Latin squares with three disjoint subsquares of the same order

2025/10/01 by Kemp, Tara, Lefevre, James G.
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.00364

Abstract

Given an integer partition P = (h₁h₂… h_k) of n, a realization of P is a latin square with disjoint subsquares of orders h₁,h₂,…,h_k. Most known results restrict either k or the number of different integers in P. There is little known for partitions with arbitrary k and subsquares of at least three orders. It has been conjectured that if h₁=h₂=h₃≥ h₄≥…≥ h_k then a realization of P always exists. We prove this conjecture, and thus show the existence of realizations for many general partitions.

Citations

• Further Results on Latin Squares with Disjoint Subsquares using Rational Outline Squares

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