Convertible and m-convertible matrices

Abstract

In this article, we consider the calculation of the permanent of a (0, 1)-matrix using determinants. We investigate when determinants of more than one signing of the original are used to calculate the permanent and we show that non-convertible matrices require at least four different signings. Then, we loosen the restriction that the matrices used to convert the permanent are signings of the original and use this to reduce the number of determinants necessary to convert the permanent of the all 1's matrix by considering a particular partition of the set S _n of permutations of {1, … , n}. Finally, we construct a sequence of maximal convertible matrices with a small number of nonzero entries, thus lowering the possible upper bound for the number of nonzero entries of such a matrix, relative to the order.

Keywords:

• permanent
• convertible matrix
• m-convertible matrix
• sign-nonsingular matrix

AMS Subject Classifications::

• 15A15
• 05C50
• 15B35