Strictly increasing and decreasing sequences in subintervals of words and a conjecture of Guo and Poznanovi\'c

TL;DR

This paper proves a conjecture about chains in 01-fillings of moon polyominoes by establishing a novel correspondence between words and pairs of increasing tableaux, revealing sequence length properties in subintervals.

Contribution

It introduces a new correspondence between words and pairs of increasing tableaux using K-infusion, and develops variants of RSK and Knuth equivalence to prove the conjecture.

Findings

Proves Guo and Poznanović's conjecture on chains in 01-fillings.

Establishes a correspondence linking sequence lengths to tableaux.

Develops new variants of RSK and Knuth equivalence.

Abstract

We prove a conjecture of Guo and Poznanovi\'{c} concerning chains in certain 01-fillings of moon polyominoes. A key ingredient of our proof is a correspondence between words $w$ and pairs $(\mathcal{W}(w), \mathcal{M}(w))$ of increasing tableaux such that $\mathcal{M}(w)$ determines the lengths of the longest strictly increasing and strictly decreasing sequences in every subinterval of $w$. We define this correspondence by using Thomas and Yong's K-infusion operator and then use it to obtain the bijections that prove the conjecture of Guo and Poznanovi\'{c}. In constructing our bijections we introduce new variants of the RSK correspondence and Knuth equivalence.