Another combinatorial proof of a result of Zagier and Stanley

Ricky X. F. Chen, Christian M. Reidys

Published 2015-02-26Version 1

In this paper, we present another combinatorial proof for the result of Zagier and Stanley, that the number of $n$-cycles $\omega$, for which $\omega(12\cdots n)$ has exactly $k$ cycles is $0$, if $n-k$ is odd and $\frac{2C(n+1,k)}{n(n+1)}$, otherwise, where $C(n,k)$ is the unsigned Stirling number of the first kind. To this end, we generalize permutations to plane permutations and study exceedances via a natural transposition action on plane permutations. Furthermore, based on a "reflection principle" argument, we give a refinement of a recurrence satisfied by the numbers counting one-face hypermaps which was recently obtained by Chapuy by counting bipartite unicellular maps.