115 research outputs found
Parity of transversals of Latin squares
We introduce a notion of parity for transversals, and use it to show that in
Latin squares of order , the number of transversals is a multiple of
4. We also demonstrate a number of relationships (mostly congruences modulo 4)
involving , where is the number of diagonals of a given
Latin square that contain exactly different symbols.
Let denote the matrix obtained by deleting row and column
from a parent matrix . Define to be the number of transversals
in , for some fixed Latin square . We show that for all and . Also, if has odd order then the
number of transversals of equals mod 2. We conjecture that for all .
In the course of our investigations we prove several results that could be of
interest in other contexts. For example, we show that the number of perfect
matchings in a -regular bipartite graph on vertices is divisible by
when is odd and . We also show that for all , when is an integer matrix of odd
order with all row and columns sums equal to
A lower bound on the maximum permanent in Λnk
AbstractLet Pnk be the maximum value achieved by the permanent over Λnk, the set of (0,1)-matrices of order n with exactly k ones in each row and column. Brègman proved that Pnk⩽k!n/k. It is shown here that Pnk⩾k!tr! where n=tk+r and 0⩽r<k. From this simple bound we derive that (Pnk)1/n∼k!1/k whenever k=o(n) and deduce a number of structural results about matrices which achieve Pnk. These include restrictions for large n and k on the number of components which may be drawn from Λk+ck for a constant c⩾1.Our results can be directly applied to maximisation problems dealing with the number of extensions to Latin rectangles or the number of perfect matchings in regular bipartite graphs
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