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

Abstract : We look at the number of permutations $\beta$ of $[N]$ with $m$ cycles such that $(1 2 \ldots N) \beta^{-1}$ 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 :
Document type :
Conference papers
Domain :
Complete list of metadata

Cited literature [10 references]

https://hal.inria.fr/hal-01186243
Contributor : Coordination Episciences Iam <>
Submitted on : Monday, August 24, 2015 - 3:44:24 PM
Last modification on : Tuesday, November 10, 2020 - 4:38:14 PM
Long-term archiving on: : Wednesday, November 25, 2015 - 4:33:14 PM

### File

dmAN0151.pdf
Publisher files allowed on an open archive

### Identifiers

• HAL Id : hal-01186243, version 1

### Citation

Valentin Féray, Ekaterina A. Vassilieva. Linear coefficients of Kerov's polynomials: bijective proof and refinement of Zagier's result. 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), 2010, San Francisco, United States. pp.713-724. ⟨hal-01186243⟩

Record views