lindstr¨om–Gessel–Viennot lemma [12]. In Section 4 we give a bijective proof ofTheorem 1.6. In Section 5 we find a connection of our bijection with the bijections due to Novelli,Pak, and Stoyanovskii [20] and Krattenthaler [14]. Finally, in Section 6 we provide some openproblems. Acknowledgements.
The authors want to thank the University of California, Berkeley, wherethis work was started and BIRS (Banff Canada), where this work was completed. We want tothank personally the Director of BIRS that accepted to extend our stay at BIRS after the workshop“Asymptotic Algebraic Combinatorics” in March 2019. Both authors would like to thank CurtisGreene and Carla Savage for their precious comments and advice during the elaboration of thispaper, Brendon Rhoades, Travis Scrimshaw and U-Keun Song for helpful discussions, and theanonymous referees for their careful reading and helpful comments. J.S.K. was supported by NRFgrants
Jacobi–Trudi identity
In this section we interpret an n -lecture hall tableau as non-intersecting lattice paths and givea Jacobi–Trudi type identity for the generating function L nλ/µ ( x ) for n -lecture hall tableaux of agiven shape. We then prove Theorem 1.1 and Proposition 1.2.The paths we consider are on an infinite directed graph embedded in the plane R defined asfollows. Definition 2.1.
The lecture hall graph G = ( V, E ) is a directed graph on the vertex set V = (cid:26)(cid:18) i, ji + 1 (cid:19) : i, j ∈ N (cid:27) , whose edge set E consists of • (nearly) horizontal edges from ( i, k + ri +1 ) to ( i + 1 , k + ri +2 ) for i, k ∈ N and 0 ≤ r ≤ i ,and • vertical edges from ( i, k + r +1 i +1 ) to ( i, k + ri +1 ) for i, k ∈ N and 0 ≤ r ≤ i .See Figure 2 for an example of the lecture hall graph G . We note that in [9] a slightly differentgraph is used to describe lecture hall tableaux, however, both graphs can equally be used for thispurpose.We now consider (directed) paths in the lecture hall graph. A path in G is a (possibly infinite)sequence P of vertices of G such that ( u, v ) is a directed edge of G for every two consecutiveelements u and v in P . If P is a finite path ( u ℓ , u ℓ − , . . . , u ), we say that P is a path from u ℓ to SYLVIE CORTEEL AND JANG SOO KIM ∞ ... ......01234 · · ·· · · x x x x x x x x x x x x x x x Figure 3.
Non-intersecting paths in G . For each horizontal edge, its weight isshown above it. u . If P is an infinite path ( . . . , u , u , u ) for u i = ( a i , b i ), i ≥
1, such that lim i →∞ a i = a , wesay that P is a path from ( a, ∞ ) to ( a , b ).From now on every path considered in this section will be either a finite path or an infinitepath in G satisfying the above limit condition.We define the weight wt( P ) of a path P to be the product of its edge weights, where the weightof the horizontal edge from ( i, k + ri +1 ) to ( i + 1 , k + ri +2 ) is defined to be x k and the weight of everyvertical edge is defined to be 1. A sequence ( P , . . . , P k ) of paths is said to be non-intersecting if they do not share any vertex. The weight of the system ( P , . . . , P k ) of paths is defined tobe the product Q ki =1 wt( P i ) of the weights of the paths. The following lemma gives a way ofunderstanding lecture hall tableaux as non-intersecting paths. Lemma 2.2.
Let λ and µ be partitions satisfying µ ⊂ λ and ℓ = ℓ ( λ ) ≤ n . Then there exists abijection between LHT n ( λ/µ ) and the set of non-intersecting paths ( P , . . . , P ℓ ) where P i is a pathfrom ( µ i + n − i, ∞ ) to ( λ i + n − i, . This bijection is such that if L ∈ LHT n ( λ/µ ) correspondsto ( P , . . . , P ℓ ) then x ⌊ L ⌋ = ℓ Y i =1 wt( P i ) . Proof.
As in [9] the bijection between lecture hall tableaux L and non-intersecting paths ( P , . . . , P ℓ )is constructed by counting the number of regions under each horizontal edge of each path. Namely, L ( i, j ) is given by the number of regions under the ( j − µ i ) th horizontal edge of P i . Then theweight of the edge is x ⌊ L ( i,j ) / ( n − i + j ) ⌋ , so the bijection satisfies the desired property. (cid:3) Figure 3 shows the non-intersecting paths in G corresponding to the lecture hall tableau L ∈ LHT n ( λ/µ ) in Figure 1 for n = 5, λ = (6 , , ,
3) and µ = (3 , x x x x , which is equal to x ⌊ L ⌋ . The entries of ⌊ L ⌋ can be seen on the right of Figure 1.Recall that x = ( x , x , . . . ) and that | x | = x + x + · · · . The following proposition is aJacobi–Trudi type identity for L nλ/µ ( x ). Proposition 2.3.
Let λ and µ be partitions satisfying µ ⊂ λ and ℓ = ℓ ( λ ) ≤ n . Then we have L nλ/µ ( x ) = det (cid:16) L µ j + n − j +1( λ i − µ j − i + j ) ( x ) (cid:17) ≤ i,j ≤ ℓ . NUMERATION OF BOUNDED LECTURE HALL TABLEAUX 7
Proof.
This is a direct consequence of the Lindstr¨om–Gessel–Viennot lemma [12], which states thatthe weight generating function for non-intersecting paths from vertices u , u , . . . , u ℓ to vertices v , v , . . . , v ℓ of the planar graph G is det( P ( u j , v i )) ≤ i,j ≤ ℓ , where P ( u j , v i ) is the weight generating function of the paths from u j to v i . Here we choose u j = ( µ j + n − j, ∞ ), v i = ( λ i + n − i,
0) and therefore P ( u j , v i ) = L µ j + n − j +1( λ i − µ j − i + j ) ( x ). Then theproposition follows from Lemma 2.2. (cid:3) Let us now compute the entries of the matrix of the previous proposition:
Proposition 2.4.
For n, k ≥ we have L n ( k ) ( x ) = | x | k (cid:18) n + k − k (cid:19) . Proof.
Let us first recall that(7) L n ( k ) ( x ) = X L x ⌊ L ⌋ , where the sum is over the n -lecture hall tableaux L ∈ LHT n ( λ ) of shape λ = ( k ), i.e., L (1 , n ≥ L (1 , n + 1 ≥ · · · ≥ L (1 , k ) n + k − ≥ . Consider the case x = ( x , , , . . . ). Then the n -lecture hall tableaux L contributing nonzeroterms in (7) are those satisfying1 > L (1 , n ≥ L (1 , n + 1 ≥ · · · ≥ L (1 , k ) n + k − ≥ . It is easy to check that for a, b, k ∈ N , the condition 1 > ak ≥ bk +1 is equivalent to k > a ≥ b .Thus, the above condition is equivalent to n > L (1 , ≥ · · · ≥ L (1 , k ) ≥ L n ( k ) ( x ) := L n ( k ) ( x , , , . . . ) = x k (cid:18) n + k − k (cid:19) . Now consider the general case x = ( x , x , . . . ). Fix an n -lecture hall tableau L ∈ LHT n (( k ))and let j be the index such that L (1 ,j ) n + j − ≥ L (1 ,j +1) n + j <
1. Here we suppose that L , = ∞ and L ,k +1 = 0 so that the index 0 ≤ j ≤ k is always defined. We can decompose L into twolecture hall tableaux L ′ ∈ LHT n (( j )) and L ′′ ∈ LHT n + j (( k − j )) so that L ′ (1 , i ) = L (1 , i ) and L ′′ (1 , i ) = L (1 , j + i ). Then L ′ and L ′′ satisfy(8) L ′ (1 , n ≥ · · · ≥ L ′ (1 , j ) n + j − ≥ , (9) 1 > L ′′ (1 , n + j ≥ · · · ≥ L ′′ (1 , k − j ) n + k − ≥ . Conversely, for any pair of L ′ and L ′′ satisfying (8) and (9), we obtain an n -lecture hall tableau L ∈ LHT n (( k )). Moreover, the tableaux L ′ ∈ LHT n (( j )) satisfying the condition (8) are thosecontributing nonzero terms in L n ( j ) (0 , x , x , . . . ) and the tableaux L ′′ ∈ LHT n + j (( k − j )) satisfyingthe condition (9) are those contributing nonzero terms in L n + j ( k − j ) ( x ). Therefore L n ( k ) ( x , x , . . . ) = k X j =0 L n + j ( k − j ) ( x ) L n ( j ) (0 , x , x , . . . ) . Now we notice that sequences ( L ′ (1 , , . . . , L ′ (1 , j )) such that L ′ (1 , n ≥ · · · ≥ L ′ (1 , j ) n + j − ≥ , SYLVIE CORTEEL AND JANG SOO KIM are in bijection with sequences ( U ′ (1 , , . . . , U ′ (1 , j )) such that U ′ (1 , n ≥ · · · ≥ U ′ (1 , j ) n + j − ≥ , by setting U ′ (1 , i ) = L ′ (1 , i ) − n + i − ≤ i ≤ j . This implies that x ⌊ L ′ ⌋ = j Y i =1 x ⌊ L ′ ,i / ( n − i +1) ⌋ = j Y i =1 x ⌊ U ′ ,i / ( n − i +1) ⌋ +1 . We get that L n ( j ) (0 , x , x , . . . ) = L n ( j ) ( x , x , . . . ) . Therefore L n ( k ) ( x , x , . . . ) = k X j =0 L n + j ( k − j ) ( x ) L n ( j ) ( x , x , . . . ) . This gives L n ( k ) ( x , x , . . . ) = (cid:0) n + k − k (cid:1) | x | k using induction. (cid:3) Combining the two previous propositions, we obtain the main theorem in this section.
Theorem 2.5.
Let λ and µ be partitions satisfying µ ⊂ λ and ℓ = ℓ ( λ ) ≤ n . Then we have L nλ/µ ( x ) = | x | | λ/µ | det (cid:18)(cid:18) λ i + n − iµ j + n − j (cid:19)(cid:19) ≤ i,j ≤ ℓ . Proof.
By Propositions 2.3 and 2.4, we have L nλ/µ ( x ) = det (cid:18) | x | λ i − i − µ j + j (cid:18) λ i + n − iµ j + n − j (cid:19)(cid:19) ≤ i,j ≤ ℓ . By factoring out the factor | x | λ i − i for each row i and the factor | x | j − µ j for each column j , weobtain the theorem. (cid:3) Setting x = (1 m ) in Theorem 2.5, we obtain Theorem 1.1. Since | SSCT n ( λ/µ ) | = L nλ/µ (1 , , , . . . ),Theorem 2.5 implies that(10) | SSCT n ( λ/µ ) | = det (cid:18)(cid:18) λ i + n − iµ j + n − j (cid:19)(cid:19) ≤ i,j ≤ ℓ . Therefore Theorem 2.5 is equivalent to(11) L nλ/µ ( x ) = | x | | λ/µ | | SSCT n ( λ/µ ) | . Since SSCT n ( λ ) = SSYT n ( λ ), by setting µ = ∅ in (11) we obtain the following corollary. Corollary 2.6.
For a partition λ with at most n parts, we have L nλ ( x ) = | x | | λ | s λ (1 n ) . We finish this section by giving a proof of Proposition 1.2.
Proof of Proposition 1.2 .
The first equality is shown in (10). It remains to show that(12) det (cid:18)(cid:18) λ i + n − iµ j + n − j (cid:19)(cid:19) ≤ i,j ≤ ℓ = | SYT( λ/µ ) || λ/µ | ! Y x ∈ λ/µ ( n + c ( x )) . We need the following determinant formula for | SYT( λ/µ ) | due to Aitken [2], see also [26, Corol-lary 7.16.3]: | SYT( λ/µ ) | = | λ/µ | ! det (cid:18) λ i − µ j − i + j )! (cid:19) ≤ i,j ≤ ℓ . Then (12) follows immediately from Aitken’s formula with the identities:det (cid:18)(cid:18) λ i + n − iµ j + n − j (cid:19)(cid:19) ≤ i,j ≤ ℓ = ℓ ( λ ) Y i =1 ( λ i + n − i )!( µ i + n − i )! det (cid:18) λ i − µ j − i + j )! (cid:19) ≤ i,j ≤ ℓ , NUMERATION OF BOUNDED LECTURE HALL TABLEAUX 9 ωω + 1 · · · Figure 4.
The content graph G ′ .0 1 2 3 4 5 6 7 8 9 10 ωω + 1 · · · y y y y y y y y y y y y Figure 5.
Non-intersecting paths in G ′ . The weight of each horizontal edge isshown above the edge.and ℓ ( λ ) Y i =1 ( λ i + n − i )!( µ i + n − i )! = Y x ∈ λ/µ ( n + c ( x )) , which can be easily verified. (cid:3) Proof of Theorem 1.6 using Jacobi–Trudi identity
In this section, we prove Theorem 1.6 using a Jacobi–Trudi identity for the generating function(13) S nλ/µ ( y ) = X T ∈ SSCT n ( λ/µ ) y T . To this end we introduce another infinite directed graph. We use the notation ω for the smallestinfinite ordinal number, i.e, 1 < < · · · < ω . Definition 3.1.
The content graph G ′ is the directed graph G ′ = ( V ′ , E ′ ) on the vertex set V ′ = (cid:26)(cid:18) i, ω + ri + 1 (cid:19) : i ∈ N , r ∈ { , , . . . , i + 1 } (cid:27) , whose edge set E ′ consists of • (nearly) horizontal edges from ( i, ω + ri +1 ) to ( i + 1 , ω + ri +2 ) for i ∈ N and 0 ≤ r ≤ i , and • vertical edges from ( i, ω + r +1 i +1 ) to ( i, ω + ri +1 ) for i ∈ N and 0 ≤ r ≤ i .Figure 4 shows the content graph G ′ . Now to any path P ′ in G ′ , we associate a monomialwt( P ′ ) equal to the product of the weights of the edges of P ′ , where the weight of the horizontaledge from ( i, ω + ri +1 ) to ( i + 1 , ω + ri +2 ) is defined to be y r and the weight of every vertical edgeis 1.The following lemma gives a way to understand a semistandard n -content tableau as non-intersecting paths in G ′ . Lemma 3.2.
There is a bijection between
SSCT n ( λ/µ ) and the set of non-intersecting paths ( P , . . . , P ℓ ) in G ′ , where each P i starts at u i = ( µ i + n − i, ω + 1) and ends at v i = ( λ i + n − i, ω ) .The correspondence between T ∈ SSCT n ( λ/µ ) and ( P , . . . , P ℓ ) is as follows. The number ofregions under the ( j − µ i ) th horizontal step of P i is the entry T ( i, j ) . In this case we have y T = Q ni =1 wt( P i ) .Proof. This can be proved similarly to the proof of Lemma 2.2. (cid:3)
Figure 6.
A semistandard n -content tableau of shape λ/µ with n = 4, λ =(6 , ,
3) and µ = (1).0 1 2 3 4 5 6 7 8 9 ωω + 1 ... ......012 · · ·· · ·· · · Figure 7.
The extended lecture hall graph G ∗ = G ′ ⊎ G .For example, the non-intersecting paths on Figure 5 correspond to the tableau on Figure 6.Note that both have weight y y y y y .The following is a Jacobi–Trudi identity for S nλ/µ ( x ). Proposition 3.3.
Let λ and µ be partitions satisfying µ ⊂ λ and ℓ = ℓ ( λ ) ≤ n . Then we have S nλ/µ ( y ) = det (cid:16) S µ j + n − j +1( λ i − µ j − i + j ) ( y ) (cid:17) ≤ i,j ≤ ℓ . Proof.
The proof is similar to that of Proposition 2.3, hence we omit it. (cid:3)
Since SSCT n ( λ ) = SSYT n ( λ ), the definition (13) of S nλ/µ ( x ) implies that for k ≥ n ≥ S n ( k ) ( y ) = s ( k ) ( y , . . . , y n − ) = h k ( y , . . . , y n − ) , where h k ( y , . . . , y n − ) is the complete homogeneous polynomial defined by(15) h k ( y , . . . , y n − ) = X ≤ i ≤···≤ i k ≤ n − y i · · · y i k . Note that y , . . . , y n − are the only variables that actually appear in S n ( k ) ( y ) even though y =( y , y , . . . ) is an infinite sequence of variables. Using (14), Proposition 3.3 can be restated as(16) S nλ/µ ( y ) = det( h λ i − µ j − i + j ( y , . . . , y µ j + n − j )) ≤ i,j ≤ ℓ . In order to prove Theorem 1.6 we introduce yet another graph.
Definition 3.4.
The extended lecture hall graph G ∗ is the disjoint union G ′ ⊎ G of the contentgraph G ′ and the lecture hall graph G .We will draw the extended lecture hall graph G ∗ = G ′ ⊎ G with G ′ on top of G as shown inFigure 7 so that each vertex ( i, ω ) of G ′ can be considered as the “limit” of the sequence of vertices( i, , ( i, , ( i, , . . . in G .We define an ω -path to be a pair Q = ( P ′ , P ) satisfying the following conditions: • P ′ and P are paths in G ′ and G , respectively. NUMERATION OF BOUNDED LECTURE HALL TABLEAUX 11
Figure 8.
A pair (
L, S ) of tableaux L ∈ LHT n ( λ/ν ) and S ∈ SSCT n ( ν/µ ) for n = 5, λ = (6 , , , µ = (3 , ν = (4 , L and S areseparated by the thick border. • P ′ is a path from ( a, ω +1) to ( b, ω ) and P is a path from ( b, ∞ ) to ( c,
0) for some a ≤ b ≤ c .In this case we say that Q is an ω -path from ( a, ω +1) to ( c, P ) for a pathin G as in Section 2 and define the weight of an ω -path Q = ( P ′ , P ) by wt( Q ) = wt( P ′ ) wt( P ).We are now ready to prove Theorem 1.6, which states that(17) S nλ/µ ( | x | + y ) = X µ ⊂ ν ⊂ λ L nλ/ν ( x ) S nν/µ ( y ) . Proof of Theorem 1.6.
Let
LHS and
RHS be the left hand side and the right hand side of (17),respectively. By (16), we have(18)
LHS = det( h λ i − µ j − i + j ( y + | x | , . . . , y µ j + n − j + | x | )) ≤ i,j ≤ ℓ . Our strategy is to express
RHS also as a determinant that agrees with the determinant in (18)entry-wise.First, observe that
RHS = X µ ⊂ ν ⊂ λ L nλ/ν ( x ) S nν/µ ( y ) = X ( L,S ) x ⌊ L ⌋ y S , where the sum is over all pairs ( L, S ) of tableaux L ∈ LHT n ( λ/ν ) and S ∈ SSCT n ( ν/µ ) for somepartition ν with µ ⊂ ν ⊂ λ . Combining the bijections in Lemmas 2.2 and 3.2, we obtain a bijectionbetween the set of such pairs ( L, S ) and the set of non-intersecting ω -paths ( Q , . . . , Q ℓ ) such that Q i is an ω -path from u i = ( µ i + n − i, ω + 1) to v i = ( λ i + n − i, x ⌊ L ⌋ y S = wt( Q ) · · · wt( Q ℓ ), which implies that RHS = X ( L,S ) x ⌊ L ⌋ y S = X ( Q ,...,Q ℓ ) wt( Q ) · · · wt( Q ℓ ) . For example, the pair (
L, S ) of tableaux given on Figure 8 corresponds to the non-intersecting ω -paths on Figure 9.By the Lindstr¨om–Gessel–Viennot lemma, we have(19) RHS = X ( Q ,...,Q ℓ ) wt( Q ) · · · wt( Q ℓ ) = det( ˜ P ( u j , v i )) ≤ i,j ≤ ℓ , where ˜ P ( u j , v i ) is the sum of wt( Q ) for all ω -paths Q from u j = ( µ j + n − j, ω + 1) to v i =( λ i + n − i, P ( u j , v i ) = λ i − µ j − i + j X k =0 S µ j + n − j +1( k ) ( y ) L µ j + n − j + k +1( λ i − µ j − i + j − k ) ( x ) . By (14) and Proposition 2.4, we have S µ j + n − j +1( k ) ( y ) = h k ( y , . . . , y µ j + n − j ) ,L µ j + n − j + k +1( λ i − µ j − i + j − k ) ( x ) = | x | λ i − µ j − i + j − k (cid:18) λ i + n − iλ i − µ j − i + j − k (cid:19) . ωω + 1 ... ......01234 · · ·· · ·· · · x x x x x x x y y x x x y x x Figure 9.
Non-intersecting ω -paths in G ∗ . For each horizontal edge, its weightis shown above it.Therefore, by (18), (19), (20) and the above two equations, it suffices to prove the followingidentity:(21) h λ i − µ j − i + j ( y + | x | , . . . , y µ j + n − j + | x | )= λ i − µ j − i + j X k =0 h k ( y , . . . , y µ j + n − j ) | x | λ i − µ j − i + j − k (cid:18) λ i + n − iλ i − µ j − i + j − k (cid:19) . Using the definition (15) of the complete homogeneous polynomial, it is not hard to see that h t ( y + | x | , . . . , y a + | x | ) = X ≤ i ≤···≤ i t ≤ a ( y i + | x | )( y i + | x | ) . . . ( y i t + | x | )= t X k =0 h k ( y , . . . , y a ) h t − k ( | x | a + k +1 ) , where h t − k ( | x | a + k +1 ) means h t − k ( a + k +1 z }| { | x | , . . . , | x | ). Since h t − k ( | x | a + k +1 ) = | x | t − k (cid:0) a + tt − k (cid:1) , we obtain(21) from the above identity by setting a = µ j + n − j and t = λ i − µ j − i + j . The proof is nowcomplete. (cid:3) A bijective proof of the main theorem
In this section we give a bijective proof of Theorem 1.6. We first introduce some definitionsand restate the theorem accordingly.A marked tableau of shape λ/µ is a map T : λ/µ → N × ( N ∪ {∞} ). If T ( i, j ) = ( a, r ) we saythat a is a value and r is a mark . Instead of T ( i, j ) = ( a, r ), we will simply write T ( i, j ) = a r .A marked n -content tableau is a marked tableau T with a condition that if T ( i, j ) = a r , then0 ≤ a < n − i + j . For a marked tableau T of shape λ/µ and a skew shape α ⊂ λ/µ , we denoteby T | α the restriction of T to the cells in α .Let T be a marked tableau of shape λ/µ . For each ( i, j ) ∈ λ/µ , letwt ∗ ( T ( i, j )) = ( x b , if T ( i, j ) = a b and b = ∞ , y a , if T ( i, j ) = a ∞ . NUMERATION OF BOUNDED LECTURE HALL TABLEAUX 13
25 25 2116 18 21 10 48 9 2 04 4 0
258 259 2110165 186 217 108 4983 94 25 0642 43 04 Figure 10.
On the left is a lecture hall tableau L ∈ LHT n ( λ/µ ) for n = 5, λ = (6 , , ,
3) and µ = (3 , L ( i, j ) / ( n + c ( i, j )) for each entry ( i, j ) ∈ λ/µ . The diagram on the right is thecorresponding marked tableau T , given by T ( i, j ) = a r , where a and r are theunique integers satisfying L ( i, j ) = r · ( n − i + j ) + a and 0 ≤ a < n − i + j .1 ∞ ∞ ∞ Figure 11.
An extended n -lecture hall tableau L in LHT ∗ n ( λ/µ ), where n = 5, λ = (6 , , ,
2) and µ = (3 , L is wt ∗ ( L ) = x x x x y y . Thetail of L is indicated by the blue circle.The weight wt ∗ ( T ) of T is defined bywt ∗ ( T ) = Y ( i,j ) ∈ λ/µ wt ∗ ( T ( i, j )) . Consider an n -lecture hall tableau L ∈ LHT n ( λ/µ ). We construct a marked tableau T asfollows. For each cell ( i, j ) ∈ λ/µ , let T ( i, j ) = a r , where r = ⌊ L ( i, j ) / ( n + j − i ) ⌋ and a = L ( i, j ) − r · ( n + j − i ). See Figure 10. Clearly, L can be recovered from T . From now on we willidentify the lecture hall tableau L with the marked tableau T . Note that under this identificationevery mark of a lecture hall tableau is a nonnegative integer.An extended n -lecture hall tableau of shape λ/µ is a marked tableau T : λ/µ → N × ( N ∪ {∞} )satisfying the following conditions:(1) If ( i, j ) ∈ λ/µ and T ( i, j ) = a r , then 0 ≤ a < n + j − i .(2) If ( i, j ) , ( i, j + 1) ∈ λ/µ and T ( i, j ) = a r , T ( i, j + 1) = b s , then we have either r > s , or r = s and a ≥ b .(3) If ( i, j ) , ( i + 1 , j ) ∈ λ/µ and T ( i, j ) = a r , T ( i + 1 , j ) = b s , then we have either r > s , or r = s and a > b .We denote by LHT ∗ n ( λ/µ ) the set of extended n -lecture hall tableaux of shape λ/µ . See Figure 11for an example.A marked semistandard n -content tableau is a marked tableau T such that the tableau obtainedfrom T by deleting its marks is a semistandard n -content tableau. See Figure 12 for an example.We denote by SSCT ∗ n ( λ/µ ) the set of marked semistandard n -content tableaux of shape λ/µ . Fromthe definition one can easily see that(22) S nλ/µ ( | x | + y ) = X T ∈ SSCT ∗ n ( λ/µ ) wt ∗ ( T ) . Observe that if T is an extended n -lecture hall tableau, then the marks are weakly decreasingin each row and each column, and for all i ∈ N ∪ {∞} the values with mark i form a semistandard n -content tableau. Therefore, if we restrict T to the cells whose marks are not ∞ , we obtain an ∞ ∞ ∞ Figure 12.
A marked semistandard n -content tableau S in SSCT ∗ n ( λ/µ ), where n = 5, λ = (6 , , ,
2) and µ = (3 , S is wt ∗ ( S ) = x x x x y y .The head of S is indicated by the red cell. n -lecture hall tableau, which implies that(23) X T ∈ LHT ∗ n ( λ/µ ) wt ∗ ( T ) = X ν X T ∈ LHT n ( λ/ν ) x ⌊ T ⌋ X T ∈ SSCT n ( ν/µ ) y T = X ν L nλ/ν ( x ) S nν/µ ( y ) . By (22) and (23), Theorem 1.6 can be restated as follows.
Theorem 4.1.
We have X T ∈ LHT ∗ n ( λ/µ ) wt ∗ ( T ) = X T ∈ SSCT ∗ n ( λ/µ ) wt ∗ ( T ) . We will construct a weight-preserving bijection between LHT ∗ n ( λ/ν ) and SSCT ∗ n ( λ/µ ). Thebasic idea is to sort the values of L ∈ LHT ∗ n ( λ/µ ) using a variation of “jeu de taquin” accordingto a certain order of the cells in λ/µ depending on L itself. Our algorithms are inspired by thosedue to Krattenthaler [14]. Algorithm 4.2 (Value-jeu de taquin) . The value-jdt algorithm is described as follows.
Notation: φ vjdt ( P, u ) = (
Q, v ). Input:
A pair (
P, u ) of a marked tableau P of shape λ/µ and a cell u ∈ λ/µ . Output:
A pair (
Q, v ) of a marked tableau Q of shape λ/µ and a cell v ∈ λ/µ . Step 1:
Set Q = P and v = u . We call v the active cell . Step 2:
Let ( i, j ) be the coordinate of the active cell v . Let a r = Q ( i, j ), b s = Q ( i, j +1),and c t = Q ( i + 1 , j ). If ( i, j + 1) λ/µ (resp. ( i + 1 , j ) λ/µ ), then set b s = ( − (resp. c t = ( − ). If a ≥ b and a > c , then stop the process and return ( Q, v ) as theoutput. Otherwise, there are two cases. • If b − > c , then set Q ( i, j ) = ( b − s and Q ( i, j + 1) = a r as shown below,where the active cell v is the cell containing a r . Set v = ( i, j + 1) and repeatStep 2. a r b s c t → ( b − s a r c t • If c + 1 ≥ b , then set Q ( i, j ) = ( c + 1) t and Q ( i + 1 , j ) = a r as shown below,the active cell v is the cell containing a r . Set v = ( i + 1 , j ) and repeat Step 2. a r b s c t → ( c + 1) t b s a r See Figure 13 for an example of the value-jdt algorithm.
Algorithm 4.3 (Mark-jeu de taquin) . The mark-jdt algorithm is described as follows.
Notation: φ mjdt ( Q, v ) = (
P, u ). Input:
A pair (
Q, v ) of a marked tableau Q of shape λ/µ and a cell v ∈ λ/µ . Output:
A pair (
P, u ) of a marked tableau P of shape λ/µ and a cell u ∈ λ/µ . NUMERATION OF BOUNDED LECTURE HALL TABLEAUX 15 ∞ P = 0 ∞ Q = Figure 13. If u = (2 ,
3) and v = (5 , φ vjdt ( P, u ) = (
Q, v ) and φ mjdt ( Q, v ) = (
P, u ). In each diagram the positions that the active cell visitsare enclosed by the thick polygon. NWNW SENW SE
Figure 14.
The northwest corners are the cells with an “NW” and the southeastcorners are the cells with an “SE”.
Step 1:
Set P = Q and u = v . We call u the active cell . Step 2:
Let ( i, j ) be the coordinate of the active cell u . Let a r = P ( i, j ), b s = P ( i, j − c t = P ( i − , j ). If ( i, j − λ/µ (resp. ( i − , j ) λ/µ ), then set b s = ∞ ∞ (resp. c t = ∞ ∞ ). If r ≤ s and r ≤ t , then stop the process and return ( P, u ) as theoutput. Otherwise, there are two cases. • If t < r ≤ s , or s, t < r and b ≥ c −
1, then set P ( i, j ) = ( c − t and P ( i − , j ) = a r as shown below, where the active cell u is the cell containing a r . Set u = ( i − , j ) and repeat Step 2. c t a r b s → a r ( c − t b s • If s < r ≤ t , or s, t < r and c > b + 1, then set P ( i, j ) = ( b + 1) s and P ( i, j −
1) = a r as shown below, where the active cell u is the cell containing a r . Set u = ( i, j −
1) and repeat Step 2. c t a r b s → c t ( b + 1) s a r See Figure 13 for an example of the value-jdt algorithm.Let λ be a partition. An outer corner of λ is a cell u λ such that λ ∪ { u } is a partition. An inner corner of λ is a cell u ∈ λ such that λ \ { u } is a partition. For a skew shape λ/µ , a northwestcorner of λ/µ is a cell in λ/µ that is an outer corner of µ and a southeast corner of λ/µ is a cellin λ/µ that is an inner corner of λ . See Figure 14 for an example. Definition 4.4.
Let α be a skew shape and L ∈ LHT ∗ n ( α ). Suppose that r is the smallest markand a is the smallest value with mark r in L . Then the tail of L , denoted tail( L ), is defined to bethe rightmost cell ( i, j ) ∈ α with L ( i, j ) = a r . See Figure 11 for an example.Note that for distinct cells ( i, j ) , ( i ′ , j ′ ) ∈ λ/µ , if L ( i, j ) = L ( i ′ , j ′ ) = a r , then the fact that L isan element in LHT ∗ n ( λ/µ ) ensures that j = j ′ . Thus the tail of L ∈ LHT ∗ n ( λ/µ ) is well-defined. Itis clear from the definition that the tail of L is a southeast corner of λ/µ . T i = α i β i u i − v i Figure 15.
A typical diagram with T i , α i , β i , u i − , and v i . The border between α i and β i is shown with a thick path. The blue circle represents u i − and the redcircle represents v i . The dashed path represents the movement of the active cellin the process of φ vjdt ( T i − , u i − ) = ( T i , v i ). Definition 4.5.
Let β be a skew shape and S ∈ SSCT ∗ n ( β ). Suppose that r is the largest markand a is the largest value with mark r in S . Then the head of S , denoted head( S ), is defined tobe the leftmost cell ( i, j ) ∈ β with S ( i, j ) = a r . See Figure 12 for an example.By a similar argument as before, one can check that if S ∈ SSCT ∗ n ( β ), then head( S ) is well-defined. Note, however, that head( S ) is not necessarily a (northwest or southeast) corner of β .We are now ready to define a map sending an extended n -lecture hall tableau L ∈ LHT ∗ n ( λ/µ )to a marked semistandard n -content tableau S ∈ SSCT ∗ n ( λ/µ ). Recall from the definition that in L the marks are weakly decreasing along each row and column but the values are not sorted. In S , on the contrary, the values are weakly decreasing along each row and strictly decreasing alongeach column but the marks are not sorted. Our approach is, therefore, to sort the values of L inorder to obtain S , and to sort the marks of S in order to obtain L . The two sorting algorithmsare described below. See Figure 15 for an illustration of a typical situation and Figure 16 for aconcrete example of these algorithms. Algorithm 4.6 (Value-sorting) . The value-sorting algorithm is described as follows.
Notation: φ vsort ( L ) = S . Input:
An extended n -lecture hall tableau L of shape λ/µ . Output:
A marked semistandard n -content tableau S of shape λ/µ . Step 1:
Set T = L , α = λ/µ , β = ∅ , and u = tail( T ). Step 2:
For i = 1 , , . . . , | λ/µ | , define α i , β i , T i , u i , and v i recursively by( T i , v i ) = φ vjdt ( T i − , u i − ) ,α i = α i − \ { u i − } ,β i = β i − ∪ { u i − } ,u i = tail( T i | α i ) . Step 3:
Return S = T | λ/µ | as the output. Algorithm 4.7 (Mark-sorting) . The mark-sorting algorithm is described as follows.
Notation: φ msort ( S ) = L . Input:
A marked semistandard n -content tableau S of shape λ/µ . Output:
An extended n -lecture hall tableau L of shape λ/µ . Step 1:
Set T | λ/µ | = S , α | λ/µ | = ∅ , β | λ/µ | = λ/µ , and v | λ/µ | = head( T | λ/µ | ). NUMERATION OF BOUNDED LECTURE HALL TABLEAUX 17 L = 1 ∞ → ∞ → ∞ → ∞ ↓ ∞ ← ∞ ← ∞ ← ∞ S = Figure 16.
The value-sorting algorithm applied to L ∈ LHT ∗ n ( λ/µ ) returns S ∈ SSCT ∗ n ( λ/µ ), where n = 7, λ = (4 , ,
1) and µ = (1). The mark-sorting algorithmis the reverse process. Each diagram represents T i . The border between α i and β i is drawn by a thick path. The blue circle indicates u i = tail( T i | α i ) and the redcircle indicates v i = head( T i | β i ). Step 2:
For i = | λ/µ | − , | λ/µ | − , . . . ,
0, define α i , β i , T i , u i , and v i recursively by( T i , u i ) = φ mjdt ( T i +1 , v i +1 ) ,α i = α i +1 ∪ { u i +1 } ,β i = β i +1 \ { u i +1 } ,v i = head( T i | β i ) . Step 3:
Return L = T as the output.In order to show that the above algorithms are inverse to each other, we need the following twolemmas. Lemma 4.8.
Let L ∈ LHT ∗ n ( λ/µ ) . Suppose that α i , β i , T i , u i , and v i are given as in Algo-rithm 4.6. Then, for each i = 1 , , . . . , | λ/µ | , the following properties hold. (1) T i | α i ∈ LHT ∗ n ( α i ) and T i | β i ∈ SSCT ∗ n ( β i ) . In particular, T | λ/µ | ∈ SSCT ∗ n ( λ/µ ) . (2) head( T i | β i ) = v i . (3) φ mjdt ( T i , v i ) = ( T i − , u i − ) .Proof. (1): We prove this for i = 0 , , . . . , | λ/µ | by induction. Since T | α = L and T | β = ∅ ,it is true for i = 0. Let 1 ≤ i ≤ | λ/µ | and suppose that (1) is true for i −
1. Since T i − | α i − ∈ LHT ∗ n ( α i − ), we have that u i − = tail( T i − | α i − ) is a southeast corner of α i − . Hence, α i = α i − \{ u i − } and β i = β i − ∪ { u i − } are skew shapes. When we compute ( T i , v i ) = φ vjdt ( T i − , u i − ),the value-jdt algorithm does not modify the cells in α i , which implies that T i | α i = T i − | α i = L | α i ∈ LHT ∗ n ( α i ) and φ vjdt ( T i − | β i , u i − ) = ( T i | β i , v i ). It is not hard to check that in the processof φ vjdt ( T i − | β i , u i − ) to obtain T i | β i , the values of the cells in β i are weakly decreasing in eachrow and strictly decreasing in each column with only possible exceptions between the active celland the cell to the right of it and the cell below it. When the process stops these two possibleexceptions are resolved and we obtain T i | β i ∈ SSCT ∗ n ( β i ) as desired.(2): It is clear from the construction that if r is the largest mark and a is the largest valuewith mark r in T i | β i , then T i ( v i ) = a r . If v i is the only cell in β i with this property, then wehave head( T i | β i ) = v i . Otherwise, we must show that v i is the leftmost cell with this property.To this end suppose that T i − ( u i − ) = T i ( u i ) = a r , u i − = ( k, l ), u i = ( k ′ , l ′ ), and v i = ( p, q ), v i +1 = ( p ′ , q ′ ). Then it is sufficient to show that q ′ < q . Since T i − | α i − ∈ LHT ∗ n ( α i − ) and u i − = head( T i − | α i − ), we have k ′ ≥ k and j ′ < j .Let u i − = w , w , w , . . . , w d = v i be the sequence of positions of the active cell in the con-struction of φ vjdt ( T i − , u i − ) = ( T i , v i ). We claim that when we compute φ vjdt ( T i , v i ), the activecell never enters the position w t if w t +1 is south of w t , for 0 ≤ t < d . ∗ ∗ c y b x ∗ ( c + 1) y ∗ b x c y ∗∗ b x Figure 17.
The restrictions of T i − (on the left), T i (in the middle), and T i +1 (on the right) to the cells ( g, h − , ( g, h ) , ( g + 1 , h − , ( g + 1 , h ).Suppose that the claim is false. Then we can find the smallest integer m such that w m = ( g, h ), w m +1 = ( g + 1 , h ) and the active cell enters w m . Considering the relative positions of u i − and u i ,one can check that the active cell must enter w m from the east. Now we focus on the restrictionsof T i − , T i , and T i +1 to the cells ( g, h − , ( g, h ) , ( g + 1 , h − , ( g + 1 , h ) as in Figure 17. Let T i − ( g + 1 , h −
1) = b x and T i − ( g + 1 , h ) = c y . Since T i − | β i − ∈ SSCT ∗ n ( β i − ), we have b ≥ c .Considering the positions of the active cell in the process of φ vjdt ( T i − , u i − ) and φ vjdt ( T i , u i ), weobtain that T i ( g +1 , h −
1) = b x , T i ( g, h ) = ( c +1) y , T i +1 ( g +1 , h −
1) = b x , and T i +1 ( g, h −
1) = c y .Since T i +1 | β i +1 ∈ SSCT ∗ n ( β i +1 ), we have b < c , which is a contradiction to the above fact that b ≥ c . Therefore, the claim is true.By the above claim, if q ′ ≥ q , then the active cell in the process of φ vjdt ( T i , u i ) must movefrom ( z, q −
1) to ( z, q ) for some z ≥ p . Suppose that z = p . Let T i ( p + 1 , q −
1) = c t . Since T i ( p, q ) = a r , the fact that the active cell moved from ( p, q −
1) to ( p, q ) implies that a − > c .However, this means that when the active cell was in ( p, q − z > p . In this case, since T i +1 | β i +1 ∈ SSCT ∗ n ( β i +1 ) and v i +1 is strictly below and weakly to the right of v i , we have thatthe value of T i +1 ( v i +1 ) is less than the value of T i +1 ( v i ), which is a contradiction. Therefore, wemust have q ′ < q , which completes the proof of (2).(3): By the fact that T i − | β i − ∈ SSCT ∗ n ( β i − ) and T i − | β i − ∈ SSCT ∗ n ( β i ), it is clear that thereverse process of φ vjdt ( T i − , u i − ) is given by the mark-jdt algorithm. We only need to check thatthe process of φ mjdt ( T i , v i ) stops when the active cell reaches the cell u i − . Let r be the largestmark and a the largest value with mark in T i | β i . Since v i = head( T i | β i ), we have T i ( v i ) = a r and v i is the leftmost cell with this property. Therefore, the movement of the active cell in the processof φ mjdt ( T i , v i ) continues until the active cell reaches at a northwest corner of β i , which is u i . Ifthe active cell is at u i , then the fact that the mark of every cell in α i is at least r implies that theprocess of φ mjdt ( T i , v i ) stops. (cid:3) Lemma 4.9.
Let S ∈ SSCT ∗ n ( λ/µ ) . Suppose that α i , β i , T i , u i , and v i are given as in Algo-rithm 4.7. Then, for each i = 0 , , , . . . , | λ/µ | − , the following properties hold. (1) T i | α i ∈ LHT ∗ n ( α i ) and T i | β i ∈ SSCT ∗ n ( β i ) . In particular, T ∈ LHT ∗ n ( λ/µ ) . (2) tail( T i | α i ) = u i . (3) φ vjdt ( T i , u i ) = ( T i +1 , v i +1 ) .Proof. This lemma can be proved by arguments similar to those in the proof of Lemma 4.8. Weomit the proof. (cid:3)
We now give a bijective proof of Theorem 4.1.
Theorem 4.10.
The map φ vsort : LHT ∗ n ( λ/µ ) → SSCT ∗ n ( λ/µ ) is a weight-preserving bijection whose inverse is φ msort : SSCT ∗ n ( λ/µ ) → LHT ∗ n ( λ/µ ) . Proof.
Lemmas 4.8 and 4.9 imply that the two maps φ vsort and φ msort are inverses of each other.Suppose φ vsort ( L ) = S . In the process of the value-sorting algorithm, the marks and the valueswith mark ∞ are never changed. Therefore wt ∗ ( L ) = wt ∗ ( S ). (cid:3) NUMERATION OF BOUNDED LECTURE HALL TABLEAUX 19
Remark 4.11.
The bijection allows us to generate a random bounded lecture hall tableau of agiven partition shape using Krattenthaler’s random generation of a semistandard Young tableau.It will be interesting to extend this random generation to skew shapes. In [8] a different algorithmis established using a Markov chain on bounded lecture hall tableaux and coupling from the past.5.
A connection between SSCT and SYT
In this section we use the weight-preserving bijection φ vsort : LHT ∗ n ( λ/µ ) → SSCT ∗ n ( λ/µ ) andits inverse φ msort : SSCT ∗ n ( λ/µ ) → LHT ∗ n ( λ/µ ) to find a connection between | SSCT n ( λ/µ ) | and | SYT( λ/µ ) | .Recall the sets SYT( λ/µ ), SSYT n ( λ/µ ), LHT n ( λ/µ ), and SSCT n ( λ/µ ) defined in the intro-duction. We also need the following definitions.A tableau T of shape λ/µ is called standard if every integer 1 ≤ i ≤ | λ/µ | appears exactly oncein T . The set of standard tableaux of shape λ/µ is denoted by ST( λ/µ ). An n -content tableau ofshape λ/µ is a tableau T of shape λ/µ such that 0 ≤ T ( i, j ) < n − i + j for all ( i, j ) ∈ λ/µ . Theset of n -content tableaux of shape λ/µ is denoted by CT n ( λ/µ ). A hook tabloid of shape λ is amap H : λ → Z satisfying − leg( i, j ) ≤ H ( i, j ) ≤ arm( i, j ) for all ( i, j ) ∈ λ , where leg( i, j ) = λ ′ j − i and arm( i, j ) = λ i − j . We denote by HT( λ ) the set of hook tabloids of shape λ .Let us now consider the map φ vsort : LHT ∗ n ( λ/µ ) → SSCT ∗ n ( λ/µ ) restricted to the followingsets: X n ( λ/µ ) = { L ∈ LHT ∗ n ( λ/µ ) : wt ∗ ( L ) = x · · · x | λ/µ | } ,Y n ( λ/µ ) = { T ∈ SSCT ∗ n ( λ/µ ) : wt ∗ ( T ) = x · · · x | λ/µ | } . Since φ vsort is a weight-preserving bijection, we obtain the induced bijection φ vsort : X n ( λ/µ ) → Y n ( λ/µ ) . We can naturally identify L ∈ X n ( λ/µ ) with the pair ( A, R ) of tableaux of shape λ/µ : if L ( i, j ) = a r then A ( i, j ) = a and R ( i, j ) = r . Then by the condition on L , we have A ∈ CT n ( λ/µ )and R ∈ SYT( λ/µ ). This allows us to identify X n ( λ/µ ) with CT n ( λ/µ ) × SYT( λ/µ ). Similarly,we can identify Y n ( λ/µ ) with SSCT n ( λ/µ ) × ST( λ/µ ). Using this identification we can consider φ vsort as a bijection between these sets:(24) φ vsort : CT n ( λ/µ ) × SYT( λ/µ ) → SSCT n ( λ/µ ) × ST( λ/µ ) . Therefore we obtain the following corollary, which is a restatement of Proposition 1.2.
Corollary 5.1.
For any skew shape λ/µ , we have | SSCT n ( λ/µ ) | Q x ∈ λ/µ ( n + c ( x )) = | SYT( λ/µ ) || λ/µ | ! , which means that the probability that a random T ∈ CT n ( λ/µ ) is semistandard is equal to theprobability that a random T ∈ ST( λ/µ ) is a standard Young tableau. It is possible to understand the probabilistic description in Corollary 5.1 using the map (24).To this end we note that each element (
A, B ) ∈ CT n ( λ/µ ) × SYT( λ/µ ) is a fixed point of φ vsort ,i.e., φ vsort ( A, B ) = (
A, B ), if and only if A ∈ SSCT n ( λ/µ ). Similarly, each element ( A, B ) ∈ SSCT n ( λ/µ ) × ST( λ/µ ) is a fixed point of the inverse map φ msort = φ − if and only if B ∈ SYT( λ/µ ). The probability that a random A ∈ CT n ( λ/µ ) is an element in SSCT n ( λ/µ ) isclearly equal to the probability that a random pair ( A, B ) ∈ CT n ( λ/µ ) × SYT( λ/µ ) satisfies A ∈ SSCT n ( λ/µ ). In other words, this is the probability that a random pair ( A, B ) ∈ CT n ( λ/µ ) × SYT( λ/µ ) is a fixed point of φ vsort . By the same argument, we obtain that the probability that arandom B ∈ ST( λ/µ ) is an element of SYT( λ/µ ) is equal to the probability that a random pair(
A, B ) ∈ SSCT n ( λ/µ ) × ST( λ/µ ) is a fixed point of the map φ − . Since φ vsort and φ − areinverses of each other with the same set of fixed points, we obtain that the two probabilities thatwe consider are equal. CT n ( λ ) × SYT( λ ) SSYT n ( λ ) × ST( λ )SSYT n ( λ ) × HT( λ ) × SYT( λ ) φ vsort φ K φ NP S
Figure 18.
Three maps between three objects.We now consider the map (24) for the case µ = ∅ . Since SSCT n ( λ ) = SSYT n ( λ ), we have thefollowing bijection:(25) φ vsort : CT n ( λ ) × SYT( λ ) → SSYT n ( λ ) × ST( λ ) . Recall the two bijections due to Novelli–Pak–Stoyanovskii and Krattenthaler.
Theorem 5.2 (Novelli-Pak-Stoyanovskii) . For any partition λ , there is a bijection φ NP S : ST( λ ) → SYT( λ ) × HT( λ ) . Theorem 5.3 (Krattenthaler) . For any partition λ , there is a bijection φ K : CT n ( λ ) → SSYT n ( λ ) × HT( λ ) . Note that φ NP S naturally induces a bijection φ NP S : SSYT n ( λ ) × ST( λ ) → SSYT n ( λ ) × HT( λ ) × SYT( λ )by fixing the first component. Similarly φ K induces a bijection φ K : CT n ( λ ) × SYT( λ ) → SSYT n ( λ ) × HT( λ ) × SYT( λ ) . Then the three maps φ vsort , φ K , and φ NP S are bijections among three sets CT n ( λ ) × SYT( λ ),SSYT n ( λ ) × ST( λ ), and SSYT n ( λ ) × HT( λ ) × SYT( λ ), see Figure 18. These maps are not directlyrelated. It might be interesting to find any connection between these maps.6. Final remarks
Stanley [24] showed that semistandard Young tableaux and standard Young tableaux fit to-gether nicely in the framework of the P -partition theory, see also [25, Chapter 3] and [26, Chapter7]. Lecture hall tableaux are also a special case of lecture hall P -partitions introduced by Br¨and´enand Leander [7]. They found a connection between generating functions for the bounded lecturehall P -partitions and colored linear extensions of P . It will be interesting to compare our resultswith theirs. Problem 6.1.
Investigate bounded lecture hall tableaux using the results of Br¨and´en and Leander[7].Krattenthaler’s map [14] in fact gives a bijective proof of the following q -analog of (2), also dueto Stanley [23]: X T ∈ SSYT n ( λ ) q | T | = q P i ≥ ( i − λ i Y ( i,j ) ∈ λ [ n + c ( i, j )] q [ h ( i, j )] q , where [ k ] q = 1 + q + q + · · · + q k − . If we only look at the values and ignore the marks, thenour jeu de taquin slides in Algorithms 4.2 and 4.3 are essentially the same as those in [14]. Recallthat during these algorithms values are changing. Krattenthaler carefully designed his bijectionso that these value changes are consistent with the value changes in hook tabloids. Our bijection,on the contrary, does not have hook tabloids, which makes it difficult to follow the change ofvalues. If we can keep track of all the value changes, then it may be possible to find a refinementof Theorem 4.1. Problem 6.2.
Find a q -analogue of Theorem 4.1. NUMERATION OF BOUNDED LECTURE HALL TABLEAUX 21
For a partition λ = ( λ , . . . , λ ℓ ) with distinct parts, the shifted Young diagram of λ is an arrayof squares in which the i th row has λ i squares and is shifted to the right by i − Problem 6.3.
Find a formula for the number of bounded n -lecture hall tableaux of a given shiftedshape.Let δ n = ( n − , n − , . . . , ,
0) and d ( n ) λ,µ = det (cid:18)(cid:18) λ i + n − iµ j + n − j (cid:19)(cid:19) ≤ i,j ≤ n , which is by Proposition 1.2 equal to | SSCT n ( λ/µ ) | . As mentioned in the introduction Lascoux [15]used (4) to compute the Chern classes of the exterior square ∧ E and symmetric square Sym E of a vector bundle E . To be more precise, let c ( E ) = Q ni =1 (1 + y i ) be the total Chern class of E .Lascoux showed that c ( ∧ E ) = Y ≤ i For µ ⊆ δ n and λ ⊆ δ n +1 , we have | SSCT n ( δ n /µ ) | = 2 | δ n /µ | r ( n ) µ , | SSCT n ( δ n +1 /λ ) | = X µ ⊆ λ ∩ δ n λ/µ a vertical strip | δ n /µ | r ( n ) µ . The objects in SSCT n ( δ n /µ ) and those counting r ( n ) µ have somewhat similar conditions on theirentries but their shapes are complementary: δ n /µ and µ . Understanding the connection betweenthese two objects will be very interesting. Problem 6.5. Find a bijective proof of Proposition 6.4.In a forthcoming paper [8], the first author, Keating and Nicoletti show that lecture hall tableauxare in bijection with a certain dimer model on a graph whose faces are hexagons and octagons.Moreover they show that bounded lecture hall tableaux of a “large” shape exhibit the arctic curvephenomenon. References [1] R. Adin and Y. Roichman. Standard Young tableaux. In Handbook of enumerative combinatorics , DiscreteMath. Appl. (Boca Raton), pages 895–974. CRC Press, Boca Raton, FL, 2015.[2] A. C. Aitken. The monomial expansion of determinantal symmetric functions. Proc. Royal Soc. Edinburgh (A) ,61:300–310, 1943.[3] M. Beck, B. Braun, M. K¨oppe, C.D. Savage, and Z. Zafeirakopoulos. s -lecture hall partitions, self-reciprocalpolynomials, and Gorenstein cones. Ramanujan J. , 36(1-2):123–147, 2015.[4] M. Beck, B. Braun, M. K¨oppe, C.D. Savage, and Z. Zafeirakopoulos. Generating functions and triangulationsfor lecture hall cones. SIAM J. Discrete Math , vol.30, no. 3 (2016), 1470–1479.[5] M. Beck and C.D. Savage. personal communication (2009).[6] S. C. Billey, B. Rhoades, V. Tewari. Boolean product polynomials, Schur positivity, and Chern plethysm. https://arxiv.org/abs/1902.11165 [7] P. Br¨anden and M. Leander, Lecture hall P -partitions. Journal of Combinatorics , to appear (2019). https://arxiv.org/abs/1609.02790 [8] S. Corteel, D. Keating and M. Nicoletti. Arctic curves for bounded lecture hall tableaux (2019). https://arxiv.org/abs/1905.02881 .[9] S. Corteel and J. S. Kim. Lecture hall tableaux, (2018) https://arxiv.org/abs/1804.02489 .[10] S. Corteel, S. Lee, and C. D. Savage. Enumeration of sequences constrained by the ratio of consecutive parts. Sem. Lothar. Combin. , 54A:Art. B54Aa, 12 pp. (electronic), 2005/07.[11] J. S. Frame, G. d. B. Robinson, and R. M. Thrall. The hook graphs of the symmetric groups. Canadian J.Math. , 6:316–324, 1954.[12] I. M. Gessel and X. G. Viennot. Binomial determinants, paths, and hook-length formulas. Advances in Math. 58 (1985), 300–321.[13] A.N. Kirillov and T. Scrimshaw. Hook-content formula using excited Young diagrams. https://arxiv.org/abs/1904.00371 [14] C. Krattenthaler. Another involution principle-free bijective proof of Stanley’s hook-content formula. J. Com-bin. Theory Ser. A , 88(1):66–92, 1999.[15] A. Lascoux. Classes de Chern d’un produit tensoriel. C. R. Acad. Sci. Paris S´er. A-B , 286(8):385–387, 1978.[16] F. Liu and R. Stanley. The lecture hall parallelepiped. Annals of Combinatorics 18 (2014), no. 3, 473–488.[17] I. G. Macdonald. Symmetric functions and Hall polynomials . Oxford Mathematical Monographs. The Claren-don Press Oxford University Press, New York, second edition, 1995.[18] A. Morales, G. Panova, I. Pak. Hook formulas for skew shapes I. q-analogues and bijections. J. Combin. TheorySer. A , 154:350–405, 2018.[19] H. Naruse. Schubert calculus and hook formula. Talk at 73rd S´em. Lothar. Combin., Strobl, Austria, 2014;available at http://tinyurl.com/z6paqzu [20] J.-C. Novelli, I. Pak, and A. V. Stoyanovskii. A direct bijective proof of the hook-length formula. DiscreteMath. Theor. Comput. Sci. , 1(1):53–67, 1997.[21] MC. Olsen, Hilbert bases and lecture hall partitions. Ramanujan J. , 47(3):509–531, 2018.[22] C. D. Savage. The mathematics of lecture hall partitions. J. Combin. Theory Ser. A , 144:443–475, 2016.[23] R. P. Stanley. Theory and application of plane partitions. I, II. Studies in Appl. Math. , 50:167–188; ibid. 50(1971), 259–279, 1971.[24] R. P. Stanley. Ordered structures and partitions . American Mathematical Society, Providence, R.I., 1972.Memoirs of the American Mathematical Society, No. 119.[25] R. P. Stanley. Enumerative Combinatorics. Vol. 1, second ed. Cambridge University Press, 2011.[26] R. P. Stanley. Enumerative combinatorics. Vol. 2. Cambridge University Press, 1999. University of California, Berkeley, United States E-mail address : [email protected] Sungkyunkwan University, Suwon, South Korea E-mail address ::