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