On a matrix function interpolating between determinant and permanent (Q2575020)

For a permutation \(\sigma\) of \(\{1,2,\dots,n\}\) define \(\gamma(\sigma) =\prod(1-t^{| w| })\) where the product is over all orbits \(w\) of \(\sigma\) and \(| w| \) is the cardinality of \(w\). Then for an \(r\times r\) matrix \(M=[m_{ij}]\) define \(\Gamma(M)=\sum_\sigma \gamma(\sigma) \prod_i m_{i\sigma(i)}\). Considering \(\Gamma(M)\) as a polynomial in \(t\), the constant term is the permanent of \(M\) while the leading coefficient is, up to sign, the determinant of \(M\). For \(I\subseteq\{1,2,\dots,r\}\) define \(I^*=\{1,2,\dots,r\}\setminus I\) and let \(M[I]\) denote the submatrix of \(M\) formed by all cells for which the row and column indices are both in \(I\). The first result of this note is to show \[ \Gamma(M)=\sum_I(-t)^{| I| }\det M[I]\,\text{per}\,M[I^*]. \] When \(t=1\) this specialises to an old identity due to \textit{T. Muir} [A relation between permanents and determinants. Proc. Roy. Soc. Edinburgh 22, 134--136 (1897)]. The remainder of the note shows an application to the calculation of multisymmetric power sums of the (finitely many) solutions to a certain system of quadratic equations.