Selections Without Adjacency on a Rectangular Grid
Combinatorics
2014-09-16 v1
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).
Cite
@article{arxiv.1409.3869,
title = {Selections Without Adjacency on a Rectangular Grid},
author = {Jacob A. Siehler},
journal= {arXiv preprint arXiv:1409.3869},
year = {2014}
}