INCREASING TABLEAUX, NARAYANA NUMBERS AND AN INSTANCE OF THE CYCLIC SIEVING PHENOMENON

Transcription

1 INCREASING TABLEAUX, NARAYANA NUMBERS AND AN INSTANCE OF THE CYCLIC SIEVING PHENOMENON TIMOTHY PRESSEY, ANNA STOKKE AND TERRY VISENTIN Abstract We give a counting formula for the set of rectangular increasing tableaux in terms of generalized Narayana numbers We define small m-schröder paths and give a bijection between the set of increasing rectangular tableaux and small m-schröder paths, generalizing a result of Pecheni [4] Using K-jeu de taquin promotion, we give a cyclic sieving phenomenon for the set of increasing hoo tableaux 1 Introduction Let λ be a partition of a positive integer N An increasing tableau T is a λ- tableau in which both rows and columns are strictly increasing and, if N is the largest entry in the tableau, then each i with 1 i N appears at least once in T Let Inc (λ) denote the set of increasing λ-tableaux with maximum value N and let SYT(λ) = Inc 0 (λ) denote the set of standard λ-tableaux with entries in {1,,, N} In the first half of the article we focus on increasing tableaux of rectangular shape λ = (n, n,, n) = n m and will denote the corresponding sets by Inc (m n) and SYT(m n) The two-dimensional Catalan numbers enumerate SYT( n), the set of standard tableaux with two rows In [4], Pecheni gave explicit bijections between Inc ( n), small Schröder paths with diagonal steps and SYT(n, n, 1 ), giving a formula for the cardinality of Inc ( n) and showing that the total number of increasing tableaux of shape n is the nth small Schröder number The generalized Narayana numbers N(m, n, l) studied in [10] and [11] count the m-dimensional lattice paths from (0, 0,, 0) to (n, n,, n) lying in the region {(x 1, x,, x m ) 0 x m x 1 } using steps X 1 = (1, 0,, 0), X = (0, 1,, 0),, X m = (0, 0,, 1), which have l ascents An ascent in a path occurs when the path contains consecutive steps X i X j with j < i We prove that the cardinality of Inc (m n) is a linear combination of Narayana numbers in Theorem (m 1)(n 1) 4 An interesting corollary is that Inc 1 (m n) = SYT(m n) The small m-schröder numbers are given by the sequence (N m,n ()) n 0, where N m,n (t) = l=0 N(m, n, l)t l is the m-narayana polynomial In general, the m-dimensional Catalan numbers (N m,n (1)) n 0 enumerate SYT(m n) We prove that the small m-schröder number N m,n () is equal to the total number of increasing rectangular tableaux of shape m n, generalizing Pecheni s result This research was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada 1

2 TIMOTHY PRESSEY, ANNA STOKKE AND TERRY VISENTIN for Inc ( n) We define a generalized version of small Schröder paths in m- dimensional space, called small m-schröder paths, and give a bijection between small m-schröder paths and the set of increasing tableaux of shape m n Let X be a finite set, C = g a cyclic group of order N, and X(q) a polynomial with integer coefficients The triple (X, C, X(q)) is said to exhibit the cyclic sieving phenomenon (CSP) [] if for any positive integer d, we have X(ω d ) = {x X g d x = x}, where ω = e πi/n is a primitive Nth root of unity A CSP for SYT(m n) was given by Rhoades [6] using classic jeu de taquin promotion and a q-analogue of the Frame-Robinson-Thrall hoo length formula [] Using the K-jeu de taquin of Thomas and Yong [1] and a natural q-analogue of a formula that enumerates Inc ( n), Pecheni gave a CSP for Inc ( n) Rhoades [7] has recently given a representation-theoretic proof of this result The natural q- analogue of our counting formula for Inc (m n) does not, in general, serve as a CSP polynomial for the action of K-promotion on Inc (m n) In Section, we focus on proving a CSP for the set Inc (n r, 1 r ) of increasing hoo tableaux using K-promotion and a q-analogue of a formula that enumerates Inc (N r, 1 r ) This polynomial has a natural combinatorial interpretation the coefficients count arm leg inversions in increasing hoo tableaux, which are pairs (i, j) with i < j, where i belongs to the row and j the column Using a map from Inc (n r, 1 r ) to a set of standard hoo tableaux that behaves nicely with respect to K-promotion, along with results of Reiner, Stanton and White [], we exhibit a CSP for the set of increasing hoo tableaux Enumerating increasing tableaux with Narayana numbers We recall results concerning generalized Narayana numbers and generalized Schröder numbers from [11] Let C(m, n) denote the set of lattice paths in m-dimensional space that run from (0, 0,, 0) to (n, n,, n) using the steps X 1 = (1, 0,, 0), X = (0, 1,, 0),, X m = (0, 0,, 1) and lie in the region {(x 1, x,, x m ) 0 x m x m 1 x 1 } A pair of steps ɛ i ɛ i+1 on a path P = ɛ 1 ɛ ɛ mn is called an ascent if ɛ i ɛ i+1 = X j X r with r < j The set of ascents on a path P is denoted asc(p ) = {i ɛ i 1 ɛ i = X j X r for r < j} For m and 0 l (m 1)(n 1), the m-narayana number is defined as { N(m, n, l) = P C(m, n) asc(p ) = l} For m and n 1, the nth m-narayana polynomial is defined as N m,n (t) = l=0 N(m, n, l)t l

3 INCREASING TABLEAUX AND THE CYCLIC SIEVING PHENOMENON The m-dimensional Catalan numbers are given by the sequence (N m,n (1)) n 0 and these enumerate SYT(m n) By the hoo length formula, m 1 i! N m,n (1) = (mn)! (n + i)! The small m-schröder numbers are given by the sequence (N m,n ()) n 0 In the case where m =, Pecheni showed that the small -Schröder numbers enumerate the set of increasing n tableaux We will mae use of the following proposition and corollary from [11] Proposition 1 [11, Proposition 1] For m and for 0 l (m 1)(n 1), i=0 N(m, n, l) = l j=0 ( 1) l j ( mn + 1 l j ) m 1 i=0 ( n + i + j n ) ( ) 1 n + i n Corollary [11, Corollary 1] For m and n 1, N m,n (t) is a self-reciprocal polynomial of degree (m 1)(n 1) In other words, for each n, the sequence of coefficients of N m,n (t) is symmetric A path P = ɛ 1 ɛ ɛ mn C(m, n) gives a standard m n tableau by reading the path left to right and placing i in the th row of the tableau whenever ɛ i = X The condition 0 x m x m 1 x 1 ensures that if ɛ i = X, then the number of occurrences of X 1 in the sequence occurring previously is strictly greater than the number of occurrences of X, so the tableau generated by this procedure is standard In the case where m =, C(, n) consists of the paths from (0, 0) to (n, n) with horizontal and vertical steps that stay below the line y = x This is a very well-nown set of objects counted by the Catalan numbers An ascent occurs in a path P precisely when the tableau generated by it has an entry i occurring in a row above i 1 in the rectangular tableau it encodes It follows that N(m, n, l) is equal to the number of tableaux in SYT(m n) for which an entry i appears in a row above an i 1 exactly l times For a tableau T SYT(λ), let asc(t ) = {i in T i occurs in a row above i 1} To obtain a counting formula for increasing tableaux of rectangular shape, we define a map φ : Inc (m n) SYT(m n) We first define φ j : Inc j (m n) Inc j 1 (m n), for j 1 For T Inc j (m n), let a be the minimal entry that appears more than once in T Increase all entries in T that are greater than or equal to a, except for the leftmost value of a Define φ : Inc (m n) SYT(m n) as a composition φ = φ 1 φ φ 1 φ Example Below we find the image of a tableau T under φ : Inc ( ) SYT( ) T = φ φ φ = φ(t )

4 4 TIMOTHY PRESSEY, ANNA STOKKE AND TERRY VISENTIN Note that φ is not one-to-one For example, to determine the preimage of the tableau S = , we consider asc(s) = {,, 6, 8} Since =, each -element subset {a, b, c} of asc(s), with a < b < c, corresponds to an element in the preimage by first subtracting one from all entries in S that are greater than or equal to c, then subtracting one from all entries in the resulting tableau that ( are greater than or equal to b, and then repeating the process for a So there are 4 ) elements in the preimage of S; specifically φ 1 (S) = 1 4 6, , 1 4 6, Theorem 4 For 0, Inc (m n) = l= ( ) l N(m, n, l) Proof For any tableau T SYT(m n), ( φ) 1 (T ) if and only if asc(t ) l and if asc(t ) = l, then φ 1 (T ) = We have Inc (m n) = = = T SYT(m n) l= l= φ 1 (T ) ( ) l { T SYT(m n) asc(t ) = l} ( ) l N(m, n, l) Corollary The number of increasing tableaux of shape m n with exactly one repeated entry is given by Inc 1 (m n) = (m 1)(n 1) SYT(m n)

5 INCREASING TABLEAUX AND THE CYCLIC SIEVING PHENOMENON Proof By Corollary, we have N(m, n, l) = N(m, n, (m 1)(n 1) l) for 0 l (m 1)(n 1) It follows that Inc 1 (m n) = 1 = = ( l=0 (m 1)(n 1) ln(m, n, l) + l=0 (m 1)(n 1) SYT(m n) l=0 N(m, n, l) ) ((m 1)(n 1) l)n(m, n, l) Using the above result, we can use the hoo length formula for SYT(m n) to give the cardinality of Inc 1 (m n) Corollary 6 For m, the number of increasing tableaux of shape m n with maximum entry mn 1 is given by Inc 1 (m n) = (m 1)(n 1)(mn)! m 1 i=0 i! (n + i)! Pecheni revealed a relationship between Inc ( n) and small Schröder numbers [4, Theorem 11] The nth small Schröder number is equal to N,n () while the nth large Schröder number is equal to N,n () A large Schröder path is a path from (0, 0) to (n, n) with steps of the form (1, 0), (0, 1) and (1, 1) that stays below the line y = x A small Schröder path is a large Schröder path with no diagonal steps along y = x Pecheni s bijection (in a slightly modified form) between Inc ( n) and small Schröder paths is given by assigning a step ɛ i to each entry i in T Inc ( n) If i appears only in the first row, then ɛ i = (1, 0), while if i appears only in the second row, ɛ i = (0, 1) and if i appears in both the first and second row, then ɛ i = (1, 1) This gives a small Schröder path P T = ɛ 1 ɛ ɛ n and the procedure is easily reversible: given a small Schröder path from (0, 0) to (n, n), we can construct a tableau T Inc ( n) Example 7 We give the small Schröder path for T = We generalize Pecheni s result for rectangular increasing tableaux of arbitrary size and then define small m-schröder paths and give a bijection between these paths and the set of all increasing rectangular tableaux of shape m n

6 6 TIMOTHY PRESSEY, ANNA STOKKE AND TERRY VISENTIN Corollary 8 For m and n 1 we have =0 Inc (m n) = N m,n () In other words, the total number of increasing tableaux of shape m n is given by the small m-schröder number Proof We have =0 Inc (m n) = = = =0 t=0 t=0 l= ( t i=0 ( ) l N(m, n, l) ( ) ) t N(m, n, t) i t N(m, n, t) = N m,n () Sulane defined large m-schröder paths [11] as paths running from (0, 0,, 0) to (n, n,, n) with nonzero steps of the form (ξ 1, ξ,, ξ m ), with ξ i {0, 1}, that lie in the region {(x 1, x,, x m ) 0 x m x m 1 x 1 } He proved that the number of large m-schröder paths is equal to m 1 N m,n () We define a small m-schröder path to be a large m-schröder path with the property that the path does not contain any steps from (x 1,, x j 1, a, a, x j+,, x m ) to (y 1,, y j 1, a + 1, a + 1, y j+,, y m ) In other words, if after steps the path reaches position (x 1,, x m ), where x j = x j+1, then the ( + 1)th step ɛ +1 = (ξ 1,, ξ m ) cannot have ξ j = ξ j+1 = 1 For example, a small -Schröder path is a path from (0, 0, 0) to (,, ) with nonzero steps of the form (ξ 1, ξ, ξ ), ξ i {0, 1}, that lies in the region {(x, y, z) 0 z y x} and does not contain any steps from (a, a, z) to (a + 1, a + 1, z ) or from (x, b, b) to (x, b + 1, b + 1) In the case where m =, the small m-schröder paths are the usual small Schröder paths Theorem 9 There is a bijection between the collection of small m-schröder paths and the set of all increasing tableaux of shape m n Proof For an increasing tableau T with largest entry mn, define a path P T = ɛ 1 ɛ ɛ mn from (0, 0,, 0) to (n, n,, n) in the following way For each 1 i mn, let ɛ i = (ξ 1, ξ,, ξ m ) where ξ j = 1 if i appears in the jth row of T and ξ j = 0 otherwise Since T has strictly increasing columns, P T lies in the region {(x 1, x,, x m ) 0 x m x m 1 x 1 } If, after the th step ɛ, the path reaches position t then, after the ( + 1)th step, the path reaches position t +1 = t + ɛ +1 Furthermore, the subtableau of T of shape λ = t = (x 1, x,, x m ) is the portion of T that contains the entries 1,,, (See Example 10 for an illustration) If t = (x 1,, x m ) has x j = x j+1, then the subtableau of T containing the entries up to and including has x j boxes in both the jth and (j + 1)th row If ɛ +1 = (ξ 1,, ξ m ) has ξ j = ξ j+1 = 1 then

7 INCREASING TABLEAUX AND THE CYCLIC SIEVING PHENOMENON 7 t +1 = (y 1,, x j + 1, x j + 1,, y m ) so the subtableau of T containing the entries up to and including + 1 has x j + 1 boxes in both the jth and (j + 1)th row, which forces two entries equal to + 1 in column x j + 1 of T It follows that P T is a small m-schröder path Given a small m-schröder path P T, we can construct an increasing tableau T of shape m n by reversing the above procedure Example 10 For T = , P T = ɛ 1 ɛ ɛ 11 where ɛ 1 = (1, 0, 0, 0), ɛ = (1, 1, 0, 0), ɛ = (0, 0, 1, 0), ɛ 4 = (1, 1, 0, 1), ɛ = (1, 1, 0, 0), etc The steps in P T tae the path to positions t 1 = (1, 0, 0, 0), t = (, 1, 0, 0), t = (, 1, 1, 0), t 4 = (,, 1, 1), t = (4,, 1, 1), etc The position t i gives the shape λ = t i of the subtableau of T that contains the entries 1,, i Remar 11 Using the same construction as in the proof of Theorem 9, the large m-schröder paths are in one-to-one correspondence with the set of rowincreasing tableaux of shape m n where the entries are an initial segment of Z 0 or, by transpose, to the set of semistandard n m tableaux with entries an initial segment of Z 0 By [11, Proposition 10], this subset of the collection of semistandard n m tableaux has cardinality equal to m 1 N m,n () Cyclic sieving phenomena In this section, we give a CSP for increasing hoo tableaux A CSP for semistandard hoo tableaux was given in [1] We also show that the polynomial obtained by taing the natural q-analogue of the integer in Corollary 6, along with K-jeu de taquin promotion does not, in general, give a CSP for increasing rectangular tableaux, apart from the n version given in [4] Our focus is on increasing hoo tableaux and for such tableaux, K-promotion, which defines a bijection : Inc (N r, 1 r ) Inc (N r, 1 r ), can be described in the following way Given T Inc (N r, 1 r ), replace the 1 in T with a dot and repeatedly move all dots through the tableau using the rules below until every dot appears in the right-most box of the row or the lowest box in the column Then replace each dot with N and decrease all other entries in the tableau by one to obtain (T ) (1) a b a b b a if a < b if b < a, a a a

8 8 TIMOTHY PRESSEY, ANNA STOKKE AND TERRY VISENTIN Note that when = 0, K-promotion amounts to Schützenberger s jeu de taquin promotion on SYT(N r, 1 r ) For a description of K-promotion on increasing tableaux for general shapes, see [4] For a more general description of promotion and its basic properties, a survey is given in [9] Example 1 For T = 1 4, K-promotion wors as follows: T 4 = (T ) The content of a tableau T Inc (λ), where λ is a partition of N, is equal to α = (α 1,, α N ), where α i gives the number of entries equal to i in T ; we denote this by cont(t ) The symmetric group S N on N letters acts on (N )-tuples by permuting places: θ (α 1, α,, α N ) = (α θ(1), α θ(),, α θ(n ) ), where θ S N For increasing hoo tableaux, K-promotion permutes the content via the cycle σ = ( N ) S N In other words, if cont(t ) = (α 1, α,, α N ), where α 1 is necessarily equal to 1, then () cont( (T )) = (1, α,, α n, α ) = σ (α 1,, α N ), for T Inc (N r, 1 r ) The cardinality of Inc (N r, 1 r ) is given by ( )( ) N 1 r Inc (N r, 1 r ) = r To give a CSP for Inc (N r, 1 r ), we wor with a map ψ : Inc (N r, 1 r ) SYT(N r, 1 r ) that behaves nicely with respect to K-promotion This will allow us to use established results concerning SYT(N r, 1 r ) Define ψ : Inc (N r, 1 r ) SYT(N r, 1 r ) by deleting the entries in the row of T Inc (N r, 1 r ) that also appear in the column of T Then ψ is onto, but not one-to-one The following lemma follows easily from (1) Lemma If T Inc (N r, 1 r ), then ψ( (T )) = (ψ(t )) The order of promotion on Inc (N r, 1 r ) is the smallest positive integer l that satisfies l (T ) = T for all T Inc (N r, 1 r ) When = 0, there is a one-to-one correspondence between SYT(N r, 1 r ) and the set A that consists of subsets of {,, N} containing r 1 elements Define γ : SYT(N r, 1 r ) A, by defining γ(s) to be the set of entries in the first column of S that sit below the (1, 1)-box and let θ = ( 4 N) 1 S N We have γ( (S)) = θ(γ(s)) for S SYT(N r, 1 r ) so jeu de taquin promotion on S SYT(N r, 1 r ) is

9 INCREASING TABLEAUX AND THE CYCLIC SIEVING PHENOMENON 9 completely determined by considering the action of θ on the column of S follows that the order of promotion on SYT(N r, 1 r ) is equal to N 1 It Theorem The order of promotion on Inc (N r, 1 r ) is equal to N 1 Proof Let σ = ( N ) S N and suppose that T Inc (N r, 1 r ) has content α = (α 1, α,, α N ) Then the content of N 1 (T ) is equal to σ N 1 (α 1, α,, α N ) = α We have N 1 (S) = S for S SYT(N r, 1 r ) so N 1 (ψ(t )) = ψ(t ) for T Inc (N r, 1 r ) By Lemma, ψ( N 1 (T )) = ψ(t ) and since cont(t ) = cont( N 1 (T )), we have N 1 (T ) = T Furthermore, T Inc (N r, 1 r ) with content equal to (1,,,,, 1,, 1) is fixed by no less than N 1 iterations }{{}}{{} N 1 of K-promotion The following theorem is due to Reiner, Stanton and White [], where the theorem is stated in terms of -subsets of {1,,, N} under the action of the long cycle (1 N) S N Theorem 4 [, Theorem 11] The triple (SYT(N r, 1 r ), C, X 0 (q)) satisfies the cyclic sieving phenomenon, where C is the cyclic[ group of] order N 1 given N 1 by jeu de taquin promotion on SYT(λ) and X 0 (q) = r [ ] N 1 Let f 1 (q) = r, f (q) = q [ r ] q and X(q) = f 1 (q)f (q), which is a q- analogue of the formula that enumerates Inc (N r, 1 r ) In fact, X(q) has a fairly natural combinatorial interpretation An ordered pair (i, j) with i < j N will be called an inversion in T Inc (N r, 1 r ) if i appears as a row entry in T and j appears as a column entry in T Then T qa(t ) = q ( ) X(q), where the sum runs over the increasing tableaux T Inc (N r, 1 r ) and a(t ) is the number of inversions in T This follows easily from the interpretation of the q- binomial coefficients (or Gaussian coefficients) as generating functions for subsets with respect to between-set inversions Details of this interpretation are given in [] Let ω be a primitive (N 1)th root of unity Then ω m is a primitive dth root of unity where d gcd(n 1, m) = N 1 and by [, Proposition 4], ( ) (N 1)/d if d r () f 1 (ω m r/d ) = 0 otherwise In general, f (ω m ) may not be an integer but we are only concerned with the value of f (ω m ) when f 1 (ω m )f (ω m ) 0 In particular, if f 1 (ω m ) 0, then d r so q

10 10 TIMOTHY PRESSEY, ANNA STOKKE AND TERRY VISENTIN we have ( ) r/d if d (4) f (ω m /d ) = 0 otherwise, when f 1 (ω m ) 0 Lemma 6 is the main ingredient that will be used to prove a CSP for Inc (N r, 1 r ) The following example will be useful when reading the proof of Lemma 6 Example Consider ψ : Inc (, 1 6 ) SYT(, 1 6 ) Promotion on a tableau in SYT(, 1 6 ) corresponds to the action of the permutation θ = ( ) on the column entries of the tableau Since θ 4 = ( 6)( 7)(4 8)( 9), the column of a tableau in SYT(, 1 6 ) is fixed by θ 4 only when the entries in the column of the tableau below the (1, 1)-box correspond to the values in three of the four - cycles in the decomposition of θ 4 The following tableau in SYT(, 1 6 ) satisfies 4 (S) = S: S = There are ( 6 ) = 1 tableaux that map to S under ψ We wish to determine those tableaux in the preimage of S with content that is fixed by 4 In general, a tableau T Inc (, 1 6 ) has content that is fixed by 4 if and only if the content of T is equal to one of the following: (1,,1,1,1,,1,1,1), (1,1,1,1,,1,1,1,), (1,1,,1,1,1,,1,1), (1,1,1,,1,1,1,,1) If T Inc (, 1 6 ) also satisfies ψ(t ) = S, then the two elements in the row of T that are repeated in the column must belong to one of the -cycles that appear in the decomposition of θ 4, so if ψ(t ) = S and cont(t ) = cont( 4 (T )) then cont(t ) must be equal to one of the first three sequences above This completely determines those tableaux in the preimage of S that have content fixed by 4 : , ,

11 INCREASING TABLEAUX AND THE CYCLIC SIEVING PHENOMENON 11 Lemma 6 Let S SYT(N r, 1 r ) with m (S) = S and suppose that ω is a primitive (N 1)th root of unity The number of T Inc [(N ] r, 1 r ) with r ψ(t ) = S such that cont(t ) = cont( m (T )) is equal to f (ω m ) = q=ω m Proof Since ω is a primitive (N 1)th root of unity, ω m is a dth root of unity where d gcd(n 1, m) = N 1 Let θ = σ 1 = ( 4 N ) 1 S N The column of (S) is given by the action of θ on the entries of the column of S that sit below the (1, 1)-box, so θ m fixes these r elements in the column of S We can write θ m = θ 1 θ θ m, which is a product of m = gcd(n 1, m) disjoint cycles of length d Since θ m fixes the r column elements of S, we have that d divides r Let T Inc (N r, 1 r ) have content equal to α = (α 1, α,, α N ), and suppose that cont( m (T )) = α Then by (), σ m α = α and since σ m is the product of d-cycles and there are exactly entries α i that are equal to, we have that d divides Furthermore, the repeated entries in the row of T can be partitioned into /d sets of size d, where each set consists of elements from one of the d-cycles θ 1, θ,, θ m in the decomposition of θ = σ 1 Since θ m fixes the entries in the column of S, the entries below the (1, 1)-box must consist of the values from l = r/d of the d-cycles θ 1, θ,, θ m Denote this subset of d-cycles that give the column of S by θ 1, θ,, θ l If T Inc (N r, 1 r ), with ψ(t ) = S and cont(t ) = cont( m (T )), then the entries in the row of T that are repeated in the column can be partitioned into /d sets of size d, where ( each set) consists of elements from one of the d-cycles θ 1,, θ l There are exactly r/d such tableaux and by (4), this is equal to f /d (ω m ) Theorem 7 The triple (Inc (N r, 1 r ), C, X(q)) satisfies the cyclic sieving phenomenon, where C is the cyclic [ group of order ] [ N ] 1 given by K-promotion N 1 r on Inc (N r, 1 r ) and X(q) = r Proof Let X = {T Inc (N r, 1 r ) m (T ) = T } and Y = {T Inc (N r, 1 r ) m (ψ(t )) = ψ(t ) and cont(t ) = cont( m (T ))} If T X, then m (T ) = T so cont( m (T )) = cont(t ) and ψ( m (T )) = ψ(t ) By Lemma, m (ψ(t )) = ψ(t ) If T Y, then m (ψ(t )) = ψ(t ) so ψ( m (T )) = ψ(t ) Since cont( m (T )) = cont(t ), m (T ) = T Thus X = Y and by Theorem 4 and Lemma 6, Y = f 1 (ω m )f (ω m ) We close with an example that shows that a natural q-analogue of the polynomial in Corollary 6, coupled with K-jeu de taquin promotion does not give a CSP for Inc 1 ( ) Example 8 Let X(q) = (q9 1)(q 8 1)(q 7 1)(q 6 1), which is a natural (q 4 1)(q 1) (q 1) q-analogue of the integer from Corollary 6 for n = The order of promotion on Inc 1 ( ) is equal to 8 and there are four tableaux in Inc 1 ( ) that have order q q

12 1 TIMOTHY PRESSEY, ANNA STOKKE AND TERRY VISENTIN equal to two (See [4] for the definition of K-promotion for rectangular shapes) However, if ω is a primitive eighth root of unity, X(ω ) = i is not even an integer References [1] M Bennett, B Madill, and A Stoe, Jeu-de-taquin promotion and a cyclic sieving phenomenon for semistandard hoo tableaux, Discrete Math 19 (014), 6-67 [] JS Frame, G Robinson, and R M Thrall, The hoo graphs of the symmetric groups, Canadian J Math 6 (194), 16 4 [] IP Goulden and DM Jacson, Combinatorial Enumeration, Wiley Interscience, New Yor, 198 [4] O Pecheni, Cyclic sieving of increasing tableaux and small Schröder paths, J Combin Theory, Ser A 1 (014), 7 78 [] V Reiner, D Stanton, and D White, The cyclic sieving phenomenon, J Combin Theory Ser A 108 (004), 17 0 [6] B Rhoades, Cyclic sieving, promotion, and representation theory, J Combin Theory Ser A 117 (010), 8 76 [7], A sein action of the symmetric group on noncrossing partitions, arxiv: v[mathco] (01), 1 9 [8] B Sagan, The cyclic sieving phenomenon: a survey, Surveys in Combinatorics 011, London Math Soc Lecture Note Series, vol 9, Cambridge University Press, Cambridge, 011 [9] RP Stanley, Promotion and evacuation, Electron J Combin 16() (009), #R9 [10] RA Sulane, Three dimensional Narayana and Schröder numbers, Theoret Comput Sci 46 (00), [11], Generalizing Narayana and Schröder numbers to higher dimensions, Electron J Combin 11 (004), #R4 [1] H Thomas and A Yong, A jeu de taquin theory for increasing tableaux with applications to K-theoretic calculus, Algebra Number Theory (009), no, University of Winnipeg, Department of Mathematics and Statistics, Winnipeg, Manitoba, Canada RB E9 address: astoe@uwinnipegca, tvisentin@uwinnipegca