Combinatorial minors for matrix functions and their applications

TL;DR

This paper introduces combinatorial minors to generalize matrix function expansions, enabling efficient calculation of permutation enumeration, cycle indices, and related combinatorial properties for matrices.

Contribution

It develops a new approach using combinatorial minors to expand matrix functions like permanents and cycle indices, generalizing Laplace's theorem.

Findings

Generalized Laplace theorem for matrix functions

Introduced the concept of combinatorial minors

Applied to enumeration of permutations and cycle indices

Abstract

As well known, permanent of a square (0,1)-matrix $A$ of order $n$ enumerates the permutations $\beta$ of $1,2,...,n$ with the incidence matrices $B\leq A.$ To obtain enumerative information on even and odd permutations with condition $B\leq A,$ we should calculate two-fold vector $(a_1,a_2)$ with $a_1+a_2 =per A.$ More general, the introduced $\omega$-permanent, where $\omega=e^{2\pi i/m},$ we calculate as $m$-fold vector. For these and other matrix functions we generalize the Laplace theorem of their expansion over elements of the first row, using the defined so-called "combinatorial minors". In particular, in this way, we calculate the cycle index of permutations with condition $B\leq A.$