Rook Poset Equivalence of Ferrers Boards

Develin, Mike

doi:10.1007/s11083-006-9039-8

Cited by 3 publications

(4 citation statements)

References 8 publications

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“…What is true for skew Ferrers boards? • Develin [7] classified the isomorphism types of lower Bruhat intervals of 312-avoiding permutations by using the connection to rook posets discovered by Ding [8]. What are the isomorphism types of lower Bruhat intervals of permutations that avoid the patterns 4231, 35142, 42513, and 351624?…”

Section: Open Problemsmentioning

confidence: 99%

“…Really small intervals (of length 7 in A n and 5 in B n and D n ) were completely classified by Hultman [13] and Incitti [14]. Lower intervals of 312-avoiding permutations in A n were classified by Develin [7] (though he did not compute their Poincaré polynomials), and for a general lower interval [id, w] in a crystallographic Coxeter group, Björner and Ekedahl [3] showed that the coefficients of Poin [id,w] (q) are partly increasing. Reading [20] studied the cdindex and obtained a recurrence relation for the Poincaré polynomials of the intervals in any Coxeter group [19]; however he did not compute these for any particular intervals.…”

Section: Introductionmentioning

confidence: 99%

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Bruhat intervals as rooks on skew Ferrers boards

Sjöstrand

2007

Journal of Combinatorial Theory, Series A

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We characterise the permutations π such that the elements in the closed lower Bruhat interval [id, π] of the symmetric group correspond to non-taking rook configurations on a skew Ferrers board. It turns out that these are exactly the permutations π such that [id, π] corresponds to a flag manifold defined by inclusions, studied by Gasharov and Reiner.Our characterisation connects the Poincaré polynomials (rank-generating function) of Bruhat intervals with q-rook polynomials, and we are able to compute the Poincaré polynomial of some particularly interesting intervals in the finite Weyl groups A n and B n . The expressions involve q-Stirling numbers of the second kind, and for the group A n putting q = 1 yields the poly-Bernoulli numbers defined by Kaneko.

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Section: Open Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Bruhat intervals as rooks on skew Ferrers boards

Sjöstrand

2007

Journal of Combinatorial Theory, Series A

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“…Ferrers graphs/tableaux have a prominent place in the literature as they have been studied in relation to chromatic polynomials [2,18], Schubert varieties [16,15], hypergeometric series [29], permutation statistics [9,18], quantum mechanical operators [50], and inverse rook problems [23,16,15,42]. More generally, algebraic and combinatorial aspects of bipartite graphs have been studied in depth (see, e.g., [46,30] and the comprehensive monograph [51]).…”

Section: Introductionmentioning

confidence: 99%

Monomial and toric ideals associated to Ferrers graphs

Corso

Nagel

2008

Trans. Amer. Math. Soc.

104

121

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Abstract. Each partition λ = (λ 1 , λ 2 , . . . , λ n ) determines a so-called Ferrers tableau or, equivalently, a Ferrers bipartite graph. Its edge ideal, dubbed a Ferrers ideal, is a squarefree monomial ideal that is generated by quadrics. We show that such an ideal has a 2-linear minimal free resolution; i.e. it defines a small subscheme. In fact, we prove that this property characterizes Ferrers graphs among bipartite graphs. Furthermore, using a method of Bayer and Sturmfels, we provide an explicit description of the maps in its minimal free resolution. This is obtained by associating a suitable polyhedral cell complex to the ideal/graph. Along the way, we also determine the irredundant primary decomposition of any Ferrers ideal. We conclude our analysis by studying several features of toric rings of Ferrers graphs. In particular we recover/establish formulae for the Hilbert series, the Castelnuovo-Mumford regularity, and the multiplicity of these rings. While most of the previous works in this highly investigated area of research involve path counting arguments, we offer here a new and self-contained approach based on results from Gorenstein liaison theory.

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“…Ferrers graphs/tableaux have a prominent place in the literature as they have been studied in relation to chromatic polynomials [4,23], Schubert varieties [21,18], hypergeometric series [33], permutation statistics [9,23], quantum mechanical operators [64], and inverse rook problems [27,21,18]. More generally, algebraic and combinatorial aspects of bipartite graphs have been studied in depth (see, e.g., [61,35,25,13,14,49,22] and the comprehensive monographs [36,66]).…”

Section: Introductionmentioning

confidence: 99%

Blow-up algebras, determinantal ideals, and Dedekind–Mertens-like formulas

et al. 2016

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We give a combinatorial proof of the Dedekind-Mertens formula by computing the initial ideal of the content ideal of the product of two generic polynomials. As a side effect we obtain a complete classification of the rank 1 Cohen-Macaulay modules over the determinantal rings K[X]/I 2 (X). Let f, g be polynomials in one indeterminate over a commutative ring A. The Dedekind-Mertens formula relates the content ideals of f , g, and their product f g: one has c(f g)c(f) d = c(g)c(f) d+1 , d= deg g. It is the best universally valid variant of Gauß' classical formula c(f g) = c(f)c(g) for polynomials over a principal ideal domain. (The content ideal of f ∈ A[T ] is the ideal generated by the coefficients of f in A.) Content ideals and the Dedekind-Mertens formula have recently received much attention; see Glaz and Vasconcelos [8], Corso, Vasconcelos, and Villarreal [6] and Heinzer and Huneke [9], [10]. For detailed historical information about the Dedekind-Mertens formula, see [9]. The main objective of this paper is a combinatorial proof of the formula based on a Gröbner basis approach to the ideal c(f g) for polynomials with indeterminate coefficients; in fact we will determine the initial ideal of c(f g) with respect to a suitable term order. (For information on term orders and Gröbner bases we refer the reader to Eisenbud [7].) A side effect of our approach is very precise numerical information about the rank one Cohen-Macaulay modules over the determinantal ring S = K[X]/I 2 (X) where X is an m × n matrix of indeterminates and I 2 (X) the ideal generated by its 2-minors. This connection extends the ideas of [6] and was in fact suggested by them. The actual motive for our work was the need for some explicit computation modulo c(f g) in Boffi, Bruns, and Guerrieri [2], or, more precisely, modulo an ideal generalizing c(f g) slightly. Theorem 1. Let K be a field, R = K[

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Rook Poset Equivalence of Ferrers Boards

Cited by 3 publications

References 8 publications

Bruhat intervals as rooks on skew Ferrers boards

Bruhat intervals as rooks on skew Ferrers boards

Monomial and toric ideals associated to Ferrers graphs

Blow-up algebras, determinantal ideals, and Dedekind–Mertens-like formulas

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