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Bijection between trees in Stanley character formula and factorizations of a cycle. (arXiv:2210.04478v2 [math.CO] UPDATED)

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Stanley and F\'eray gave a formula for the irreducible character of the
symmetric group related to a multi-rectangular Young diagram. This formula
shows that the character is a polynomial in the multi-rectangular coordinates
and gives an explicit combinatorial interpretation for its coefficients in
terms of counting certain decorated maps (i.e., graphs drawn on surfaces). In
the current paper we concentrate on the coefficients of the top-degree
monomials in the Stanley character polynomial, which corresponds to counting
certain decorated plane trees. We give an explicit bijection between such trees
and minimal factorizations of a cycle.

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