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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.
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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, ⟨10.46298/dmtcs.2815⟩. ⟨hal-01186243⟩

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