A bijective proof of Jackson's formula for the number of factorizations of a cycle

Authors:
Gilles Schaeffer;Ekaterina Vassilieva
Affiliations:
LIX, Laboratoire d'Informatique de l'École Polytechnique, 91128 Palaiseau Cedex, France;LIX, Laboratoire d'Informatique de l'École Polytechnique, 91128 Palaiseau Cedex, France
Venue:
Journal of Combinatorial Theory Series A
Year:
2008

Citing 7
Cited 2

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Enumerative combinatorics
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Constructive combinatorics
Some combinatorial problems associated with products of conjugacy classes of the symmetric group

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Factoring N-cycles and counting maps of given genus

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A direct bijection for the Harer-Zagier formula

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A bijection for covered maps, or a shortcut between Harer-Zagier's and Jackson's formulas

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A simple model of trees for unicellular maps

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Abstract

Factorizations of the cyclic permutation (12...N) into two permutations with respectively n and m cycles, or, equivalently, unicellular bicolored maps with N edges and n white and m black vertices, have been enumerated independantly by Jackson and Adrianov using evaluations of characters of the symmetric group. In this paper we present a bijection between unicellular partitioned bicolored maps and couples made of an ordered bicolored tree and a partial permutation, that allows for a combinatorial derivation of these results. Our work is closely related to a recent construction of Goulden and Nica for the celebrated Harer-Zagier formula, and indeed we provide a unified presentation of both bijections in terms of Eulerian tours in graphs.