Rook Theory—IV. Orthogonal Sequences of Rook Polynomials

Goldman, Jay R.; Joichi, J. T.; White, Dennis E.

doi:10.1002/sapm1977563267

Cited by 23 publications

(9 citation statements)

References 5 publications

(5 reference statements)

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“…Goldman et al [14] then introduced a new version of the rook polynomial of a Ferrers board and gave it a combinatorial interpretation, which showed that it had all integer roots. Subsequent work with Ferrers boards and this rook polynomial have led to models for binomial type theorems [17] and to connections with chromatic polynomials [15], orthogonal polynomials [12,16], hypergeometric series [19], q-analogues and permutation statistics [9,11,20], statistical problems in probability [6], algebraic geometry [7], and digraph polynomials [3,5]. Today there is a very large literature about Ferrers boards.…”

Section: Introductionmentioning

confidence: 97%

Generalized Rook Polynomials

Goldman

Haglund

2000

Journal of Combinatorial Theory, Series A

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dedicated to the memory of gian-carlo rotaGeneralizing the notion of placing rooks on a Ferrers board leads to a new class of combinatorial models and a new class of rook polynomials. Connections are established with absolute Stirling numbers and permutations, Bessel polynomials, matchings, multiset permutations, hypergeometric functions, Abel polynomials and forests, and polynomial sequences of binomial type. Factorization and reciprocity theorems are proved and a q-analogue is given. Academic Press

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Section: Introductionmentioning

confidence: 97%

Generalized Rook Polynomials

Goldman

Haglund

2000

Journal of Combinatorial Theory, Series A

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“…Since the introduction of rook theory by Kaplansky and Riordan [12], the theory has thrived and developed further by revealing connections to, for instance, orthogonal polynomials [7,9], hypergeometric series [10], q-analogues and permutation statistics [2,5], algebraic geometry [3,4], and many more. Within rook theory itself, various models have been introduced, including a p, q-analogue of rook numbers [1,14,20], the j-attacking model [14], the matching model [11], the augmented rook model [13] which includes all other models as special cases, etc.…”

Section: Introductionmentioning

confidence: 99%

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

Schlosser

Yoo

2017

Advances in Applied Mathematics

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We construct elliptic extensions of the alpha-parameter rook model introduced by Goldman and Haglund and of the rook model for matchings of Haglund and Remmel. In particular, we extend the product formulas of these models to the elliptic setting. By specializing the parameter α in our elliptic extension of the alpha-parameter model and the shape of the Ferrers board in different ways, we obtain elliptic analogues of the Stirling numbers of the first kind and of the Abel polynomials, and obtain an a, q-analogue of the matching numbers. We further generalize the rook theory model for matchings by introducing l-lazy graphs which correspond to l-shifted boards, where l is a finite vector of positive integers. The corresponding elliptic product formula generalizes Haglund and Remmel's product formula for matchings already in the non-elliptic basic case.

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“…163-237]. For some recent work on rook polynomials, see Goldman, Joichi, and White [10][11][12][13][14] and Joni and Rota [18]. Let B be a subset of [n] × [n], where [n] = {1, 2, .…”

Section: Rook Polynomialsmentioning

confidence: 99%