Substring compatibility of permutation statistics

TL;DR

This paper introduces the concept of substring compatibility for permutation statistics, constructs related algebraic structures, and explores their properties and conjectures about their uniqueness.

Contribution

It defines substring-compatible permutation statistics, develops their associated algebraic structures, and proposes a conjecture on their classification.

Findings

Constructed the substring coalgebra for such statistics

Established conditions under which a Hopf algebra can be formed

Proposed a conjecture on the uniqueness of certain permutation statistics

Abstract

A permutation statistic is substring-compatible if its value on a permutation determines its value on every substring of that permutation. We construct the substring coalgebra of such a statistic, an analog of the shuffle algebra of a shuffle-compatible statistic introduced by Gessel and Zhuang. Furthermore, we show that for substring-compatible statistics that also satisfy a weak form of shuffle compatibility, the shuffle algebra and substring coalgebra can be combined to yield a Hopf algebra. Finally, we conjecture that the only nontrivial permutation statistics that are both shuffle-compatible and substring-compatible are the descent set, the peak set, and the valley set, and we describe our progress towards proving this conjecture.