# Permanents and Determinants of Latin Squares

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 2 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 non-trisotopic 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.