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Vertex degrees in grid graphs associated with 213-avoiding permutations

N. B. Huamaní

TL;DR

This work analyzes grid graphs G_pi derived from permutations in the Catalan avoidance class Av_n(213). By exploiting a Catalan decomposition around the position of the minimum, the authors derive closed generating-function identities and solve for the total number of horizontal edges H_n and the global degree counts Q_r(n) for r=1,2,3,4. They show that, under Av_n(213), degree-4 vertices dominate asymptotically with proportion tending to 1 and a precise n^{-1/2} deficit, in sharp contrast with the unrestricted case. Reversal symmetry transfers these results to Av_n(312). The methods combine gluing identities, algebraic generating functions, and global degree-sum identities to provide a complete closed-form account and asymptotics for these grid-graph statistics within the Catalan-avoiding regime.

Abstract

Given a permutation of size $n$, we consider its associated grid graph whose $i$th column has height equal to the $i$th entry, with vertical edges between consecutive levels and horizontal edges between equal levels in adjacent columns. We study global degree statistics of these graphs when the permutation is chosen from the Catalan avoidance class $\mathrm{Av}_n(213)$ (and, by reversal, also from $\mathrm{Av}_n(312)$). We first obtain an explicit closed form for the total number of horizontal edges summed over all permutations in $\mathrm{Av}_n(213)$. We then determine, for each degree $r\in\{1,2,3,4\}$, the total number of degree-$r$ vertices accumulated over the same class, yielding closed expressions in terms of central binomial coefficients and powers of four. The proofs rely on the Catalan decomposition induced by the position of the minimum entry, which leads to gluing identities and algebraic functional equations for ordinary generating functions, completed using global vertex and degree-sum identities. As a consequence, we derive asymptotic degree proportions for a uniform random permutation in $\mathrm{Av}_n(213)$: the distribution concentrates and the proportion of degree-$4$ vertices tends to $1$, with a deficit of order $n^{-1/2}$.

Vertex degrees in grid graphs associated with 213-avoiding permutations

TL;DR

This work analyzes grid graphs G_pi derived from permutations in the Catalan avoidance class Av_n(213). By exploiting a Catalan decomposition around the position of the minimum, the authors derive closed generating-function identities and solve for the total number of horizontal edges H_n and the global degree counts Q_r(n) for r=1,2,3,4. They show that, under Av_n(213), degree-4 vertices dominate asymptotically with proportion tending to 1 and a precise n^{-1/2} deficit, in sharp contrast with the unrestricted case. Reversal symmetry transfers these results to Av_n(312). The methods combine gluing identities, algebraic generating functions, and global degree-sum identities to provide a complete closed-form account and asymptotics for these grid-graph statistics within the Catalan-avoiding regime.

Abstract

Given a permutation of size , we consider its associated grid graph whose th column has height equal to the th entry, with vertical edges between consecutive levels and horizontal edges between equal levels in adjacent columns. We study global degree statistics of these graphs when the permutation is chosen from the Catalan avoidance class (and, by reversal, also from ). We first obtain an explicit closed form for the total number of horizontal edges summed over all permutations in . We then determine, for each degree , the total number of degree- vertices accumulated over the same class, yielding closed expressions in terms of central binomial coefficients and powers of four. The proofs rely on the Catalan decomposition induced by the position of the minimum entry, which leads to gluing identities and algebraic functional equations for ordinary generating functions, completed using global vertex and degree-sum identities. As a consequence, we derive asymptotic degree proportions for a uniform random permutation in : the distribution concentrates and the proportion of degree- vertices tends to , with a deficit of order .
Paper Structure (20 sections, 19 theorems, 63 equations, 1 figure)

This paper contains 20 sections, 19 theorems, 63 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Grid graphs
  4. Local degree criteria by level
  5. Pattern avoidance, Catalan numbers, and reversal
  6. Catalan decomposition by the position of the minimum
  7. Horizontal edges
  8. A gluing identity and a functional equation
  9. Closed form
  10. Degree counts: global definitions
  11. Degree-$1$ vertices
  12. A local criterion in external columns
  13. Boundary counts in $\mathrm{Av}_n(213)$
  14. External and internal columns
  15. Degree-$4$ vertices
  16. ...and 5 more sections

Key Result

Lemma 2.2

Let $\pi\in S_n$ and consider a column $i$ of height $b=\pi_i$. (i) If $2\le i\le n-1$ (internal column) and $a=\pi_{i-1}$, $c=\pi_{i+1}$, then for $1\le s\le b$ (ii) If $i=1$ (left external column) and $b=\pi_1$, $c=\pi_2$, then for $1\le s\le b$ Similarly, if $i=n$ and $a=\pi_{n-1}$, then

Figures (1)

  • Figure 1: Grid graphs $G_\pi$ for $n=4$: (a) $\pi=4132$, (b) $\pi=3412$ (both in $\mathrm{Av}_4(213)$), and (c) $\pi=2134$, which contains the pattern $213$.

Theorems & Definitions (40)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • proof
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • ...and 30 more