Linear coefficients of Kerov's polynomials: Bijective proof and refinement of Zagier's result

Valentin Féray; Ekaterina A. Vassilieva

Linear coefficients of Kerov's polynomials: Bijective proof and refinement of Zagier's result

Valentin Féray
, Ekaterina A. Vassilieva

SCRIME - LaBRI, Université Bordeaux 1

Research output: Contribution to conference › Paper › peer-review

Abstract

We look at the number of permutations β of [N] with m cycles such that (1 2.. N) β ^-1 is a long cycle. These numbers appear as coefficients of linear monomials in Kerov's and Stanley's character polynomials. D. Zagier, using algebraic methods, found an unexpected connection with Stirling numbers of size N + 1. We present the first combinatorial proof of his result, introducing a new bijection between partitioned maps and thorn trees. Moreover, we obtain a finer result, which takes the type of the permutations into account.

Original language	English
Pages	713-724
Number of pages	12
Publication status	Published - 1 Dec 2010
Event	22nd International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'10 - San Francisco, CA, United States Duration: 2 Aug 2010 → 6 Aug 2010

Conference

Conference	22nd International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'10
Country/Territory	United States
City	San Francisco, CA
Period	2/08/10 → 6/08/10

Keywords

Bicolored maps
Kerov's character polynomials
Long cycle factorization

Cite this

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title = "Linear coefficients of Kerov's polynomials: Bijective proof and refinement of Zagier's result",

abstract = "We look at the number of permutations β of [N] with m cycles such that (1 2.. N) β -1 is a long cycle. These numbers appear as coefficients of linear monomials in Kerov's and Stanley's character polynomials. D. Zagier, using algebraic methods, found an unexpected connection with Stirling numbers of size N + 1. We present the first combinatorial proof of his result, introducing a new bijection between partitioned maps and thorn trees. Moreover, we obtain a finer result, which takes the type of the permutations into account.",

keywords = "Bicolored maps, Kerov's character polynomials, Long cycle factorization",

author = "Valentin F{\'e}ray and Vassilieva, \{Ekaterina A.\}",

year = "2010",

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language = "English",

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note = "22nd International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'10 ; Conference date: 02-08-2010 Through 06-08-2010",

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T1 - Linear coefficients of Kerov's polynomials

T2 - 22nd International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC'10

AU - Féray, Valentin

AU - Vassilieva, Ekaterina A.

PY - 2010/12/1

Y1 - 2010/12/1

N2 - We look at the number of permutations β of [N] with m cycles such that (1 2.. N) β -1 is a long cycle. These numbers appear as coefficients of linear monomials in Kerov's and Stanley's character polynomials. D. Zagier, using algebraic methods, found an unexpected connection with Stirling numbers of size N + 1. We present the first combinatorial proof of his result, introducing a new bijection between partitioned maps and thorn trees. Moreover, we obtain a finer result, which takes the type of the permutations into account.

AB - We look at the number of permutations β of [N] with m cycles such that (1 2.. N) β -1 is a long cycle. These numbers appear as coefficients of linear monomials in Kerov's and Stanley's character polynomials. D. Zagier, using algebraic methods, found an unexpected connection with Stirling numbers of size N + 1. We present the first combinatorial proof of his result, introducing a new bijection between partitioned maps and thorn trees. Moreover, we obtain a finer result, which takes the type of the permutations into account.

KW - Bicolored maps

KW - Kerov's character polynomials

KW - Long cycle factorization

UR - https://www.scopus.com/pages/publications/84860503085

M3 - Paper

AN - SCOPUS:84860503085

SP - 713

EP - 724

Y2 - 2 August 2010 through 6 August 2010

ER -

Linear coefficients of Kerov's polynomials: Bijective proof and refinement of Zagier's result

Abstract

Conference

Keywords

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