Results 1 to 10 of about 142 (85)
Invisible permutations and rook placements on a Ferrers board
Discrete Mathematics, 1995 We study non-attacking rook placements on a Ferrers board by extending them to certain permutation matrices, called invisible permutations. By introducing a length function on rook placements, we define a rook length polynomial \(\text{RL}_ k(\lambda, q)\) for a given number \(k\) of rooks and a Ferrers board \(F_ \lambda\). We prove that \(\text{R}_ k(
Kequan Ding
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Descents of Permutations in a Ferrers Board [PDF]
The Electronic Journal of Combinatorics, 2012 The classical Eulerian polynomials are defined by setting $$A_n(t)= \sum_{\sigma \in \mathfrak{S}_n} t^{1+\mathrm{des}(\sigma)}= \sum_{k=1}^n A_{n,k} t^k$$where $A_{n,k}$ is the number of permutations of length $n$ with $k-1$ descents. Let $A_n(t, q) = \sum_{\pi \in \mathfrak{S}_n} t^{1+{\rm des}(\pi)}q^{{\rm inv}(\pi)} $ be the $\mathrm{inv}$ $q ...
Chunwei Song, Catherine Yan 0001
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Derangements on a Ferrers board [PDF]
Discrete Mathematics, Algorithms and Applications, 2015 We study the derangement number on a Ferrers board B = (n × n) - λ with respect to an initial permutation M, that is, the number of permutations on B that share no common points with M. We prove that the derangement number is independent of M if and only if λ is of rectangular shape.
William Linz, Catherine Yan 0001
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Rook Poset Equivalence of Ferrers Boards [PDF]
Order, 2006 A natural construction due to K. Ding yields Schubert varieties from Ferrers boards. The poset structure of the Schubert cells in these varieties is equal to the poset of maximal rook placements on the Ferrers board under the Bruhat order. We determine when two Ferrers boards have isomorphic rook posets. Equivalently, we give an exact categorization of
Mike Develin
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Bijections on m-level Rook Placements [PDF]
Discrete Mathematics & Theoretical Computer Science, 2014 Partition the rows of a board into sets of $m$ rows called levels. An $m$-level rook placement is a subset of squares of the board with no two in the same column or the same level.
Kenneth Barrese, Bruce Sagan
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Cycles and sorting index for matchings and restricted permutations [PDF]
Discrete Mathematics & Theoretical Computer Science, 2013 We prove that the Mahonian-Stirling pairs of permutation statistics $(sor, cyc)$ and $(∈v , \mathrm{rlmin})$ are equidistributed on the set of permutations that correspond to arrangements of $n$ non-atacking rooks on a fixed Ferrers board with $n$ rows ...
Svetlana Poznanović
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On criteria for rook equivalence of Ferrers boards [PDF]
European Journal of Combinatorics, 2019 In [2] we introduced a new notion of Wilf equivalence of integer partitions and proved that rook equivalence implies Wilf equivalence. In the present paper we prove the converse and thereby establish a new criterion for rook equivalence. We also refine two of the standard criteria for rook equivalence and establish another new one involving what we ...
Jonathan Bloom, Dan Saracino
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Simplicial complexes of triangular Ferrers boards [PDF]
Journal of Algebraic Combinatorics, 2012 We study the simplicial complex that arises from non-attacking rook placements on a subclass of Ferrers boards that have $a_i$ rows of length $i$ where $a_i>0$ and $i\leq n$ for some positive integer $n$. In particular, we will investigate enumerative properties of their facets, their homotopy type, and homology.
Clark, Eric, Zeckner, Matthew
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Rook Theory. I.: Rook Equivalence of Ferrers Boards [PDF]
Proceedings of the American Mathematical Society, 1975 We introduce a new tool, the factorial polynomials, to study rook equivalence of Ferrers boards. We provide a set of invariants for rook equivalence as well as a very simple algorithm for deciding rook equivalence of Ferrers boards. We then count the number of Ferrers boards rook equivalent to a given Ferrers board.
Goldman, Jay R. +2 more
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A random walk on the rook placements on a Ferrer's board [PDF]
The Electronic Journal of Combinatorics, 1996 Let $B$ be a Ferrers board, i.e., the board obtained by removing the Ferrers diagram of a partition from the top right corner of an $n\times n$ chessboard. We consider a Markov chain on the set $R$ of rook placements on $B$ in which you can move from one placement to any other legal placement obtained by switching the columns in which two rooks ...
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