Selections Without Adjacency on a Rectangular Grid

TL;DR

This paper derives formulas and identities for counting non-adjacent square selections in rectangular grids, focusing on 2-row and 3-row cases, and proves a unimodality property to optimize selection size.

Contribution

It provides explicit formulas, polynomial representations, and a unimodality theorem for the number of non-adjacent selections in 2x n and 3x n grids.

Findings

Formulas for T(2,n;k) and T(3,n;k)

Polynomial degree and coefficients for T(2,n;k)

Unimodality theorem for maximizing T(2,n;k)

Abstract

Using T(m,n;k) to denote the number of ways to make a selection of k squares from an (m x n) rectangular grid with no two squares in the selection adjacent, we give a formula for T(2,n;k), prove some identities satisfied by these numbers, and show that T(2,n;k) is given by a degree k polynomial in n. We give simple formulas for the first few (most significant) coefficients of the polynomials. We give corresponding results for T(3,n;k) as well. Finally we prove a unimodality theorem which shows, in particular, how to choose k in order to maximize T(2,n;k).