Rook theory. V rook polynomials, Möbius inversion and the umbral calculus

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

doi:10.1016/0097-3165(76)90067-4

Cited by 24 publications

(11 citation statements)

References 8 publications

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“…The original rook polynomial defined in Riordan's book [40] counts the number of ways of arranging nonattacking rooks on a board (a board is a finite subset of N × N). Goldman, Joichi and White [26] conjectured that such rook polynomials have only real zeros. Wilf extended the concept of rook polynomial to the case of arbitrary matrix instead of a board (a board is identified with a (0, 1)-matrix) and Nijenhuis showed that the extended rook polynomials of nonnegative matrices have only real zeros by establishing a result similar to Theorem 2.3 (see [37]).…”

Section: Matching Polynomialsmentioning

confidence: 99%

A unified approach to polynomial sequences with only real zeros

Liu

Wang

2007

Advances in Applied Mathematics

117

103

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We give new sufficient conditions for a sequence of polynomials to have only real zeros based on the method of interlacing zeros. As applications we derive several well-known facts, including the reality of zeros of orthogonal polynomials, matching polynomials, Narayana polynomials and Eulerian polynomials. We also settle certain conjectures of Stahl on genus polynomials by proving them for certain classes of graphs, while showing that they are false in general.Comment: 19 pages, Advances in Applied Mathematics, in pres

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Section: Matching Polynomialsmentioning

confidence: 99%

A unified approach to polynomial sequences with only real zeros

Liu

Wang

2007

Advances in Applied Mathematics

117

103

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“…Our next result generalizes a Mo bius inversion formula for factorial polynomials due to Goldman, Joichi and White [16]. For simplicity we consider only the case of acyclic digraphs, although the generalization to arbitrary digraphs is straightforward.…”

Section: Reciprocitymentioning

confidence: 85%

The Path-Cycle Symmetric Function of a Digraph

Chow

1996

Advances in Mathematics

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“…The theorem below, which is present in [16], will be useful for us. Let where µ denotes the Möbius function of Π m .…”

Section: Rook Placementsmentioning

confidence: 99%

Gessel polynomials, rooks, and extended Linial arrangements

Tewari

2019

Journal of Combinatorial Theory, Series A

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We study a family of polynomials associated with ascent-descent statistics on labeled rooted plane k-ary trees introduced by Gessel, from a rook-theoretic perspective. We generalize the excedance statistic on permutations to maximal nonattacking rook placements on certain rectangular boards by decomposing them into boards of staircase shape. We then relate the number of maximal nonattacking rook placements on certain skew boards to the number of regions in extended Linial arrangements by establishing a relation between the factorial polynomial of those boards to the characteristic polynomial of extended Linial arrangements. Furthermore, we give a combinatorial interpretation of the number of bounded regions in extended Linial arrangements in the setting of labeled rooted plane k-ary trees. Finally, using the work of Goldman-Joichi-White, we identify graphs whose chromatic polynomials equal the characteristic polynomials of extended Linial arrangements upto a straightforward normalization.

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Rook theory. V rook polynomials, Möbius inversion and the umbral calculus

Cited by 24 publications

References 8 publications

A unified approach to polynomial sequences with only real zeros

A unified approach to polynomial sequences with only real zeros

The Path-Cycle Symmetric Function of a Digraph

Gessel polynomials, rooks, and extended Linial arrangements

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