# Latin squares with disjoint subsquares of two orders

Latin squares with disjoint subsquares of two orders Let n1,…,nk∈Z+ and n1+⋯+nk=n. The integer partition (n1,…,nk) of n is said to be realized if there is a latin square of order n with pairwise disjoint subsquares of order ni for each 1≤i≤k. In this paper, we construct latin squares realizing partitions of the form (as,bt); that is, partitions with s parts of size a and t parts of size b, where a<b. Heinrich (1982) showed that (1) if s≥3 and t≥3, then there is a latin square realizing (as,bt), (2) (as,b) is realized if and only if (s−1)a≥b, and (3) (a,bt) is realized if and only if t≥3. In this paper, we resolve the open cases. We show that (a2,bt) is realized if and only if t≥3 and (as,b2) is realized if and only if as≥b. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Combinatorial Designs Wiley

# Latin squares with disjoint subsquares of two orders

, Volume 26 (5) – Jan 1, 2018
18 pages

/lp/wiley/latin-squares-with-disjoint-subsquares-of-two-orders-roC7GPM6qS
Publisher
Wiley Subscription Services, Inc., A Wiley Company
ISSN
1063-8539
eISSN
1520-6610
D.O.I.
10.1002/jcd.21570
Publisher site
See Article on Publisher Site

### Abstract

Let n1,…,nk∈Z+ and n1+⋯+nk=n. The integer partition (n1,…,nk) of n is said to be realized if there is a latin square of order n with pairwise disjoint subsquares of order ni for each 1≤i≤k. In this paper, we construct latin squares realizing partitions of the form (as,bt); that is, partitions with s parts of size a and t parts of size b, where a<b. Heinrich (1982) showed that (1) if s≥3 and t≥3, then there is a latin square realizing (as,bt), (2) (as,b) is realized if and only if (s−1)a≥b, and (3) (a,bt) is realized if and only if t≥3. In this paper, we resolve the open cases. We show that (a2,bt) is realized if and only if t≥3 and (as,b2) is realized if and only if as≥b.

### Journal

Journal of Combinatorial DesignsWiley

Published: Jan 1, 2018

Keywords: ; ; ;

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