منابع مشابه
m-Level rook placements
Goldman, Joichi, and White proved a beautiful theorem showing that the falling factorial generating function for the rook numbers of a Ferrers board factors over the integers. Briggs and Remmel studied an analogue of rook placements where rows are replaced by sets of m rows called levels. They proved a version of the factorization theorem in that setting, but only for certain Ferrers boards. We...
متن کاملBijections on m-level rook placements
Suppose the rows of a board are partitioned into sets of m rows called levels. An m-level rook placement is a subset of the board where no two squares are in the same column or the same level. We construct explicit bijections to prove three theorems about such placements. We start with two bijections between Ferrers boards having the same number of m-level rook placements. The first generalizes...
متن کاملPattern Avoiding Permutations & Rook Placements
First, we look at the distribution of permutation statistics in the context of pattern-avoiding permutations. The first part of this chapter deals with a recursively defined bijection of Robertson [6] between 123and 132-avoiding permutations. We introduce the general notion of permutation templates and pivots in order to give a non-recursive pictorial reformulation of Robertson’s bijection. Thi...
متن کاملTwo vignettes on full rook placements
Using bijections between pattern-avoiding permutations and certain full rook placements on Ferrers boards, we give short proofs of two enumerative results. The first is a simplified enumeration of the 3124, 1234avoiding permutations, obtained recently by Callan via a complicated decomposition. The second is a streamlined bijection between 1342-avoiding permutations and permutations which can be...
متن کاملPatterns in matchings and rook placements
Extending the notion of pattern avoidance in permutations, we study matchings and set partitions whose arc diagram representation avoids a given configuration of three arcs. These configurations, which generalize 3-crossings and 3-nestings, have an interpretation, in the case of matchings, in terms of patterns in full rook placements on Ferrers boards. We enumerate 312-avoiding matchings and pa...
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series A
سال: 2014
ISSN: 0097-3165
DOI: 10.1016/j.jcta.2014.01.006