Computational recreations in Mathematica
Computational recreations in Mathematica
Asymptotic enumeration methods
Handbook of combinatorics (vol. 2)
Proceedings of the 6th conference on Formal power series and algebraic combinatorics
Most Latin squares have many subsquares
Journal of Combinatorial Theory Series A
Diagonally cyclic latin squares
European Journal of Combinatorics
Completing partial latin squares with two cyclically generated prescribed diagonals
Journal of Combinatorial Theory Series A
The number of transversals in a Latin square
Designs, Codes and Cryptography
Applying fast simulation to find the number of good permutations
Cybernetics and Systems Analysis
The cycle structure of two rows in a random Latin square
Random Structures & Algorithms
Estimating the number of good permutations by a modified fast simulation method
Cybernetics and Systems Analysis
Estimating the number of latin rectangles by the fast simulation method
Cybernetics and Systems Analysis
On completing three cyclically generated transversals to a latin square
Finite Fields and Their Applications
Compound orthomorphisms of the cyclic group
Finite Fields and Their Applications
Note: Cyclotomic orthomorphisms of finite fields
Discrete Applied Mathematics
| Hi-index | 0.04 |
It is well known that if n is even, the addition table for the integers modulo n (which we denote by B"n) possesses no transversals. We show that if n is odd, then the number of transversals in B"n is at least exponential in n. Equivalently, for odd n, the number of diagonally cyclic latin squares of order n, the number of complete mappings or orthomorphisms of the cyclic group of order n, the number of magic juggling sequences of period n and the number of placements of n non-attacking semi-queens on an nxn toroidal chessboard are at least exponential in n. For all large n we show that there is a latin square of order n with at least (3.246)^n transversals. We diagnose all possible sizes for the intersection of two transversals in B"n and use this result to complete the spectrum of possible sizes of homogeneous latin bitrades. We also briefly explore potential applications of our results in constructing random mutually orthogonal latin squares.