Extending Results on Wilf-Equivalence of Partial Shuffles

TL;DR

This paper provides an alternative proof that all partial shuffles of the same size are Wilf-equivalent, and explores how this equivalence is maintained when including decreasing patterns, with additional enumerative results.

Contribution

It offers a new iterative proof of Wilf-equivalence among partial shuffles and extends the understanding of pattern avoidance with decreasing permutations.

Findings

All partial shuffles of the same size are Wilf-equivalent.

Wilf-equivalence is preserved when including decreasing patterns.

Enumerative results for avoidance classes with partial shuffles and decreasing permutations.

Abstract

In 2020, Bloom and Sagan defined subsets of the symmetric group $\mathfrak{S}_n$ called partial shuffles, and proved a formula for the Schur expansion of the pattern quasisymmetric function associated with a partial shuffle. In their proof, they establish that any two partial shuffles of the same size are Wilf-equivalent. We give an alternative proof of this fact, using an iterative approach. We also show that Wilf-equivalence is preserved on including a decreasing pattern in partial shuffles, and we provide some enumerative results for avoidance classes whose bases consist of a partial shuffle and a decreasing permutation.