Permanents and determinants of latin squares

Donovan, Diane, Johnson, Kenneth and Wanless, Ian M. (2016) Permanents and determinants of latin squares. Journal of Combinatorial Designs, 24 3: 132-148. doi:10.1002/jcd.21418

Author Donovan, Diane
Johnson, Kenneth
Wanless, Ian M.
Title Permanents and determinants of latin squares
Journal name Journal of Combinatorial Designs   Check publisher's open access policy
ISSN 1520-6610
Publication date 2016-03-01
Year available 2014
Sub-type Article (original research)
DOI 10.1002/jcd.21418
Open Access Status Not Open Access
Volume 24
Issue 3
Start page 132
End page 148
Total pages 17
Place of publication Hoboken, NJ United States
Publisher John Wiley & Sons
Collection year 2017
Language eng
Formatted abstract
Let L be a latin square of indeterminates. We explore the determinant and permanent of L and show that a number of properties of L can be recovered from the polynomials det(L) and per(L). For example, it is possible to tell how many transversals L has from per(L), and the number of 2 × 2 latin subsquares in L can be determined from both det(L) and per(L). More generally, we can diagnose from inline image or per(L) the lengths of all symbol cycles. These cycle lengths provide a formula for the coefficient of each monomial in det(L) and per(L) that involves only two different indeterminates. Latin squares A and B are trisotopic if B can be obtained from A by permuting rows, permuting columns, permuting symbols, and/or transposing. We show that nontrisotopic latin squares with equal permanents and equal determinants exist for all orders n > 9 that are divisible by 3. We also show that the permanent, together with knowledge of the identity element, distinguishes Cayley tables of finite groups from each other. A similar result for determinants was already known.
Keyword Latin square
Q-Index Code C1
Q-Index Status Provisional Code
Institutional Status UQ

Document type: Journal Article
Sub-type: Article (original research)
Collections: School of Mathematics and Physics
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