Enumeration of Graded (3+1)-Avoiding Posets

TL;DR

This paper enumerates graded (3+1)-avoiding posets under various definitions, providing structural theorems, generating functions, and asymptotic growth rates to advance understanding in combinatorics.

Contribution

It introduces a comprehensive enumeration of graded (3+1)-avoiding posets, including structural characterizations and asymptotic analysis, which was previously open.

Findings

Enumeration formulas for graded (3+1)-avoiding posets

Structural theorems characterizing these posets

Asymptotic growth rates of the enumeration

Abstract

The notion of (3+1)-avoidance has shown up in many places in enumerative combinatorics. The natural goal of enumeration of all (3+1)-avoiding posets remains open. In this paper, we enumerate graded (3+1)-avoiding posets for both reasonable definitions of the word "graded." Our proof consists of a number of structural theorems followed by some generating function magic. We also provide asymptotics for the growth rate of the number of graded (3 + 1)-avoiding posets.