Combinatorial multinomial matrices and multinomial Stirling numbers

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.