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Journal article
Latin squares with disjoint subsquares of two orders
Journal of Combinatorial Designs, Vol.26, pp.219-236
26
2017
Web of Science ID: WOS:000434970700002
Abstract
Let n₁,..., nₖ ∈ ℤ⁺ and n₁+...+nₖ =n The integer partition (n₁,..., nₖ) is said to be realized if there is a latin square of order n with pairwise disjoint subsquares of order n 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.
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Details
- Title
- Latin squares with disjoint subsquares of two orders
- Publication Details
- Journal of Combinatorial Designs, Vol.26, pp.219-236
- Resource Type
- Journal article
- Publisher
- John Wiley & Sons, Inc.; United States
- Series
- 26
- Copyright
- © 2017 Wiley Periodicals, Inc.
- Identifiers
- WOS:000434970700002; 99380090767106600
- Academic Unit
- Hal Marcus College of Science and Engineering ; Mathematics and Statistics
- Language
- English