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-- The matrix function )() (the sum of all the subpermanents of order t of a square matrix A of order n), which was introduced by Tverberg [1], is evaluated for the first time on linear spans of two or three substitution matrices and those matrices which are close to them. The results of the present paper, which are a generalization of the classical Touchard-Kaplansky formulae for the menage problem and results of Mine [7), Moser [8], and others, are obtained by the method of allocation index developed by the author. As a supplement an explicit formula is proved for the numbers of the generalized (linear) menage problem of order 3. Canfield and Wormald [10] have recently ascertained the existence of a recurrent formula of rather high order for these numbers. 1. INTRODUCTION The sum <() of all the subpermanents oi older t ( all the permanent* of submatrices of order i) of an matrix A = {atj} was first considered by Tverberg [1]. A great number of works (for references see [2,3]), dealing with the generalized van der Waerden conjecture [1] which emerged during two decades, was mainly devoted to qualitative investigations of ^() in the class
Discrete Mathematics and Applications – de Gruyter
Published: Jan 1, 1992
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