Cycles and sorting index for matchings and restricted permutations

We prove that the Mahonian-Stirling pairs of permutation statistics (\sor, \cyc) and (\inv, \mathrm{rlmin}) are equidistributed on the set of permutations that correspond to arrangements of $n$ non-atacking rooks on a Ferrers board with $n$ rows and $n$ columns. The proofs are combinatorial and use bijections between matchings and Dyck paths and a new statistic, sorting index for matchings, that we define. We also prove a refinement of this equidistribution result which describes the minimal elements in the permutation cycles and the right-to-left minimum letters. Moreover, we define a sorting index for bicolored matchings and use it to show analogous equidistribution results for restricted permutations of type $B_n$ and $D_n$.Comment: 23 page

Maximal increasing sequences in fillings of almost-moon polyominoes

It was proved by Rubey that the number of fillings with zeros and ones of a given moon polyomino that do not contain a northeast chain of size $k$ depends only on the set of columns of the polyomino, but not the shape of the polyomino. Rubey's proof is an adaption of jeu de taquin and promotion for arbitrary fillings of moon polyominoes. In this paper we present a bijective proof for this result by considering fillings of almost-moon polyominoes, which are moon polyominoes after removing one of the rows. Explicitly, we construct a bijection which preserves the size of the largest northeast chains of the fillings when two adjacent rows of the polyomino are exchanged. This bijection also preserves the column sum of the fillings. We also present a bijection that preserves the size of the largest northeast chains, the row sum and the column sum if every row of the fillings has at most one 1.Comment: 18 page

Major Index for 01-Fillings of Moon Polyominoes

We propose a major index statistic on 01-fillings of moon polyominoes which, when specialized to certain shapes, reduces to the major index for permutations and set partitions. We consider the set F(M, s; A) of all 01-fillings of a moon polyomino M with given column sum s whose empty rows are A, and prove that this major index has the same distribution as the number of north-east chains, which are the natural extension of inversions (resp. crossings) for permutations (resp. set partitions). Hence our result generalizes the classical equidistribution results for the permutation statistics inv and maj. Two proofs are presented. The first is an algebraic one using generating functions, and the second is a bijection on 01-fillings of moon polyominoes in the spirit of Foata's second fundamental transformation on words and permutations.Comment: 28 pages, 14 figure

Crossings and Nestings of Two Edges in Set Partitions

AMS subject classifications. 05A18, 05A15 Let π and λ be two set partitions with the same number of blocks. Assume π is a partition of [n]. For any integer l, m ≥ 0, let T (π, l) be the set of partitions of [n + l] whose restrictions to the last n elements are isomorphic to π, and T (π, l, m) the subset of T (π, l) consisting of those partitions with exactly m blocks. Similarly define T (λ, l) and T (λ, l, m). We prove that if the statistic cr (ne), the number of crossings (nestings) of two edges, coincides on the sets T (π, l) and T (λ, l) for l = 0, 1, then it coincides on T (π, l, m) and T (λ, l, m) for all l, m ≥ 0. These results extend the ones obtained by Klazar on the distribution of crossings and nestings for matchings. 1 Introduction and Statement of Main Result In a recent paper [5], Klazar studied distributions of the numbers of crossings and nestings of two edges in (perfect) matchings. All matchings form an infinite tree T rooted at the empty matching ∅, in which the children of a matching M are the matchings obtained from M by adding to M in all possible ways a new first edge. Given two matchings M and N on [2n], Klazar decided whe