Inequalities for the q-permanent. II Original Research Article

For a complex number q, the q-permanent of an n × n complex matrix A = ((aij)), written perq(A), is defined as image, where imagen is the symmetric group of degree n, and l(σ) the number of inversions of σ [i.e., the number of pairs, i, j such that 1 less-than-or-equals, slant i < j less-than-or-equals, slant n and σ(i) > σ(j)]. The function is of interest in that it includes both the determinant and the permanent as special cases. It is known that if A is positive semidefinite and if −1 less-than-or-equals, slant q less-than-or-equals, slant 1, then perq(A) greater-or-equal, slanted 0. We obtain results for the q-permanent, a few of which are generalizations of some results of Ando.