A survey on partially ordered patterns

doi:10.1017/CBO9780511902499.006

A survey on partially ordered patterns

Published online by Cambridge University Press: 05 October 2010

Edited by

Nik Ruškuc and

Sergey Kitaev: Affiliation:
The Mathematics Institute Reykjavík University IS-103 Reykjavík, Iceland
Steve Linton: Affiliation:
University of St Andrews, Scotland
Nik Ruškuc: Affiliation:
University of St Andrews, Scotland
Vincent Vatter: Affiliation:
Dartmouth College, New Hampshire

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Summary

Abstract

The paper offers an overview over selected results in the literature on partially ordered patterns (POPs) in permutations, words and compositions. The POPs give rise in connection with co-unimodal patterns, peaks and valleys in permutations, Horse permutations, Catalan, Narayana, and Pell numbers, bi-colored set partitions, and other combinatorial objects.

Introduction

An occurrence of a pattern τ in a permutation π is defined as a subsequence in π (of the same length as τ) whose letters are in the same relative order as those in τ. For example, the permutation 31425 has three occurrences of the pattern 1-2-3, namely the subsequences 345, 145, and 125. Generalized permutation patterns (GPs) being introduced in allow the requirement that some adjacent letters in a pattern must also be adjacent in the permutation. We indicate this requirement by removing a dash in the corresponding place. Say, if pattern 2-31 occurs in a permutation π, then the letters in π that correspond to 3 and 1 are adjacent. For example, the permutation 516423 has only one occurrence of the pattern 2-31, namely the subword 564, whereas the pattern 2-3-1 occurs, in addition, in the subwords 562 and 563. Placing “[” on the left (resp., “]” on the right) next to a pattern p means the requirement that p must begin (resp., end) from the leftmost (resp., rightmost) letter.

Type: Chapter
Information: Permutation Patterns , pp. 115 - 136

DOI: https://doi.org/10.1017/CBO9780511902499.006 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2010

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A survey on partially ordered patterns

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