Increasing and Decreasing Sequences in Fillings of Moon Polyominoes

Maximal 0-1 fillings of moon polyominoes with restricted chain-lengths and rc-graphs

We show that maximal 0-1-fillings of moon polynomials with restricted chain lengths can be identified with certain rc-graphs, also known as pipe dreams. In particular, this exhibits a connection between maximal 0-1-fillings of Ferrers shapes and Schubert polynomials. Moreover, it entails a bijective proof showing that the number of maximal fillings of a stack polyomino S with no north-east chains longer than k depends only on k and the multiset of column heights of S. Our main contribution is a slightly stronger theorem, which in turn leads us to conjecture that the poset of rc-graphs with cov…

Hecke增长图，以及堆叠多联骨牌填充中的最大递增和递减序列

Hecke growth diagrams, and maximal increasing and decreasing sequences in fillings of stack polyominoes

本文通过建立堆叠多联骨牌$01$-填充与拐角标记的双射，推广了增长图技术，为最长递增递减链对称分布定理提供了新证明与视角。

Hecke insertion and maximal increasing and decreasing sequences in fillings of stack polyominoes

We prove that the number of 01-fillings of a given stack polyomino (a polyomino with justified rows whose lengths form a unimodal sequence) with at most one 1 per column which do not contain a fixed-size northeast chain and a fixed-size southeast chain, depends only on the set of row lengths of the polyomino. The proof is via a bijection between fillings of stack polyominoes which differ only in the position of one row and uses the Hecke insertion algorithm by Buch, Kresch, Shimozono, Tamvakis, and Yong and the jeu de taquin for increasing tableaux of Thomas and Yong. Moreover, our bijection g…

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 $n$ rows and $m$ 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 $S \subseteq \{1,2,..., n\}$ and $\mathcal{R}(S)$ be the union of rows whose indices are in $S$. For any filling $M$, the top-mixed (resp. bottom-mixed) statistic $α(S; M)$ (resp. $β(S; M)$) is the sum of the numbe…

Growth diagrams, and increasing and decreasing chains in fillings of Ferrers shapes

We put recent results by Chen, Deng, Du, Stanley and Yan on crossings and nestings of matchings and set partitions in the larger context of the enumeration of fillings of Ferrers shape on which one imposes restrictions on their increasing and decreasing chains. While Chen et al. work with Robinson-Schensted-like insertion/deletion algorithms, we use the growth diagram construction of Fomin to obtain our results. We extend the results by Chen et al., which, in the language of fillings, are results about $0$-$1$-fillings, to arbitrary fillings. Finally, we point out that, very likely, these resu…

词子区间中的严格递增和严格递减序列以及Guo和Poznanović的一个猜想

Strictly increasing and decreasing sequences in subintervals of words and a conjecture of Guo and Poznanović

我们证明了Guo和Poznanović关于月亮多联骨牌01填充中链的猜想，通过构造词与递增表对之间的对应关系，并利用K-融合算子获得双射，同时引入RSK对应和Knuth等价的新变体。