-- Let HR(n, r) be equal to the number of matrices with non-negative integer elements such that all row sums and all column sums are equal to r and all elements with indices from a set A are equal to zero. We investigate the properties of the function HR(H, r) and give a combinatorial interpretation of the obtained results. 1. INTRODUCTION In  Kenji Mano investigated the number H(n,r) of different ways to allocate rar objects of ra types with r objects of each type among ra persons such that each person receives r objects. The number #(n,r) may be interpreted as the number of ra ra matrices (a tj ) with non-negative integer elements which satisfy the conditions In all subsequent papers H(n,r) is the number of such matrices. In  the following hypothesis (ADG hypothesis) was proposed: for any ra and r where = \ \ and ct depend onraand i only. Representation (1.2) was proved by Stanley [3, 4] and Ehrhart . The literature on the ADG hypothesis and its generalizations is quite extensive, we note only the papers [1-12, 14-18]. In  a combinatorial approach to evaluation of H(n,r) was suggested. Let R be
Discrete Mathematics and Applications – de Gruyter
Published: Jan 1, 1993
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