2011/05/16 by Shevelev, Vladimir
#15A15 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1105.3154
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≤ A. To obtain enumerative information on even and odd permutations with condition B≤ A, we should calculate two-fold vector (a1,a2) with a1+a2 =per A. More general, the introduced ω-permanent, where ω=e2πi/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≤ A.