Multivariate Rogers-Szegö polynomials and flags in finite vector spaces
aa r X i v : . [ m a t h . C O ] N ov Multivariate Rogers-Szeg¨o polynomials and flags in finite vectorspaces
C. Ryan Vinroot
Abstract
We give a recursion for the multivariate Rogers-Szeg¨o polynomials, along with anotherrecursive functional equation, and apply them to compute special values. We also considerthe sum of all q -multinomial coefficients of some fixed degree and length, and give a recursionfor this sum which follows from the recursion of the multivariate Rogers-Szeg¨o polynomials,and generalizes the recursion for the Galois numbers. The sum of all q -multinomial coeffi-cients of degree n and length m is the number of flags of length m − n -dimensional vector space over a field with q elements. We give a combinatorial proof ofthe recursion for this sum of q -multinomial coefficients in terms of finite vector spaces.2010 Mathematics Subject Classification:
Key words and phrases:
Rogers-Szeg¨o polynomials, finite vector spaces, Galois numbers,flags of subspaces
For a parameter q = 1, and a positive integer n , let ( q ) n = (1 − q )(1 − q ) · · · (1 − q n ), and( q ) = 1. For non-negative integers n and k , with n ≥ k , the q -binomial coefficient or Gaussianpolynomial , denoted (cid:0) nk (cid:1) q , is defined as (cid:0) nk (cid:1) q = ( q ) n ( q ) k ( q ) n − k .The Rogers-Szeg¨o polynomial in a single variable, denoted H n ( t ), is defined as H n ( t ) = n X k =0 (cid:18) nk (cid:19) q t k . The Rogers-Szeg¨o polynomials first appeared in papers of Rogers [12, 13] which led up to thefamous Rogers-Ramanujan identities, and later were independently studied by Szeg¨o [15]. Theyare important in combinatorial number theory ([1, Ex. 3.3–3.9] and [4, Sec. 20]), symmet-ric function theory [16], and are key examples of orthogonal polynomials [2]. They also haveapplications in mathematical physics [8, 9].The Rogers-Szeg¨o polynomials satisfy the recursion (see [1, p. 49]) H n +1 ( t ) = (1 + t ) H n ( t ) + t ( q n − H n − ( t ) . (1.1)Letting t = 1, we have H n (1) = P nk =0 (cid:0) nk (cid:1) q , which, when q is the power of a prime, is thetotal number of subspaces of an n -dimensional vector space over a field with q elements. Thenumbers G n = H n (1) are the Galois numbers , and from (1.1), satisfy the recursion G n +1 =2 G n + ( q n − G n − . The Galois numbers were studied from the point of view of finite vectorspaces by Goldman and Rota [5], and have been studied extensively, for example, in [11, 6].1or non-negative integers k , k , . . . , k m such that k + · · · + k m = n , we define the q -multinomial coefficient of length m as (cid:18) nk , k , . . . , k m (cid:19) q = ( q ) n ( q ) k ( q ) k · · · ( q ) k m , so that (cid:0) nk (cid:1) q = (cid:0) nk,n − k (cid:1) q . Define the homogeneous Rogers-Szeg¨o polynomial in m variables for m ≥
2, denoted ˜ H n ( t , t , . . . , t m ), by˜ H n ( t , t , . . . , t m ) = X k + ··· + k m = n (cid:18) nk , . . . , k m (cid:19) q t k · · · t k m m , and define the Rogers-Szeg¨o polynomial in m − H n ( t , . . . , t m − ), by H n ( t , . . . t m − ) = ˜ H ( t , . . . , t m − , . The homogeneous multivariate Rogers-Szeg¨o polynomials were first defined by Rogers [12] interms of their generating function, and several of their properties are given by Fine [4, Section21]. The definition of the multivariate Rogers-Szeg¨o polynomial H n is given by Andrews in [1,Chap. 3, Ex. 17], along with a generating function, although there is little other study of thesepolynomials elsewhere in the literature (however, there is a non-symmetric version of a bivariateRogers-Szeg¨o polynomial [3]). In Section 2, we give a recursion for the multivariate Rogers-Szeg¨opolynomials which generalizes (1.1). The result, given in Theorem 2.1 below, follows quicklyfrom the generating function for the multivariate Rogers-Szeg¨o polynomials, although seems notto have been noted before. We also give a few other properties of the multivariate Rogers-Szeg¨opolynomials in Section 2 which complement those given in [4, Section 21].Finally, in Section 3, we concentrate on the value H n (1 , , . . . , H n ( t , . . . , t m − ) when we let t = · · · = t m − = 1. This is the sum of all q -multinomialcoefficients of length m , which we denote by G ( m ) n , so G ( m ) n = X k + ··· + k m = n (cid:18) nk , . . . , k m (cid:19) q . These generalize the Galois numbers G n = G (2) n , and in particular, G ( m ) n is the total numberof flags of subspaces of length m of an n -dimensional vector space over a field with q elementswhen q is the power of a prime. The main task of Section 3 is to study the numbers G ( m ) n inthe context of finite vector spaces, independent of the Rogers-Szeg¨o polynomials, in the spiritof the study of Goldman and Rota. A recursion for the numbers G ( m ) n is immediately obtainedin Corollary 3.1 from the recursion of the multivariate Rogers-Szeg¨o polynomials, for whichwe provide a combinatorial proof in terms of finite vector spaces. We prove the recursion forthe generalized Galois numbers by obtaining a recursive formula for q -multinomial coefficients,which itself implies the recursion for the multivariate Rogers-Szeg¨o polynomials. For any a, r = 1, and n ≥
1, we define ( a ; r ) n by( a ; r ) n = (1 − a )(1 − ar ) · · · (1 − ar n − ) , a ; r ) = 1. Define ( a ; r ) ∞ by the formal infinite product( a ; r ) ∞ = ∞ Y i =0 (1 − ar i ) . For our fixed parameter q = 1, and any a = 1, define ( a ) n = ( a ; q ) n and ( a ) ∞ = ( a ; q ) ∞ , so( q ) n = (1 − q )(1 − q ) · · · (1 − q n ) for n ≥ ∞ X n =0 x n ( r ; r ) n = ( x ; r ) − ∞ . (2.1)We may use (2.1) to give a generating function for the multivariate Rogers-Szeg¨o polynomials.The following is stated in [1, Ex. 3.17], but we include the proof here for the sake of self-containment. Lemma 2.1.
The multivariate Rogers-Szeg¨o polynomials have the following generating function: ∞ X n =0 H n ( t , . . . , t m − )( q ) n x n = ( t x ) − ∞ · · · ( t m − x ) − ∞ ( x ) − ∞ . Proof.
By (2.1), we have( t x ) − ∞ · · · ( t m − x ) − ∞ ( x ) − ∞ = ∞ X k =0 t k x k ( q ) k · · · ∞ X k m − =0 t k m − m − x k m − ( q ) k m − ∞ X k m =0 x k m ( q ) k m = ∞ X n =0 X k + ··· + k m = n t k · · · t k m − m − ( q ) k · · · ( q ) k m x n = ∞ X n =0 X k + ··· + k m = n (cid:18) nk , . . . , k m (cid:19) q t k · · · t k m − m − x n ( q ) n = ∞ X n =0 H n ( t , . . . , t m − )( q ) n x n , as claimed.For any finite set of variables X , we let e i ( X ) denote the i th elementary symmetric poly-nomial in the variables X . We can now give the recursion of the multivariate Rogers-Szeg¨opolynomials which generalizes (1.1). Theorem 2.1.
The Rogers-Szeg¨o polynomials in m − variables satisfy the following recursion: H n +1 ( t , . . . , t m − ) = m − X i =0 e i +1 ( t , . . . , t m − , − i ( q ) n ( q ) n − i H n − i ( t , . . . , t m − ) . Proof.
Let F ( x, t , . . . , t m − ) = ∞ X n =0 H n ( t , . . . , t m − )( q ) n x n = ( t x ) − ∞ · · · ( t m − x ) − ∞ ( x ) − ∞ ,
3y Lemma 2.1. Since (1 − t i x )( t i x ) − ∞ = ( t i xq ) − ∞ and (1 − x )( x ) − ∞ = ( xq ) − ∞ , we have(1 − t x ) · · · (1 − t m − x )(1 − x ) F ( x, t , . . . , t m − ) = ( t xq ) − ∞ · · · ( t m − xq ) − ∞ ( xq ) − ∞ = F ( xq, t , . . . , t m − ) . (2.2)By the definition of the elementary symmetric polynomials, we have(1 − t x ) · · · (1 − t m − x )(1 − x ) = m X i =0 ( − i e i ( t , . . . , t m − , x i . From this and (2.2), we have m X i =0 ( − i e i ( t , . . . , t m − , x i ! ∞ X n =0 H n ( t , . . . , t m − )( q ) n x n = ∞ X n =0 q n H n ( t , . . . , t m − )( q ) n x n , which may be re-written as ∞ X n =0 (1 − q n ) H n ( t , . . . , t m − )( q ) n x n = m − X i =0 ( − i e i +1 ( t , . . . , t m − , x i +1 ! ∞ X n =0 H n ( t , . . . , t m − )( q ) n x n . Comparing the coefficients of x n +1 in both sides of the above expression, we obtain H n +1 ( t , . . . , t m − )( q ) n = m − X i =0 e i +1 ( t , . . . , t m − ,
1) ( − i ( q ) n − i H n − i ( t , . . . , t m − ) , which yields the desired result.Note that we have ( − i ( q ) n / ( q ) n − i = ( q n − q n − − · · · ( q n − i +1 − H n +1 ( t , t ) = (1 + t + t ) H n ( t , t )+( t t + t + t )( q n − H n − ( t , t )+ t t ( q n − q n − − H n − ( t , t ) . When the Rogers-Szeg¨o polynomial H n ( t ) in a single variable is evaluated at t = −
1, we getthe following identity for the alternating sum of q -binomial coefficients originally due to Gauss: H n ( −
1) = n X k =0 (cid:18) nk (cid:19) q ( − k = ( ( q ) n ( q ; q ) n/ = Q j 2, and let n ≥ 0, then H n ( ω, ω , . . . , ω m − ) = ( ( q ) n ( q m ; q m ) n/m = Q j 1) = 0for 1 ≤ i ≤ m − 1, while e m ( ω, . . . , ω m − , 1) = ( − m +1 , since these roots of unity are the rootsof x m − 1. We have H ( t , . . . , t m − ) = 1, and for 0 < n ≤ m − 1, we have H n ( t , . . . , t m − )4s a symmetric polynomial in t , . . . , t m − , 1, of degree n < m , with zero constant term. Wemay thus write H n ( t , . . . , t m − ) as a polynomial in e i ( t , . . . , t m − , < i < n . Now, H n ( ω, . . . , ω m − ) can be written as a polynomial in e i ( ω, . . . , ω m − , ≤ i ≤ m − 1, whichare all 0. It follows that H n ( ω, . . . , ω m − ) = 0 for these n . By Theorem 2.1 and the values ofthe elementary symmetric polynomials, if n ≥ m then H n ( ω, . . . , ω m − ) = m − X i =0 e i +1 ( ω, . . . , ω m − , − i ( q ) n − ( q ) n − − i H n − − i ( ω, . . . , ω m − )= (1 − q n − )(1 − q n − ) · · · (1 − q n − m +1 ) H n − m ( ω, . . . , ω m − ) . The values (2.4) now follow by induction.Another value of the Rogers-Szeg¨o polynomial of a single variable is H n ( q / ) = ( q ) n ( q / ; q / ) n = ( q / ; − q / ) n = n Y j =1 (1 + q j/ ) . This is generalized in [4] with the value H n ( q /m , q /m , . . . , q ( m − /m ) = ( q ) n ( q /m ; q /m ) n = n Y j =1 (1 + q j/m + · · · + q j ( m − /m ) . (2.5)Fine also gives a generalization of both (2.4) and (2.5), and applies it to obtain a bi-basic identity[4, 21.4].The Rogers-Szeg¨o polynomial H n ( t ) also takes the value H n ( − q ) = ( q ) n ( q ; q ) ⌊ n/ ⌋ = Y j ≤ n,j odd (1 − q j ) , (2.6)which is applied in finding identities involving Hall-Littlewood functions in [16], for example. Ageneralization of (2.6) for the multivariate Rogers-Szeg¨o polynomials is not covered above, andso we obtain one now. Let ω = e πi/m be a primitive m -th root of unity, where m ≥ 2, and let n ≥ 0. Then H n ( ωq, ω q, . . . , ω m − q ) = ( q ) n ( q m ; q m ) ⌊ n/m ⌋ = Y j ≤ n,m ∤ j (1 − q j ) , (2.7)which we calculate as follows. By Lemma 2.1 and (2.1), we have ∞ X n =0 H n ( ωq, . . . , ω m − q )( q ) n x n = 1( x ) ∞ ( xωq ) ∞ · · · ( xω m − q ) ∞ = 11 − x ∞ Y j =1 m − Y l =0 − xω l q j = 11 − x ∞ Y j =1 − x m q mj = 1 + x + · · · + x m − (1 − x m )(1 − x m q m )(1 − x m q m ) · · · = 1 + x + · · · + x m − ( x m ; q m ) ∞ = (1 + x + · · · + x m − ) ∞ X k =0 q m ; q m ) k x mk . Comparing the coefficients of x n , we obtain (2.7).5he value (2.6) of H n ( − q ) could also be computed by using (2.3) along with the functionalequation [4, 20.64b] H n ( tq ) = H n ( t ) − t (1 − q n ) H n − ( t ) . (2.8)The next result, which generalizes (2.8) to multivariate Rogers-Szeg¨o polynomials, has a verysimilar form and proof to Theorem 2.1. Theorem 2.2. Let m ≥ , and let J ⊆ { , . . . , m − } , where | J | 6 = 0 . For ≤ i ≤ m − , define s i = t i q if i ∈ J , and s i = t i otherwise. Let e i ( t J ) be the i th elementary symmetric polynomialin the set of variables t J = { t j | j ∈ J } . Then for n ≥ | J | , H n ( s , . . . , s m − ) = | J | X i =0 e i ( t J )( − i ( q ) n ( q ) n − i H n − i ( t , . . . , t m − ) . Proof. Let F ( x, t , . . . , t m − ) be the generating function for H n ( t , . . . , t m − ) as in the proof ofTheorem 2.1. Then we have Y j ∈ J (1 − t j x ) F ( x, t , . . . , t m − ) = Y j ∈ J ( t j qx ) − ∞ Y i J, ≤ i ≤ m − ( t i x ) − ∞ ( x ) − ∞ = F ( x, s , . . . , s m − ) . (2.9)Since Q j ∈ J (1 − t j x ) = P | J | i =0 ( − i e i ( t J ) x i , it follows from (2.9) that we have ∞ X n =0 H n ( s , . . . , s m − )( q ) n x n = | J | X i =0 ( − i e i ( t J ) x i ∞ X n =0 H n ( t , . . . , t m − )( q ) n x n . Comparing the coefficient of x n in both sides of the above gives the result.Now we may compute the value (2.7) by applying Theorem 2.2 in the following way. Notethat since ω, ω , . . . , ω m − , are the roots of x m + x m − + · · · + 1, then e i ( ω, . . . , ω m − ) = ( − i for 0 ≤ i ≤ m − 1. If we are able to compute the values H i ( ωq, . . . , ω m − q ) for i < m , then weuse Theorem 2.2 with J = { , . . . , m − } and t i = ω i to obtain, when n ≥ m , H n ( ωq, . . . , ω m − q ) = m − X i =1 ( q ) n ( q ) n − i H n − i ( ω, . . . , ω m − ) . The values (2.7) then follow for n ≥ m when plugging in the values (2.4). We can compute H n ( ωq, . . . , ω m − q ) for n < m using Theorem 2.2 as well. For n = 1, we begin by taking J = { } and t i = ω i to obtain H ( ωq, ω , . . . , ω m − ) = ω ( q − J = { } , t = ωq , and t i = ω i for i > 1. Applying Theorem 2.2 then gives H ( ωq, ω q, ω , . . . , ω m − ) = ( ω + ω )( q − H ( ωq, ω q, . . . , ω m − q ) = ( ω + · · · + ω m − )( q − 1) = (1 − q ) . The values for 1 < n < m may be computed similarly.6 Flags in finite vector spaces Now let q be the power of a prime, and let F q denote a finite field with q elements. If V isan n -dimensional vector space over F q , then the q -binomial coefficient (cid:0) nk (cid:1) q is the number of k -dimensional subspaces of V (see [7, Thm. 7.1] or [14, Prop. 1.3.18]). When we evaluate theRogers-Szeg¨o polynomial at t = 1, we obtain H n (1) = n X k =0 (cid:18) nk (cid:19) q , which is the total number of subspaces of an n -dimensional vector space over F q . We definethe Galois numbers as G n = H n (1). As mentioned in the introduction, the recursion for theRogers-Szeg¨o polynomials (1.1) gives the following recursion for the Galois numbers, which wasstudied by Goldman and Rota [5]: G n +1 = 2 G n + ( q n − G n − , G = 1 , G = 2 . (3.1)The recursion (3.1) was proved bijectively by counting subspaces of finite vector spaces byNijenhuis, Solow, and Wolf [11]. The proof in [11] is obtained by proving the following resultbijectively, from which (3.1) follows. Lemma 3.1. For integers n ≥ k ≥ , we have (cid:18) n + 1 k (cid:19) q = (cid:18) nk (cid:19) q + (cid:18) nk − (cid:19) q + ( q n − (cid:18) n − k − (cid:19) q . We now consider the meaning of a q -multinomial coefficient in terms of vector spaces over F q . It follows from the definition of a q -multinomial coefficient and the fact that (cid:0) nk (cid:1) q = (cid:0) nn − k (cid:1) q that we have (cid:18) nk , k , . . . , k m (cid:19) q = (cid:18) nk (cid:19) q (cid:18) n − k k (cid:19) q · · · (cid:18) n − k − · · · − k m − k m − (cid:19) q = (cid:18) nn − k (cid:19) q (cid:18) n − k n − k − k (cid:19) q · · · (cid:18) n − k − · · · − k m − n − k − · · · − k m − − k m − (cid:19) q . So, if V is an n -dimensional vector space over F q , the q -multinomial coefficient (cid:0) nk ,...,k m (cid:1) q is equalto the number of ways to choose an ( n − k )-dimensional subspace W of V , an ( n − k − k )-dimensional subspace W of W , and so on, until finally we choose an ( n − k − · · · − k m − )-dimensional subspace W m − of some ( n − k − · · · − k m − )-dimensional subspace W m − (see also[10, Sec. 1.5]). That is, W m − ⊆ W m − ⊆ · · · ⊆ W ⊆ W is a flag of subspaces of V of length m − 1, where dim W i = n − P ij =1 k j .If we evaluate the Rogers-Szeg¨o polynomial in m − t = t = · · · = t m − = 1,we obtain H n (1 , , . . . , 1) = X k + ··· + k m = n (cid:18) nk , . . . , k m (cid:19) q , which, by the discussion above, counts the total number of flags of subspaces of length m − n -dimensional F q -vector space. We denote this quantity by G ( m ) n , so that the Galoisnumber G n = G (2) n . We may apply Theorem 2.1 to obtain a recursion for the numbers G ( m ) n ,generalizing the recursion in (3.1), by noticing that the number of terms in the elementarysymmetric polynomial e i +1 ( t , . . . , t m − , 1) is (cid:0) mi +1 (cid:1) .7 orollary 3.1. The numbers G ( m ) n satisfy the following recursion, for n ≥ m − : G ( m ) n +1 = m − X i =0 (cid:18) mi + 1 (cid:19) ( − i ( q ) n ( q ) n − i G ( m ) n − i . In this section, we prove Corollary 3.1 combinatorially in terms of finite vector spaces, byproving an analog of Lemma 3.1.We need some notation. Let k denote the m -tuple ( k , . . . , k m ), and write the corresponding q -multinomial coefficient as (cid:18) nk , . . . , k m (cid:19) q = (cid:18) nk (cid:19) q . For a subset J ⊆ { , . . . , m } , let e J denote the m -tuple ( e , . . . , e m ), where e i = (cid:26) i ∈ J ,0 if i J .For example, if m = 3, J = { , } , and k = ( k , k , k ) , then (cid:18) nk − e J (cid:19) q = (cid:18) nk − , k , k − (cid:19) q . The following is our generalization of Lemma 3.1. Lemma 3.2. For m ≥ , and any k , . . . , k m > such that k + · · · + k m = n + 1 , we have (cid:18) n + 1 k , . . . , k m (cid:19) q = X J ⊆{ ,...,m } , | J | > ( − | J |− ( q ) n ( q ) n −| J | +1 (cid:18) n + 1 − | J | k − e J (cid:19) q Before proving Lemma 3.2, we explain why it implies Corollary 3.1. First note that we mayget a version of Lemma 3.2 which allows any of the k i = 0 as follows. If we want l of the k i ’sto be 0, we start with applying Lemma 3.2 to a q -multinomial coefficient of length m − l , usingthe m − l nonzero k i ’s, and note that the equation in Lemma 3.2 is not affected by inserting0’s into the appropriate positions of all the q -multinomial coefficients in both sides. That is, ifsome k i = 0, we may still apply Lemma 3.2, while ignoring these k i , or equivalently, we mayapply Lemma 3.2 to the q -multinomial coefficient obtained by removing the k i ’s which are 0,and re-inserting these 0’s in all q -multinomial coefficients in the sum the end.Now consider the sum of all q -multinomial coefficients of the form (cid:0) n +1 k ,...,k m (cid:1) q , while applyingthe more general version of Lemma 3.2 just discussed. For any i , 0 ≤ i ≤ m − 1, each term (cid:0) n +1 k ,...,k m (cid:1) q in G ( m ) n +1 may be obtained by adding 1 to i + 1 of the l i ’s in terms in G ( m ) n − i of the form (cid:0) n − il ,...,l m (cid:1) q in exactly (cid:0) mi +1 (cid:1) ways. By Lemma 3.2, these terms contribute exactly what we needto conclude Corollary 3.1. By a similar argument, we may see that in fact the recursion for themultinomial Rogers-Szeg¨o polynomials in Theorem 2.1 also follows from Lemma 3.2. Proof of Lemma 3.2. We will prove this by induction on m , where the base case m = 2 is givenby Lemma 3.1. Fix V to be an ( n + 1)-dimensional vector space over F q . We know (cid:0) n +1 k ,...,k m (cid:1) q isthe number of flags of subspaces of V , W m − ⊂ · · · ⊂ W ⊂ W , where dim W i = n +1 − P ij =1 k j .We must show that the right-hand side of the claimed identity in Lemma 3.2 also counts these8ags. We have, by choosing first the subspace W and then the rest of the flag, and applyingLemma 3.1, (cid:18) n + 1 k , . . . , k m (cid:19) q = (cid:18) n + 1 n + 1 − k (cid:19) q (cid:18) n + 1 − k k , . . . , k m (cid:19) q = (cid:18) nn + 1 − k (cid:19) q + (cid:18) nn − k (cid:19) q + ( q n − (cid:18) n − n − k (cid:19) q ! (cid:18) n + 1 − k k , . . . , k m (cid:19) q . (3.2)We have (cid:0) nn +1 − k (cid:1) q (cid:0) n +1 − k k ,...,k m (cid:1) q = (cid:0) nk − ,k ,...,k m (cid:1) q , which is the term corresponding to the subset J = { } ⊂ { , . . . , m } . This may be thought of as the total number of ways of choosing our flagso that W is contained in some fixed n -dimensional subspace of V . By our induction hypothesis,the number of remaining flags is given by (cid:18) nn − k (cid:19) q + ( q n − (cid:18) n − n − k (cid:19) q ! X I ⊆{ ,...,m − } , | I | > ( − | I |− ( q ) n − k ( q ) n − k −| I | +1 (cid:18) n + 1 − k − | I | k ′ − e I (cid:19) q , (3.3)where k ′ = ( k , . . . , k m ). We have (cid:18) nn − k (cid:19) q ( q ) n − k ( q ) n − k −| I | +1 = ( q ) n ( q ) n −| I | +1 (cid:18) n + 1 − | I | n − k − | I | + 1 (cid:19) q , and ( q n − (cid:18) n − n − k (cid:19) q ( q ) n − k ( q ) n − k −| I | +1 = ( − 1) ( q ) n ( q ) n −| I | (cid:18) n − | I | n − k − | I | + 1 (cid:19) q . Given I ⊂ { , . . . , m − } , | I | > 0, write I + 1 = { i + 1 | i ∈ I } ⊆ { , . . . , m } . If we let k = ( k , . . . , k m ), we now have (cid:18) nn − k (cid:19) q ( − | I |− ( q ) n − k ( q ) n − k −| I | +1 (cid:18) n + 1 − k − | I | k ′ − e I (cid:19) q = ( − | J |− ( q ) n ( q ) n −| J | +1 (cid:18) n + 1 − | J | k − e J (cid:19) q , (3.4)where J = I + 1, | J | = | I | , and( q n − (cid:18) n − n − k (cid:19) q ( − | I |− ( q ) n − k ( q ) n − k −| I | +1 (cid:18) n + 1 − k − | I | k ′ − e I (cid:19) q = ( − | J |− ( q ) n ( q ) n −| J | +1 (cid:18) n + 1 − | J | k − e J (cid:19) q , (3.5)where J = { } ∪ ( I + 1), | J | = | I | + 1. As I ranges over nonempty subsets of { , . . . , m − } , I + 1 and { } ∪ ( I + 1) range over all nonempty subsets of { , . . . , m } other than { } . Finally,we substitute (3.4) and (3.5) into (3.3), and we see that the sum of all of these terms, alongwith (cid:0) n +1 k − ,k ,...,k m (cid:1) corresponding to J = { } , gives the desired result.While the inductive proof of Lemma 3.2 above works nicely, it somewhat disguises the waywe count our flags to see the result bijectively. We conclude with an explanation of this count.First, we need to understand the combinatorial proof of Lemma 3.1 appearing in [11], which maybe summarized as follows. Fix V to be an ( n + 1)-dimensional F q -vector space as before. Thereare (cid:0) n +1 k (cid:1) q ways to choose a k -dimensional subspace W of V . Fix a basis { v , v , . . . , v n +1 } of V .Any k -dimensional subspace W can be written as span( W ′ , v ) where W ′ is a ( k − V ′ = span( v , . . . , v n ). We may choose W in three distinct ways. If v ∈ V ′ , then W is a subspace of V ′ , for which there are (cid:0) nk (cid:1) q choices. Call this a Type 1 subspace of V . Ifwe take v to be a scalar multiple of v n +1 , then W is determined by W ′ , for which there are (cid:0) nk − (cid:1) q choices. We call this a Type 2 subspace of V . Finally, if v is neither in V ′ nor a scalarmultiple of v n +1 , then we call W a Type 3 subspace of V , and it can be shown that there are( q n − (cid:0) n − k − (cid:1) q choices for W , giving Lemma 3.1.We now fix a basis of every subspace U of V , so that we may speak of subspaces of Type1, 2, or 3 of U . Consider a flag of subspaces of V = W , W m − ⊂ · · · ⊂ W ⊂ W , such that ifwe define k i for 1 ≤ i ≤ m by P ij =1 k j = n + 1 − dim W i , then each k i > 0. The total numberof such flags is (cid:0) n +1 k ,...,k m (cid:1) q , and these flags may also be counted in the following way. We maychoose W to be a Type 1 subspace of V , or we may choose every W i to be a Type 2 or Type3 subspace of W i − for i ≤ m − 1, or we may choose W i to be a Type 2 or Type 3 subspace of W i − for i ≤ r − r < m and W r a Type 1 subspace of W r − . These cases accountfor all possibilities for such a flag of V . For a nonempty J ⊆ { , . . . , m } , let r be the maximumelement of J . Then a closer look at the proof of Lemma 3.2 reveals that( − | J |− ( q ) n ( q ) n −| J | +1 (cid:18) n + 1 − | J | k − e J (cid:19) q is the number of ways to choose our flag such that W j is a Type 3 subspace of W j − for j ∈ J and j < r , W i is a Type 2 subspace of W i − for i J and i < r , and W r is a Type 1 subspace of W r − if r < m . These account for all 2 m − Acknowledgments. The author thanks Prof. George Andrews for helpful and encourag-ing comments. This research was supported by NSF grant DMS-0854849. References [1] G. 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Phys.-Math. Kl. , (1926), 242–252.[16] S. O. Warnaar, Rogers-Szeg¨o polynomials and Hall-Littlewood symmetric functions, J.Algebra (2006), no. 2, 810–830. Department of MathematicsCollege of William and MaryP. O. Box 8795Williamsburg, VA 23187 e-mail : [email protected]@math.wm.edu