Counting permutations by cyclic peaks and valleys

Chak On Chow; Shi Mei Ma; Toufik Mansour; Mark Shattuck

Counting permutations by cyclic peaks and valleys

Chak On Chow, Shi Mei Ma, Toufik Mansour, Mark Shattuck

Department of Mathematics

University of Haifa

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

Abstract

In this paper, we study the generating functions for the number of permutations having a prescribed number of cyclic peaks or valleys. We derive closed form expressions for these functions by use of various algebraic methods. When restricted to the set of derangements (i.e., fixed point free permutations), the evaluation at −1 of the generating function for the number of cyclic valleys gives the Pell number. We provide a bijective proof of this result, which can be extended to the entire symmetric group.

Original language	American English
Pages (from-to)	43-54
Number of pages	12
Journal	Annales Mathematicae et Informaticae
Volume	43
State	Published - 2014

Keywords

Cyclic valleys
Derangements
Involutions
Pell numbers

All Science Journal Classification (ASJC) codes

General Computer Science
General Mathematics

Cite this

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abstract = "In this paper, we study the generating functions for the number of permutations having a prescribed number of cyclic peaks or valleys. We derive closed form expressions for these functions by use of various algebraic methods. When restricted to the set of derangements (i.e., fixed point free permutations), the evaluation at −1 of the generating function for the number of cyclic valleys gives the Pell number. We provide a bijective proof of this result, which can be extended to the entire symmetric group.",

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AU - Ma, Shi Mei

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AU - Shattuck, Mark

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AB - In this paper, we study the generating functions for the number of permutations having a prescribed number of cyclic peaks or valleys. We derive closed form expressions for these functions by use of various algebraic methods. When restricted to the set of derangements (i.e., fixed point free permutations), the evaluation at −1 of the generating function for the number of cyclic valleys gives the Pell number. We provide a bijective proof of this result, which can be extended to the entire symmetric group.

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KW - Involutions

KW - Pell numbers

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JO - Annales Mathematicae et Informaticae

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Counting permutations by cyclic peaks and valleys

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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