Enumeration of non-oriented maps via integrability
Algebraic Combinatorics, Volume 5 (2022) no. 6, pp. 1363-1390.

In this note, we examine how the BKP structure of the generating series of several models of maps on non-oriented surfaces can be used to obtain explicit and/or efficient recurrence formulas for their enumeration according to the genus and size parameters.

Using techniques already known in the orientable case (elimination of variables via Virasoro constraints or Tutte equations), we naturally obtain recurrence formulas with non-polynomial coefficients. This non-polynomiality reflects the presence of shifts of the charge parameter in the BKP equation. Nevertheless, we show that it is possible to obtain non-shifted versions, meaning pure ODEs for the associated generating functions, from which recurrence relations with polynomial coefficients can be extracted. We treat the cases of triangulations, general maps, and bipartite maps.

These recurrences with polynomial coefficients are conceptually interesting but bigger to write than those with non-polynomial coefficients. However they are relatively nice-looking in the case of one-face maps. In particular we show that Ledoux’s recurrence for non-oriented one-face maps can be recovered in this way, and we obtain the analogous statement for the (bivariate) bipartite case.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.268
Classification: 05A15,  05C30,  37K10
Keywords: combinatorial maps, nonorientable surfaces, enumeration, BKP hierarchy, zonal polynomials, matrix integrals
Bonzom, Valentin 1; Chapuy, Guillaume 2; Dołęga, Maciej 3

1 Université Sorbonne Paris Nord LIPN, CNRS, UMR 7030 F-93430 Villetaneuse France
2 CNRS, IRIF UMR 8243 Université Paris Cité.
3 Institute of Mathematics Polish Academy of Sciences ul. Śniadeckich 8 00-956 Warszawa Poland.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Bonzom, Valentin; Chapuy, Guillaume; Dołęga, Maciej. Enumeration of non-oriented maps via integrability. Algebraic Combinatorics, Volume 5 (2022) no. 6, pp. 1363-1390. doi : 10.5802/alco.268. https://alco.centre-mersenne.org/articles/10.5802/alco.268/

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