(a,b)-rectangle patterns in permutations and words

TL;DR

This paper introduces and analyzes $(a,b)$-rectangle patterns in permutations and words, providing enumeration formulas, distribution results, and confirming several conjectures related to pattern occurrences and combinatorial structures.

Contribution

It generalizes the concept of pattern occurrences in permutations to $(a,b)$-rectangle patterns, extends these notions to words and matrices, and proves several conjectures in combinatorics.

Findings

Enumeration formulas for pattern-avoiding signed permutations

Distribution results for rectangle patterns on words with small alphabets

Confirmation of conjectures on LEGO walls and enumeration of specific integer sequences

Abstract

In this paper, we introduce the notion of a $(a,b)$-rectangle pattern on permutations that not only generalizes the notion of successive elements (bonds) in permutations, but is also related to mesh patterns introduced recently by Br\"and\'en and Claesson. We call the $(k,k)$-rectangle pattern the $k$-box pattern. To provide an enumeration result on the maximum number of occurrences of the 1-box pattern, we establish an enumerative result on pattern-avoiding signed permutations. Further, we extend the notion of $(k,\ell)$-rectangle patterns to words and binary matrices, and provide distribution of $(1,\ell)$-rectangle patterns on words; explicit formulas are given for up to 7 letter alphabets where $\ell \in \{1,2\}$, while obtaining distributions for larger alphabets depends on inverting a matrix we provide. We also provide similar results for the distribution of bonds over words. As a corollary to our studies we confirm a conjecture of Mathar on the number of "stable LEGO walls" of width 7 as well as prove three conjectures due to Hardin and a conjecture due to Barker. We also enumerate two sequences published by Hardin in the On-Line Encyclopedia of Integer Sequences.