A $p,q$-analogue of a Formula of Frobenius

Abstract

Garsia and Remmel (JCT. A 41 (1986), 246-275) used rook configurations to give a combinatorial interpretation to the $q$-analogue of a formula of Frobenius relating the Stirling numbers of the second kind to the Eulerian polynomials. Later, Remmel and Wachs defined generalized $p,q$-Stirling numbers of the first and second kind in terms of rook placements. Additionally, they extended their definition to give a $p,q$-analogue of rook numbers for arbitrary Ferrers boards. In this paper, we use Remmel and Wach's definition and an extension of Garsia and Remmel's proof to give a combinatorial interpretation to a $p,q$-analogue of a formula of Frobenius relating the $p,q$-Stirling numbers of the second kind to the trivariate distribution of the descent number, major index, and comajor index over $S_n$. We further define a $p,q$-analogue of the hit numbers, and show analytically that for Ferrers boards, the $p,q$-hit numbers are polynomials in $(p,q)$ with nonnegative coefficients.

Elliptic extensions of the alpha-parameter model and the rook model for matchings

“…, n} with k cycles, or the signless Stirling numbers of the first kind, denoted by c(n, k). Hence, r(1) n−k (a, b; q, p; St n ) can be defined as an elliptic analogue of c(n, k). Let us use the notation c a,b;q,p (n, k) := r(1) n−k (a, b; q, p; St n ).…”

“…Hence, r(1) n−k (a, b; q, p; St n ) can be defined as an elliptic analogue of c(n, k). Let us use the notation c a,b;q,p (n, k) := r(1) n−k (a, b; q, p; St n ). The recurrence relation in Proposition 3.1 gives a recurrence relation for c a,b;q,p (n, k):c a,b;q,p (n + 1, k) = [n] aq −2n ,bq −n c a,b;q,p (n, k) + W aq −2n ,bq −n (n)c a,b;q,p (n, k − 1), (3.4)with the initial conditions c a,b;q,p (0, 0) = 1 and c a,b;q,p (n, k) = 0 for k < 0 or k > n. Furthermore, if we consider the truncated staircase board St(r) n = B(b 1 , .…”

“…[z] a,b;q,p ([z + an] aq −2an ,bq −2an ;q,p ) n−1 = (a, b; q, p; A n )([z] a,b;q,p ) k .In particular, r(1) n−k (a, b; q, p; A n ) has a nice closed form expressionr (1) n−k (a, b; q, p; A n ) = n − 1 k − 1 (W aq −2an ,bq −an ;q,p (an)) k−1 ([an] aq −2an ,bq −an ;q,p ) n−k which can easily be proved by the recurrence relation (3.2). For a detailed description of the combinatorial interpretation for f n−k (A n ) and more general cases, see [17, Section 5.3].…”

Elliptic Rook and File Numbers

“…For a concrete example of a rook configuration mapped to a placement of elements in tubes, see Figure 5, where we have chosen n = 8, r = 2, k = 4. We consider a placement , the rooks being in the cells (9, 6), (3,5), (6,3), and (8, 1) (from top to bottom). We start with putting the numbers 1 and 2 into the (the first) two distinct tubes.…”

“…Among other results, they were in particular able to extend the product formula in (1.1) to the q-case. In 1991 Wachs and White [44] introduced a (p, q)-analogue of rook numbers which was later studied in more detail by Briggs and Remmel [3] and by Remmel and Wachs [32]. In 2001, Haglund and Remmel [23] considered a suitably modified statistic involving rook cancellation on shifted Ferrers boards.…”