Refined enumeration of planar Eulerian orientations

TL;DR

The paper advances the refined enumeration of planar Eulerian orientations by introducing a three-parameter generating function $Q(t,\omega,v)$ and establishing a robust functional-equation framework via patches, dualities, and Ambjørn–Budd bijections. It derives a compact single-catalytic-variable characterisation for $Q$, then solves three two-parameter specializations: $(\omega=0)$ counting by edges and vertices, $(\omega=1)$ counting by vertices and clockwise faces, and $(v=1)$ corresponding to the six-vertex model with a theta-function representation. In the two-parameter cases, the authors obtain differential-algebraic expressions (and prove D-algebraicity) for the generating functions, along with explicit radius-of-convergence results and combinatorial interpretations in terms of trees. For the six-vertex case, they develop an intricate theta-function/elliptic-function framework, including an Ansatz linking $M(x)$ to a shifted meromorphic function, and provide a purely formal-power-series proof that avoids complex-analytic machinery. Collectively, the work unifies and extends prior results, delivers new proofs, and reveals structural invariants and algebraicity phenomena in refined map-enumeration problems with potential connections to combinatorial probabilistic models and statistical mechanics.

Abstract

We address the enumeration of Eulerian orientations of 4-valent planar maps according to three parameters: the number of vertices, the number of alternating vertices (having in/out/in/out incident edges), and the number of clockwise oriented faces. This is a refinement of the six vertex model studied by Kostov, then Zinn-Justin and Elvey Price, where one only considers the first two parameters. Via a bijection of Ambjorn and Budd, our problem is equivalent to the enumeration of Eulerian partial orientations of general planar maps, counted by the number of edges, the number of undirected edges, and the number of vertices. We first derive from combinatorial arguments a system of functional equations characterising the associated trivariate series $Q(t,ω,v)$. We then derive from this system a compact characterisation of this series. We use it to determine $Q(t,ω,v)$ in three two-parameter cases. The first two cases correspond to setting the variable $ω$ counting alternating vertices (or undirected edges after the AB bijection) to $0$ or $1$: when $ω=0$ we count Eulerian orientations of general planar maps by edges and vertices, and when $ω=1$ we count Eulerian orientations of quartic maps by vertices and clockwise faces. The final forms of these two series, namely $Q(t,0, v)$ and $Q(t,1,v)$, refine those obtained by the authors in an earlier paper for $v=1$. The third case that we solve, namely $v=1$ (but $ω$ arbitrary), is the standard six-vertex model, for which we provide a new proof of the formula of Elvey Price and Zinn-Justin involving Jacobi theta functions. This new derivation remains purely in the world of formal power series, not relying on complex analysis. Our results also use a more direct approach to solving the functional equations, in contrast to the guess and check approaches used in previous work.

Figures (21)

• Figure 1: Left: a rooted planar map, which is 4-valent (or: quartic). Right: the same map, equipped with an Eulerian orientation.
• Figure 2: The planar Eulerian orientations with at most two edges. On the right are the four quartic Eulerian orientations with one vertex.
• Figure 3: Left: a rooted planar map, with the root edge and root corner shown. Right: the dual map, in dashed edges.
• Figure 4: A labelled map, rooted from $0$ to $1$.
• Figure 5: A partial Eulerian orientation (solid edges) and the corresponding dual weakly labelled map (dashed edges). The labelling rule is shown on the right.

Theorems & Definitions (83)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: Thms. 1.1 and 6.1 in elvey-zinn
• Proposition 1.4
• Definition 2.1
• Lemma 2.2
• Proposition 2.3
• Corollary 2.4
• Remark 2.5: 3-coloured quadrangulations
• Remark 2.6