... K)=−w(G,K).In the second case the entries in the i-th column (i, j, i, 1) satisfy the inequalities i +1≤ j ≤ n,whilethej-th column (on the right of the i-thcolumn) is of the form (aj,bj,j,0). Then ... Let (k,l)beanotherlink of G such that k ∈ K.Thenk<iby definition of i. On the otherhand, l = j.Ifl = i,then(k, l)remainsalinkofG.Ifl = i,then(bk,k)=(bi,ai) and the link (k,i)inG ... denotes the permanent of a matrix A, and the matrix prod-uct XtP that appears on the right side of TURis taken with the assump-tion that the xi,j and pi,jcommute.Since the proof of this...