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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 $A_n(x)=sum_{ggeq 0}A(n,g)x^{n+1-2g}$ 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.
Ordinary maps satisfy topological recursion for a certain spectral curve $(x, y)$. We solve a conjecture from arXiv:1710.07851 that claims that fully simple maps, which are maps with non self-intersecting disjoint boundaries, satisfy topological recu
The operator nabla, introduced by Garsia and the author, plays a crucial role in many aspect of the study of diagonal harmonics. Besides giving several new formulas involving this operator, we show how one is lead to representation theoretic explanat
The effectiveness of shortcut/skip-connection has been widely verified, which inspires massive explorations on neural architecture design. This work attempts to find an effective way to design new network architectures. It is discovered that the main
We introduce the notion of fully simple maps, which are maps with non self-intersecting disjoint boundaries. In contrast, maps where such a restriction is not imposed are called ordinary. We study in detail the combinatorics of fully simple maps with
Simple boundary expressions for the k-th power of the cotangent line class on the moduli space of stable 1-pointed genus g curves are found for k >= 2g. The method is by virtual localization on the moduli space of maps to the projective line. As a co