Place-difference-value patterns: A generalization of generalized permutation and word patterns

TL;DR

This paper introduces place-difference-value patterns (PDVP), a highly general framework that encompasses all known permutation and word pattern types, enabling new theoretical developments and connections in combinatorics.

Contribution

The paper proposes PDVP, a unifying pattern concept that generalizes existing permutation and word patterns, and explores its potential for new combinatorial insights.

Findings

PDVP covers all common permutation and word pattern definitions.

Examples of PDVP demonstrate patterns not describable by previous definitions.

Raises bijective questions linking PDVP to other combinatorial objects.

Abstract

Motivated by study of Mahonian statistics, in 2000, Babson and Steingrimsson introduced the notion of a "generalized permutation pattern" (GP) which generalizes the concept of "classical" permutation pattern introduced by Knuth in 1969. The invention of GPs led to a large number of publications related to properties of these patterns in permutations and words. Since the work of Babson and Steingrimsson, several further generalizations of permutation patterns have appeared in the literature, each bringing a new set of permutation or word pattern problems and often new connections with other combinatorial objects and disciplines. For example, Bousquet-Melou et al. introduced a new type of permutation pattern that allowed them to relate permutation patterns theory to the theory of partially ordered sets. In this paper we introduce yet another, more general definition of a pattern, called place-difference-value patterns (PDVP) that covers all of the most common definitions of permutation and/or word patterns that have occurred in the literature. PDVPs provide many new ways to develop the theory of patterns in permutations and words. We shall give several examples of PDVPs in both permutations and words that cannot be described in terms of any other pattern conditions that have been introduced previously. Finally, we raise several bijective questions linking our patterns to other combinatorial objects.