On various formulas counting one-face maps

In this note, we make some connections between the Lehman-Walsh formula, the Harer-Zagier formula and Chapuy's recurrence, all counting one-face maps. First, we prove a new expression for the Harer-Zagier formula combinatorially. Extracting the coefficients from the new expression, we then obtain a new explicit formula involving convolution of the Stirling numbers of the first kind for the number, $A(n,g)$, of one-face maps of genus $g$. We next prove that the new formula equals the Lehman-Walsh formula by constructing two involutions on pairs of permutations. Finally, we connect our new expression to a result of Stanley, and obtain the log-concavity of $A(n,g)$. In addition, thanks to the symmetry of the new expression, a straightforward computation leads to the recursion for $A(n,g)$ obtained by Chapuy recently.