From Hertzsprung's problem to pattern-rewriting systems

TL;DR

This paper generalizes the enumeration of permutation patterns inspired by Hertzsprung's problem, introduces pattern-rewriting systems, and solves open problems related to permutation equivalences.

Contribution

It develops a framework combining cluster and transfer-matrix methods to analyze Hertzsprung patterns and applies pattern-rewriting systems to permutation classification.

Findings

Derived joint distribution formulas for Hertzsprung pattern occurrences.

Counted permutations under pattern-replacement equivalences.

Solved open problems on permutation pattern rewriting systems.

Abstract

Drawing on a problem posed by Hertzsprung in 1887, we say that a given permutation $\pi\in\mathcal{S}_n$ contains the Hertzsprung pattern $\sigma\in\mathcal{S}_k$ if there is factor $\pi(d+1)\pi(d+2)\cdots\pi(d+k)$ of $\pi$ such that $\pi(d+1)-\sigma(1) =\cdots = \pi(d+k)-\sigma(k)$. Using a combination of the Goulden-Jackson cluster method and the transfer-matrix method we determine the joint distribution of occurrences of any set of (incomparable) Hertzsprung patterns, thus substantially generalizing earlier results by Jackson et al. on the distribution of ascending and descending runs in permutations. We apply our results to the problem of counting permutations up to pattern-replacement equivalences, and using pattern-rewriting systems -- a new formalism similar to the much studied string-rewriting systems -- we solve a couple of open problems raised by Linton et al. in 2012.