The Harer-Zagier and Jackson formulas and new results for one-face bipartite maps
Proceedings of the American Mathematical Society, 152(10) (2024), 4245--4259 The study of bipartite maps (or Grothendieck's dessins d'enfants) is closely connected with geometry, mathematical physics and free probability. Here we study these objects from their permutation factorization for...
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Format | Journal Article |
Language | English |
Published |
07.02.2023
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Subjects | |
Online Access | Get full text |
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Summary: | Proceedings of the American Mathematical Society, 152(10) (2024),
4245--4259 The study of bipartite maps (or Grothendieck's dessins d'enfants) is closely
connected with geometry, mathematical physics and free probability. Here we
study these objects from their permutation factorization formulation using a
novel character theory approach. We first present some general symmetric
function expressions for the number of products of two permutations
respectively from two arbitrary, but fixed, conjugacy classes indexed by
$\alpha$ and $\gamma$ which produce a permutation with $m$ cycles. Our next
objective is to derive explicit formulas for the cases where $\alpha$
corresponds to full cycles, i.e., one-face bipartite maps. We prove a
far-reaching explicit formula, and show that the number for any $\gamma$ can be
iteratively reduced to that of products of two full cycles, which implies an
efficient dimension-reduction algorithm for building a database of all these
numbers. Note that the number for products of two full cycles can be computed
by the Zagier-Stanley formula. Also, in a unified way, we easily prove the
celebrated Harer-Zagier formula and Jackson's formula, and we obtain explicit
formulas for several new families as well. |
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DOI: | 10.48550/arxiv.2302.03695 |