Combinatorial minors for matrix functions and their applications

As well known, permanent of a square (0,1)-matrix $A$ of order $n$ enumerates the permutations $β$ 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 $ω$-permanent, where $ω=e^{2πi/m},$ 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 $B\leq A.$

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