Séminaire Lotharingien de Combinatoire, 91B.102 (2024), 9 pp.
Theo Douvropoulos
Counting Unicellular Maps under Cyclic Symmetries
Abstract.
We count unicellular maps (matchings of the edges of a 2n-gon) of arbitrary genus with respect to the 2n-rotation symmetries of the polygon. An associated generating function that keeps track of the number of symmetric vertices of the resulting map generalizes the celebrated Harer-Zagier formula.
The answer to this enumerative question is not in the form of the usual cyclic sieving phenomenon (CSP), but does recover in the leading terms (genus-0 maps) a well known CSP for the Catalan numbers. The approach is representation theoretic, in that we relate symmetric unicellular maps with factorizations of the Coxeter element in a reflection group of type G(m,1,n).
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
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