   ISSN 1088-6826(online) ISSN 0002-9939(print)

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
DOI: https://doi.org/10.1090/S0002-9939-1990-0965944-4
MathSciNet review: 965944
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Abstract: Fred C. Barnett and James R. Weaver considered the stochastic matrix (1) ${\left [ {\left ( {\begin {array}{*{20}{c}} n \\ j \\ \end {array} } \right ){{\left ( {\frac {{n - i}}{n}} \right )}^{n - j}}{{\left ( {\frac {i}{n}} \right )}^j}} \right ]_{i,j = 0, \ldots ,n}}$ 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) ${\left [ {\left ( {\begin {array}{*{20}{c}} n \\ {{\beta _1},{\beta _2}, \ldots ,{\beta _q}} \\ \end {array} } \right ){{\left ( {\frac {{{\alpha _1}}}{n}} \right )}^{{\beta _1}}}{{\left ( {\frac {{{\alpha _2}}}{n}} \right )}^{{\beta _2}}} \cdots {{\left ( {\frac {{{\alpha _q}}}{n}} \right )}^{{\beta _q}}}} \right ]_{{\alpha _1} + {\alpha _2} + \cdots + {\alpha _q} = n,{\beta _1} + {\beta _2} + \cdots + {\beta _q} = n}}$ 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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