Permanental Inequalities and Unit Interval Orders

Given an n × n matrix A = (a i, j ) and I, J ⊆ [1, n] := {1, . . ., n}, let A I,J = (a i, j ) i∈I, j∈J denote the (I, J)-submatrix of A. We consider a class of 0-1 totally nonnegative (TNN) matrices arising as antiadjacency matrices of unit interval orders, and show that for each matrix A in this class, the inequalities..,n , and B n be the parity alternating permutations {w = w 1 • • • w n ∈ S n | i and w i have the same parity}.We find a bijective map f n : A n → B n such that w ≤ f n (w) under the Bruhat order for n ≤ 13 and conjecture this holds for all n.The existence of the bijective map f n ensures that inequalities ( * ) are satisfied for all TNN matrices A when h = ⌊ n 2 ⌋.We also conjecture the inequalities ( * ) to hold for all TNN matrices and all h = 1, . . ., n -1.