Geodesic
Linear coefficients of Kerov's polynomials: bijective proof and refinement of Zagier's result
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) (2010).

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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. 
@article{DMTCS_2010_special_259_a10,
     author = {F\'eray, Valentin and Vassilieva, Ekaterina A.},
     title = {Linear coefficients of {Kerov's} polynomials: bijective proof and refinement of {Zagier's} result},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)},
     year = {2010},
     doi = {10.46298/dmtcs.2815},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2815/}
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Féray, Valentin; Vassilieva, Ekaterina A. Linear coefficients of Kerov's polynomials: bijective proof and refinement of Zagier's result. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) (2010). doi : 10.46298/dmtcs.2815. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2815/

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