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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Combinatorial multinomial matrices and multinomial Stirling numbers
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by Daniel S. Moak PDF
Proc. Amer. Math. Soc. 108 (1990), 1-8 Request permission

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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Additional Information
  • © Copyright 1990 American Mathematical Society
  • 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