Signs in the Laplace expansions and the parity of the distinguished representatives (Q1332413)

The author solves the problem of the numbers of positive and negative terms (\(P_{n,k}\) and \(N_{n,k}\), respectively) in the Laplace expansion of an arbitrary determinant of \(n\)th order along \(k\) rows. Explicit expressions and Pascal-like triangles are found for the numbers \(P_{n,k}\), \(N_{n,k}\) and their difference \(D_{n,k}= P_{n,k}- N_{n,k}\), together with their generating functions, e.g. \[ D_{n,k}= {1\over 2} \bigl(1+ (-1)^{k(n-1)}\bigr)\left({\lfloor n/2\rfloor\atop \lfloor k/2\rfloor}\right) \] and \[ D(X,Y)= \sum_{n,k\geq 0} D_{n,k} X^ k Y^ n= (1+ Y+ XY)/(1- Y^ 2- X^ 2 Y^ 2). \] Applications to permanents, Pfaffians and distinguished coset representations of parabolic subgroups of the symmetric group (as Coxeter group) are mentioned.