Counting permutations by the number of successions within cycles

Toufik Mansour; Mark Shattuck

doi:https://doi.org/10.1016/j.disc.2015.12.004

Counting permutations by the number of successions within cycles

Toufik Mansour, Mark Shattuck

Department of Mathematics

University of Haifa

Research output: Contribution to journal › Article › peer-review

Abstract

In this note, we consider the problem of counting (cycle) successions, i.e., occurrences of adjacent consecutive elements within cycles, of a permutation expressed in the standard form. We find an explicit formula for the number of permutations having a prescribed number of cycles and cycle successions, providing both algebraic and combinatorial proofs. As an application of our ideas, it is possible to obtain explicit formulas for the joint distribution on S_n for the statistics recording the number of cycles and adjacencies of the form j,j+d where d>0 which extends earlier results.

Original language	American English
Pages (from-to)	1368-1376
Number of pages	9
Journal	Discrete Mathematics
Volume	339
Issue number	4
DOIs	https://doi.org/10.1016/j.disc.2015.12.004
State	Published - 6 Apr 2016

Keywords

Combinatorial proof
Permutation
Stirling number of first kind
Succession

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Discrete Mathematics and Combinatorics

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https://doi.org/10.1016/j.disc.2015.12.004

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N2 - In this note, we consider the problem of counting (cycle) successions, i.e., occurrences of adjacent consecutive elements within cycles, of a permutation expressed in the standard form. We find an explicit formula for the number of permutations having a prescribed number of cycles and cycle successions, providing both algebraic and combinatorial proofs. As an application of our ideas, it is possible to obtain explicit formulas for the joint distribution on Sn for the statistics recording the number of cycles and adjacencies of the form j,j+d where d>0 which extends earlier results.

AB - In this note, we consider the problem of counting (cycle) successions, i.e., occurrences of adjacent consecutive elements within cycles, of a permutation expressed in the standard form. We find an explicit formula for the number of permutations having a prescribed number of cycles and cycle successions, providing both algebraic and combinatorial proofs. As an application of our ideas, it is possible to obtain explicit formulas for the joint distribution on Sn for the statistics recording the number of cycles and adjacencies of the form j,j+d where d>0 which extends earlier results.

KW - Combinatorial proof

KW - Permutation

KW - Stirling number of first kind

KW - Succession

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EP - 1376

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 4

ER -

Counting permutations by the number of successions within cycles

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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