The Hafnian master theorem (Q2158281)

The permanent of a square matrix \(A=(a_{ij})_{1 \le i,j \le n}\) is defined as \[\operatorname{per} A= \sum_{\sigma \in S_n} a_{1 \sigma(1)} a_{2 \sigma(2)} \cdots a_{n \sigma(n)},\] where the sum extends over all elements \(\sigma\) of the symmetric group \(S_n\). The famous master theorem by \textit{P. A. MacMahon} [Combinatory analysis. Vol. 2. Cambridge: Univ. Press (1915; JFM 46.0118.07)] can be formulated as follows: Let \(Z\) be the diagonal matrix with diagonal entries \(z_1,z_2,\dots\). If we expand \((\det (I-ZA))^{-1}\) in terms of \(z_j\)'s, then the coefficients are given as permanents with matrix entries from \(A\). In this article, an extension of that formula involving Hafnians is obtained. The Hafnian of a symmetric matrix \(S=(s_{ij})_{1 \le i,j \le 2m}\) is defined as \[ \operatorname{Hf} S= \frac{1}{2^m m!} \sum_{\sigma \in S_{2m}} s_{\sigma(1) \sigma(2)} s_{\sigma(3) \sigma(4)} \dots s_{\sigma(2m-1) \sigma(2m)}. \] This is the sign-less version of the Pfaffian. The main theorem in this article states that all coefficients of \(z_j\)'s in the expansion of \((\det(I-\mathbf{Z} S))^{-1/2}\) are given by Hafnians. Here \(\mathbf{Z}\) is the direct sum of matrices \(\left(\begin{smallmatrix} 0 & z_j \\ z_j & 0 \end{smallmatrix}\right)\). An alternative extension of the master theorem was studied in [\textit{S. Matsumoto}, Linear Algebra Appl. 403, 369--398 (2005; Zbl 1077.15008)].