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Mixed Dimer Models for Euler and Catalan Numbers

Andrew Claussen, Nicholas Ovenhouse

TL;DR

The paper introduces mixed dimer covers on border-strip (snake) graphs and uncovers deep connections to classical combinatorial sequences: Euler numbers for straight snakes and Catalan numbers for zigzag snakes, with refined Entringer and ballot numbers obtained via boundary multiplicities. It develops matrix-product recurrences and explicit bijections to alternating and 132-avoiding permutations, then extends to q-analogs through rank-generating functions tied to the middle order on permutations. A distributive lattice structure is established for these dimer lattices, enabling q-analogs $E_n(q)$ and $C_n(q)$ and revealing poset isomorphisms with permutation orders. The work further situates these results in broader combinatorial models (networks, rhombus tilings, order ideals) and generalizes to arbitrary snake graphs, providing a cohesive framework linking dimers, lattice paths, and permutation theory with potential for further exploration of standard labelings and dualities.

Abstract

We study the enumeration of mixed dimer covers on skew Young diagrams of ribbon shape (also called border strips or snake graphs). For the two extreme cases of straight and zigzag shapes, we show that the number of mixed dimer covers are given by the Euler and Catalan numbers. We also give q-analogs by showing that the rank generating functions of the partial orders on mixed dimer covers agree with certain q-Euler and q-Catalan numbers. These q-analogs are a consequence of an isomorphism between the partial order on mixed dimer covers and the so-called middle order on certain classes of permutations.

Mixed Dimer Models for Euler and Catalan Numbers

TL;DR

The paper introduces mixed dimer covers on border-strip (snake) graphs and uncovers deep connections to classical combinatorial sequences: Euler numbers for straight snakes and Catalan numbers for zigzag snakes, with refined Entringer and ballot numbers obtained via boundary multiplicities. It develops matrix-product recurrences and explicit bijections to alternating and 132-avoiding permutations, then extends to q-analogs through rank-generating functions tied to the middle order on permutations. A distributive lattice structure is established for these dimer lattices, enabling q-analogs and and revealing poset isomorphisms with permutation orders. The work further situates these results in broader combinatorial models (networks, rhombus tilings, order ideals) and generalizes to arbitrary snake graphs, providing a cohesive framework linking dimers, lattice paths, and permutation theory with potential for further exploration of standard labelings and dualities.

Abstract

We study the enumeration of mixed dimer covers on skew Young diagrams of ribbon shape (also called border strips or snake graphs). For the two extreme cases of straight and zigzag shapes, we show that the number of mixed dimer covers are given by the Euler and Catalan numbers. We also give q-analogs by showing that the rank generating functions of the partial orders on mixed dimer covers agree with certain q-Euler and q-Catalan numbers. These q-analogs are a consequence of an isomorphism between the partial order on mixed dimer covers and the so-called middle order on certain classes of permutations.

Paper Structure

This paper contains 22 sections, 17 theorems, 31 equations, 22 figures, 3 tables.

Table of Contents

  1. Introduction and Main Results
  2. Euler and Entringer Numbers from Straight Snakes
  3. Mixed Dimer Covers on Straight Snakes
  4. Euler and Entringer Numbers
  5. An Explicit Bijection
  6. Genocchi Numbers from Straight Snakes
  7. Catalan and Ballot Numbers from Zigzag Snakes
  8. Catalan and Ballot Numbers
  9. Enumeration for Zigzag Snakes
  10. Distributive Lattice Structures and $q$-Analogs
  11. The Middle Order on Permutations
  12. Rank Functions for Mixed Dimer Lattices
  13. Other Combinatorial Models
  14. Networks
  15. Single Dimer Covers
  16. ...and 7 more sections

Key Result

Theorem 1

For an integer sequence $m = (m_0,m_1,\dots,m_n)$, define the matrix $M_m$ by Then the $(i,j)$-entry of $M_m$ is equal to $\left| \Omega^{ij}_{\mathbf{n}_{m}}(\mathcal{G}^s_n) \right|$. In particular, the $(1,1)$-entry is $\left| \Omega_{\mathbf{n}_m}(\mathcal{G}^s_n) \right|$, the total number of mixed dimer covers on $\mathcal{G}^s_n$ with vertex multiplicities $m$.

Figures (22)

  • Figure 1: Examples of graphs $\mathcal{G}_4^s$ (left) and $\mathcal{G}_5^z$ (right), with their standard vertex labelings.
  • Figure 2: The graph $\mathcal{G}^s_3$
  • Figure 3: The bijection $\Psi \colon \mathcal{D}_{n,k} \to \bigcup_{i > n-k} \mathcal{D}_{n-1,i}$. Vertex labels are in blue, and edge labels are the multiplicities in a mixed dimer cover.
  • Figure 4: The bijection $F \colon \mathrm{Alt}_7 \to \Omega(\mathcal{G}^s_{5})$. The multiplicities of the vertical edges are given by $\delta_i = L_i-L_{i-1}$ and $\Delta_i = L_{i-1}-L_i-1$, where $L(\sigma) = (L_1,L_2,\dots,L_7)$ is the Lehmer code of an alternating permutation. Note that the first and last vertical edges also follow this pattern if we define $L_0 := 0$.
  • Figure 5: The 5 elements of $\Omega(\mathcal{G}^s_2)$.
  • ...and 17 more figures

Theorems & Definitions (70)

  • Definition 1
  • Remark 1
  • Definition 2
  • Definition 3
  • Remark 2
  • Definition 4
  • Example 1
  • Theorem 1
  • proof
  • Example 2
  • ...and 60 more