[Paper Review] Simple recurrence formulas to count maps on orientable surfaces

S. R. Carrell, Guillaume Chapuy|arXiv (Cornell University)|Feb 25, 2014

Advanced Combinatorial Mathematics21 references4 citations

TL;DR

This paper presents a new, simple recurrence formula for counting rooted maps on orientable surfaces by edges and genus, derived from the KP equation and a Tutte-type equation. The key contribution is a highly efficient computational method that generalizes the Harer-Zagier formula and enables fast enumeration of maps, including by faces, with applications to fixed-genus generating functions and bipartite quadrangulations.

ABSTRACT

We establish a simple recurrence formula for the number $Q_g^n$ of rooted orientable maps counted by edges and genus. We also give a weighted variant for the generating polynomial $Q_g^n(x)$ where $x$ is a parameter taking the number of faces of the map into account, or equivalently a simple recurrence formula for the refined numbers $M_g^{i,j}$ that count maps by genus, vertices, and faces. These formulas give by far the fastest known way of computing these numbers, or the fixed-genus generating functions, especially for large $g$. In the very particular case of one-face maps, we recover the Harer-Zagier recurrence formula. Our main formula is a consequence of the KP equation for the generating function of bipartite maps, coupled with a Tutte equation, and it was apparently unnoticed before. It is similar in look to the one discovered by Goulden and Jackson for triangulations, and indeed our method to go from the KP equation to the recurrence formula can be seen as a combinatorial simplification of Goulden and Jackson's approach (together with one additional combinatorial trick). All these formulas have a very combinatorial flavour, but finding a bijective interpretation is currently unsolved.

Motivation & Objective

• To derive a simple recurrence formula for counting rooted maps on orientable surfaces by genus and number of edges.
• To extend the recurrence to include face count as a parameter via a generating polynomial.
• To provide a computationally efficient method for enumerating maps, especially for large genus.
• To generalize the Harer-Zagier formula for one-face maps as a special case.
• To offer a combinatorial framework that may guide future bijective constructions for map enumeration.

Proposed method

• Derivation of the recurrence from the KP equation for the generating function of bipartite maps.
• Application of a Tutte-type equation to relate map structures to their generating functions.
• Use of a combinatorial simplification of Goulden and Jackson’s approach to derive closed-form recurrences.
• Introduction of a generating polynomial $ Q_g^n(x) $ where $ x $ tracks the number of faces.
• Use of a degenerate Tutte equation to close the system and obtain a finite recurrence.
• Verification of the recurrence through initial conditions and consistency checks with known formulas like Harer-Zagier.

Experimental results

Research questions

• RQ1 Can a simple recurrence formula be derived for counting rooted maps on orientable surfaces by genus and edges, without relying on complex generating function manipulations?
• RQ2 How can the generating function for maps be refined to include face count, and what recurrence governs this refined count?
• RQ3 Is the Harer-Zagier recurrence for one-face maps a special case of a broader, unified recurrence formula?
• RQ4 Can the KP equation and Tutte equation be combined in a way that yields a closed-form recurrence for map enumeration?
• RQ5 What combinatorial or bijective interpretation, if any, underlies the derived recurrence formula?

Key findings

• The recurrence formula for $ Q_g^n $, the number of rooted maps of genus $ g $ with $ n $ edges, is the fastest known method for computing these numbers, especially for large $ g $.
• The formula generalizes to a generating polynomial $ Q_g^n(x) $, where the exponent of $ x $ tracks the number of faces, enabling enumeration by edges, genus, and faces.
• When specialized to one face ($ f=1 $), the recurrence reduces exactly to the Harer-Zagier formula, confirming consistency with a well-known result.
• The recurrence includes a double sum over lower-genus and lower-edge components, reflecting a recursive decomposition of maps via a Tutte-type equation.
• The method provides a combinatorial simplification of Goulden and Jackson’s approach to triangulations, suggesting a broader applicability to other map models.
• Despite the formula’s simplicity and efficiency, a bijective interpretation remains unknown, highlighting a key open problem in the field.