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Alternating Sign Matrices and Hypermatrices, and a Generalization of Latin Square
An alternating sign matrix, or ASM, is a -matrix where the
nonzero entries in each row and column alternate in sign. We generalize this
notion to hypermatrices: an hypermatrix is an
{\em alternating sign hypermatrix}, or ASHM, if each of its planes, obtained by
fixing one of the three indices, is an ASM. Several results concerning ASHMs
are shown, such as finding the maximum number of nonzeros of an ASHM, and properties related to Latin squares. Moreover, we
investigate completion problems, in which one asks if a subhypermatrix can be
completed (extended) into an ASHM. We show several theorems of this type.Comment: 39 page
On the number of transversals in latin squares
The logarithm of the maximum number of transversals over all latin squares of
order is greater than
Enumerating extensions of mutually orthogonal Latin squares
Two n×n Latin squares L1,L2 are said to be orthogonal if, for every ordered pair (x, y) of symbols, there are coordinates (i, j) such that L1(i,j)=x and L2(i,j)=y. A k-MOLS is a sequence of k pairwise-orthogonal Latin squares, and the existence and enumeration of these objects has attracted a great deal of attention. Recent work of Keevash and Luria provides, for all fixed k, log-asymptotically tight bounds on the number of k-MOLS. To study the situation when k grows with n, we bound the number of ways a k-MOLS can be extended to a (k+1)-MOLS. These bounds are again tight for constant k, and allow us to deduce upper bounds on the total number of k-MOLS for all k. These bounds are close to tight even for k linear in n, and readily generalise to the broader class of gerechte designs, which include Sudoku squares
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Embedding partial Latin squares in Latin squares with many mutually orthogonal mates
In this paper it is shown that any partial Latin square of order can be embedded in a Latin square of order at most which has at least mutually orthogonal mates. Further, for any , it is shown that a pair of orthogonal partial Latin squares of order can be embedded in a set of mutually orthogonal Latin squares (MOLS) of order a polynomial with respect to . A consequence of the constructions is that, if denotes the size of the largest set of MOLS of order , then . In particular, it follows that , improving the previously known lower bound
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