Pattern Avoiding Linear Extensions of Rectangular Posets

TL;DR

This paper explores pattern avoiding linear extensions of rectangular posets, connecting them to standard tableaux, and provides enumeration results for various pattern avoidances using combinatorial methods.

Contribution

It introduces new enumeration techniques for pattern avoiding linear extensions of rectangular posets, including Fuss-Catalan number connections and transfer matrix methods.

Findings

Linear extensions avoiding certain patterns are enumerated.

The EN rectangular order avoiding 1243 is counted by Fuss-Catalan numbers.

Enumeration of linear extensions avoiding 2143 using transfer matrix method.

Abstract

Inspired by Yakoubov's 2015 investigation of pattern avoiding linear extensions of the posets called combs, we study pattern avoiding linear extensions of rectangular posets. These linear extensions are closely related to standard tableaux. For positive integers $s$ and $t$ we consider two natural rectangular partial orders on $\{1,2,\ldots,s t\}$, which we call the NE rectangular order and the EN rectangular order. First we enumerate linear extensions of both rectangular orders avoiding most sets of patterns of length three. Then we use both a generating tree and a bijection to show that the linear extensions of the EN rectangular order which avoid 1243 are counted by the Fuss-Catalan numbers. Next we use the transfer matrix method to enumerate linear extensions of the EN rectangular order which avoid 2143. Finally, we open an investigation of the distribution of the inversion number on pattern avoiding linear extensions.