Extending Results on Wilf-Equivalence of Partial Shuffles

TL;DR

The paper addresses Wilf-equivalence for partial shuffles in $\mathfrak{S}_n$, building on Bloom–Sagan by providing an alternative iterative proof that all partial shuffles of the same size are Wilf-equivalent. It introduces the $S$-map and a sequence of lemmas to establish a bijection between avoidance classes $\mathrm{Av}_n(\Pi(a,b))$ and $\mathrm{Av}_n(\Pi(a-1,b+1))$, then extends the Wilf-equivalence to any pair with the same total size via reverse-complement symmetry. It further shows that Wilf-equivalence is preserved when a decreasing pattern $\delta_m$ is added to the basis, and provides enumeration results: for large $n$, $|\mathrm{Av}_n(\Pi(a,b),\delta_m)|$ is a polynomial in $n$ of degree $(a+b-2)(m-2)$, with leading Catalan coefficient $C_{a+b-2}$ when $m=3$. The work combines permutation diagrams, injective mappings, symmetry, and peg-permutation geometry to derive both structural and enumerative consequences for these avoidance classes.

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.

Figures (12)

• Figure 1: The permutation $582916743\in \mathrm{Av}_9(\Pi(3,1))$. Of the patterns from $\Pi(2,2)$, the smallest $a$ element is shown in blue, where $a = 3$, and all associated $a-1$ elements are shown in red.
• Figure 2: If $\underline{a}$ is the smallest $a$ from any pattern in $\Pi(a-1,b+1)$ appearing in a permutation $\pi$, and $r$ plays the role of the rightmost $a-2$, then nothing may occur in the shaded region, and any element $q$ such that $r<q<\underline{a}$ is an $a-1$ element associated with $\underline{a}$.
• Figure 3: $\pi = 582916743 \in \mathrm{Av}(\Pi(3,1))$ and its image $S(\pi)=683912754$.
• Figure 4: Left: If $S(\underline{a})$ is the $a-1$ element of $\sigma_{a,b}$, and $m<t$, then $\pi$ contains $\sigma_{a,b}$. Right: If $m>t$ in $\pi$, then $m$ cannot be below $S(\underline{a})$ in $S(\pi)$.
• Figure 5: Left: Suppose $S(\underline{a})$ is the $a$ element of $\sigma_{a,b}$. If $m$ is to the right of $(a-1)_\sigma$, then $\pi$ contains $\sigma_{a,b}$. Right: If $m$ is to the left of $(a-1)_\sigma$ then $(a-1)_\sigma$ is an $a-1$ associated with $\underline{a}$ in $\pi$.

Theorems & Definitions (33)

• Example 2.1
• Lemma 2.2
• proof
• Definition 2.3
• Example 2.4
• Corollary 2.5
• Definition 2.6
• Lemma 2.7
• proof
• Lemma 2.8