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On the number of antichains in $\{0,1,2\}^n$

Matthew Jenssen, Jinyoung Park, Michail Sarantis

TL;DR

This work resolves the Dedekind-type enumeration problem for the 3-ary hypercube by delivering precise multiplicative asymptotics for α([3]^n), identifying the three central layers as the dominant contributors, and characterizing the typical structure of a random antichain. The authors introduce a novel graph-container lemma that handles irregular two-layer and three-layer graphs, combine it with isoperimetric inequalities and a polymer-cluster expansion framework, and validate a central-limit theorem for defect counts supported by a detailed cumulant analysis. By reducing the counting to the central layers and performing a refined enumeration of configurations there, they obtain an explicit asymptotic formula involving T1 and T2, with T2=o(T1), yielding a sharp growth rate of α([3]^n) beyond logarithmic scales. The results have connections to high-dimensional partitions and Ramsey-type problems, and the techniques extend to other product posets, highlighting the broader impact of container methods in irregular combinatorial settings.

Abstract

We provide precise asymptotics for the number of antichains in the poset $\{0,1,2\}^n$, answering a question of Sapozhenko. Finding improved estimates for this number was also a problem suggested by Noel, Scott, and Sudakov, who obtained asymptotics for the logarithm of the number. Key ingredients for the proof include a graph-container lemma to bound the number of expanding sets in a class of irregular graphs, isoperimetric inequalities for generalizations of the Boolean lattice, and methods from statistical physics based on the cluster expansion.

On the number of antichains in $\{0,1,2\}^n$

TL;DR

This work resolves the Dedekind-type enumeration problem for the 3-ary hypercube by delivering precise multiplicative asymptotics for α([3]^n), identifying the three central layers as the dominant contributors, and characterizing the typical structure of a random antichain. The authors introduce a novel graph-container lemma that handles irregular two-layer and three-layer graphs, combine it with isoperimetric inequalities and a polymer-cluster expansion framework, and validate a central-limit theorem for defect counts supported by a detailed cumulant analysis. By reducing the counting to the central layers and performing a refined enumeration of configurations there, they obtain an explicit asymptotic formula involving T1 and T2, with T2=o(T1), yielding a sharp growth rate of α([3]^n) beyond logarithmic scales. The results have connections to high-dimensional partitions and Ramsey-type problems, and the techniques extend to other product posets, highlighting the broader impact of container methods in irregular combinatorial settings.

Abstract

We provide precise asymptotics for the number of antichains in the poset , answering a question of Sapozhenko. Finding improved estimates for this number was also a problem suggested by Noel, Scott, and Sudakov, who obtained asymptotics for the logarithm of the number. Key ingredients for the proof include a graph-container lemma to bound the number of expanding sets in a class of irregular graphs, isoperimetric inequalities for generalizations of the Boolean lattice, and methods from statistical physics based on the cluster expansion.
Paper Structure (28 sections, 42 theorems, 205 equations, 1 figure)

This paper contains 28 sections, 42 theorems, 205 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results and strategy
  3. Connection to high-dimensional partitions and a Ramsey-type problem
  4. Methods
  5. Graph containers and cluster expansion
  6. Isoperimetric inequalities
  7. Organization
  8. Preliminaries
  9. Tools from probability
  10. Polymer models and cluster expansion
  11. Isoperimetry
  12. Classical results for $[t]^n$
  13. Isoperimetric inequalities for $[t]^n$
  14. Graph containers
  15. Preliminaries for the proof of \ref{["lem:container.graph'"]}
  16. ...and 13 more sections

Key Result

Theorem 1.1

where and

Figures (1)

  • Figure 1: The bracket configuration of $x=(0,2,1,3,2,1) \in [4]^6$. Its bracket structure $B(x)$ is $(((\ | \ ))(\ | \ )((\ | \ )))\ | \ **(\ | \ )**$.

Theorems & Definitions (79)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 2.2: Kotecký and Preiss kp
  • Lemma 3.1: Anderson anderson1968divisors
  • Lemma 3.2: Anderson anderson1968divisors and Harper harper1974morphology
  • Definition 3.3
  • ...and 69 more