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A survey on partially ordered patterns

Published online by Cambridge University Press:  05 October 2010

By
Sergey Kitaev
Edited by
Steve Linton ,
Nik Ruškuc  and
Vincent Vatter
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
Publisher: Cambridge University Press
Print publication year: 2010

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References

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