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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
DOI: https://doi.org/10.1090/S0002-9939-1990-0965944-4
MathSciNet review: 965944
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Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [1] F. C. Barnett and J. R. Weaver, Eigenvalues and eigenvectors of a certain stochastic matrix, Linear and Multilinear Algebra 13 (1983), 345-350. MR 704783 (84k:15010a)
  • [2] D. A. Cohen, Basic techniques of combinatorial theory, John Wiley and Sons, Inc., New York, 1978. MR 533589 (81i:05001)
  • [3] K. J. Heuvers, L. J. Cummings and K. P. S. Bhaskara Rao, A characterization of the permanent function by the Binet-Cauchy Theorem, Linear Algebra Appl. 101 (1988), 49-72. MR 941295 (89h:15013)
  • [4] J. Riordan, Combinatorial identities, John Wiley and Sons, Inc., New York, 1968. MR 0231725 (38:53)
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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-0965944-4
Article copyright: © Copyright 1990 American Mathematical Society

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