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

- Fred C. Barnett, James R. Weaver, and J. S. Frame,
*Eigenvalues and eigenvectors of a certain stochastic matrix*, Linear and Multilinear Algebra**13**(1983), no. 4, 345–350. MR**704783**, DOI https://doi.org/10.1080/03081088308817533 - Daniel I. A. Cohen,
*Basic techniques of combinatorial theory*, John Wiley & Sons, New York-Chichester-Brisbane, 1978. MR**533589** - Konrad 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**, DOI https://doi.org/10.1016/0024-3795%2888%2990142-5 - John Riordan,
*Combinatorial identities*, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0231725** - Robert Shelton, Konrad J. Heuvers, Daniel Moak, and K. P. S. Bhaskara Rao,
*Multinomial matrices*, Discrete Math.**61**(1986), no. 1, 107–114. MR**850935**, DOI https://doi.org/10.1016/0012-365X%2886%2990033-6

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Article copyright:
© Copyright 1990
American Mathematical Society