16 research outputs found
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
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 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
Increasing and Decreasing Sequences in Fillings of Moon Polyominoes
We present an adaptation of jeu de taquin and promotion for arbitrary
fillings of moon polyominoes. Using this construction we show various symmetry
properties of such fillings taking into account the lengths of longest
increasing and decreasing chains. In particular, we prove a conjecture of Jakob
Jonsson. We also relate our construction to the one recently employed by
Christian Krattenthaler, thus generalising his results.Comment: fixed typo
Mixed Statistics on 01-Fillings of Moon Polyominoes
We establish a stronger symmetry between the numbers of northeast and
southeast chains in the context of 01-fillings of moon polyominoes. Let \M be
a moon polyomino with rows and columns. Consider all the 01-fillings of
\M in which every row has at most one 1. We introduce four mixed statistics
with respect to a bipartition of rows or columns of \M. More precisely, let
and be the union of rows whose
indices are in . For any filling , the top-mixed (resp. bottom-mixed)
statistic (resp. ) is the sum of the number of
northeast chains whose top (resp. bottom) cell is in , together
with the number of southeast chains whose top (resp. bottom) cell is in the
complement of . Similarly, we define the left-mixed and
right-mixed statistics and , where is a subset
of the column index set . Let be any of these
four statistics , , and , we show that the joint distribution of the pair is symmetric and independent of the subsets . In
particular, the pair of statistics is
equidistributed with (\se(M),\ne(M)), where \se(M) and are the
numbers of southeast chains and northeast chains of , respectively.Comment: 20 pages, 6 figure
Research on Combinatorial Statistics: Crossings and Nestings in Discrete Structures
We study the distribution of combinatorial statistics that exhibit a structure of crossings and nesting in various discrete structures, in particular, in set partitions, matchings, and fillings of moon polyominoes with entries 0 and 1. Let pi and y be two set partitions with the same number of blocks. Assume pi is a partition of [n]. For any integers l, m >̲ 0, let T (pi, l) be the set of partitions of [n + l] whose restrictions to the last n elements are isomorphic to pi, and T (pi, l, m) the subset of T (pi, l) consisting of those partitions with exactly m blocks. Similarly define T (pi, l) and T (y, l, m). We prove that if the statistic cr (ne), the number of crossings (nestings) of two edges, coincides on the sets T (pi, l) and T (pi, l) for l = 0; 1, then it coincides on T (pi, l, m) and T (y, l, m) for all l, m >̲ 0. These results extend the ones obtained by Klazar on the distribution of crossings and nestings for matchings. Moreover, we give a bijection between partially directed paths in the symmetric wedge y = +̲ x and matchings, which sends north steps to nestings. This gives a bijective proof of a result of E. J. Janse van Rensburg, T. Prellberg, and A. Rechnitzer that was first discovered through the corresponding generating functions: the number of partially directed paths starting at the origin confined to the symmetric wedge y = +̲ x with k north steps is equal to the number of matchings on [2n] with k nestings. Furthermore, 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