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Proceedings of the American Mathematical Society
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Combinatorial multinomial matrices and multinomial Stirling numbers


Author: Daniel S. Moak
Journal: Proc. Amer. Math. Soc. 108 (1990), 1-8
MSC: Primary 05A10; Secondary 15A15, 39B60, 92A15
MathSciNet review: 965944
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Abstract: Fred C. Barnett and James R. Weaver considered the stochastic matrix (1)

$\displaystyle {\left[ {\left( {\begin{array}{*{20}{c}} n \\ j \\ \end{array} } ... ...ght)}^{n - j}}{{\left( {\frac{i}{n}} \right)}^j}} \right]_{i,j = 0, \ldots ,n}}$

when modeling the spread of a viral infection through a population, where the virus has two forms. This can be generalized to viruses with $ q$ forms using the matrix (2)

$\displaystyle {\left[ {\left( {\begin{array}{*{20}{c}} n \\ {{\beta _1},{\beta ... ... + \cdots + {\alpha _q} = n,{\beta _1} + {\beta _2} + \cdots + {\beta _q} = n}}$

These matrices also appear in a different context when Konrad J. Heuvers, et al, studied the characterization of the permanent function by the Cauchy-Binet formula. In this paper, the eigenvalues and inverse of the matrix (2) are given and the existence of a basis of right eigenvectors is established.

In the process the inverse of a generalized multinomial coefficient matrix is found.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1990-0965944-4
PII: S 0002-9939(1990)0965944-4
Article copyright: © Copyright 1990 American Mathematical Society