# Combinatorial approach to enumeration of doubly stochastic non-negative integer square matrices

Combinatorial approach to enumeration of doubly stochastic non-negative integer square matrices -- 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 [1] 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 [2] 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 [6]. The literature on the ADG hypothesis and its generalizations is quite extensive, we note only the papers [1-12, 14-18]. In [12] a combinatorial approach to evaluation of H(n,r) was suggested. Let R be http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discrete Mathematics and Applications de Gruyter

# Combinatorial approach to enumeration of doubly stochastic non-negative integer square matrices

Discrete Mathematics and Applications, Volume 3 (6) – Jan 1, 1993
14 pages

/lp/de-gruyter/combinatorial-approach-to-enumeration-of-doubly-stochastic-non-sHx0mH3k2B
Publisher
de Gruyter
ISSN
0924-9265
eISSN
1569-3929
DOI
10.1515/dma.1993.3.6.649
Publisher site
See Article on Publisher Site

### Abstract

-- 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 [1] 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 [2] 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 [6]. The literature on the ADG hypothesis and its generalizations is quite extensive, we note only the papers [1-12, 14-18]. In [12] a combinatorial approach to evaluation of H(n,r) was suggested. Let R be

### Journal

Discrete Mathematics and Applicationsde Gruyter

Published: Jan 1, 1993

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