Permanents and Determinants of Latin Squares. Issue 3 (19th November 2014)

• Record Type: Journal Article
• Authors: Donovan, Diane; Johnson, Kenneth; Wanless, Ian M.
• Is Part Of: Journal of combinatorial designs. Volume 24:Issue 3(2016:Mar.)
• Page Start: 132
• Page End: 148
• Publication Date: 2014-11-19
• Subjects: Latin square -- quasigroup -- determinant -- permanent -- transversal; Combinatorial designs and configurations -- Periodicals; Configurations et schémas combinatoires -- Périodiques; 511.6
• DOI: 10.1002/jcd.21418
• Languages: English
• ISSNs: 1063-8539

Abstract

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 det ( L ) 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.