Projects per year
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.
Original language  English 

Pages (fromto)  132148 
Number of pages  17 
Journal  Journal of Combinatorial Designs 
Volume  24 
Issue number  3 
DOIs  
Publication status  Published  1 Mar 2016 
Keywords
 determinant
 Latin square
 permanent
 quasigroup
 transversal
Projects
 1 Finished

Towards the prime power conjecture
Australian Research Council (ARC), Monash University
27/02/12 → 31/12/16
Project: Research