Asymptotics of the number of 2-threshold functions

TL;DR

This paper derives the asymptotic count of 2-threshold functions on a rectangular grid, revealing their growth rate as the grid size increases, which advances understanding of their combinatorial complexity.

Contribution

The paper provides the first asymptotic formula for the number of 2-threshold functions on a grid, specifically for the case k=2, highlighting their growth rate as grid dimensions increase.

Findings

Number of 2-threshold functions grows as (25/12π^4) m^4 n^4 + o(m^4 n^4)

Asymptotic behavior established for the case k=2

Enhances understanding of threshold function enumeration

Abstract

A $k$-threshold function on a rectangular grid of size $m \times n$ is the conjunction of $k$ threshold functions on the same domain. In this paper, we focus on the case $k=2$ and show that the number of two-dimensional 2-threshold functions is~$\dfrac{25}{12\pi^4} m^4 n^4 + o(m^4n^4)$.