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In this paper, we begin with the Lehman-Walsh formula counting one-face mapsand construct two involutions on pairs of permutations to obtain a new formulafor the number $A(n,g)$ of one-face maps of genus $g$. Our new formula is inthe form of a convolution of the Stirling numbers of the first kind whichimmediately implies a formula for the generating function $A_n(x)=\sum_{g\geq0}A(n,g)x^{n+1-2g}$ other than the well-known Harer-Zagier formula. Byreformulating our expression for $A_n(x)$ in terms of the backward shiftoperator $E: f(x)\rightarrow f(x-1)$ and proving a property satisfied bypolynomials of the form $p(E)f(x)$, we easily establish the recursion obtainedby Chapuy for $A(n,g)$. Moreover, we give a simple combinatorial interpretationfor the Harer-Zagier recurrence.

New formulas counting one-face maps and Chapuy's recursion