Latin squares with three disjoint subsquares of the same order
Tara Kemp, James G. Lefevre
TL;DR
This work addresses the problem of realizing integer partitions $P=(h_1h_2\dots h_k)$ as latin squares of order $n=\sum h_i$ that contain pairwise disjoint subsquares of orders $h_i$. It proves the conjecture that if $h_1=h_2=h_3\ge h_4\ge\dots\ge h_k$, then a realization $RP(h_1^3h_4\dots h_k)$ exists, thereby establishing realizations for a wide class of partitions. The authors develop two complementary methods: a Circulant construction that explicitly builds realizations under precise numeric conditions, and a Frequency-array framework that combines smaller realizations into larger ones via outline arrays, enabling broader existence results and refinements for incomplete latin squares. Together, these approaches advance the theory of partitioned incomplete latin squares (PILS) and provide constructive realizations for many partitions beyond previously known cases, with practical implications for related combinatorial designs.
Abstract
Given an integer partition $P = (h_1h_2\dots h_k)$ of $n$, a realization of $P$ is a latin square with disjoint subsquares of orders $h_1,h_2,\dots,h_k$. 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 $h_1=h_2=h_3\geq h_4\geq\dots\geq h_k$ then a realization of $P$ always exists. We prove this conjecture, and thus show the existence of realizations for many general partitions.