A weak composition is an infinite sequence M = (M(1),M(2), ...) of nonnegative integers whose sum is finite. The partially ordered set (or poset) WCOMP of weak compositions is defined by the order relation N <= M for M,N in WCOMP whenever N(i) <= M(i) for all i. In this work we study the graded Segre powers WCOMP^{\circ s} of WCOMP. We use the lexicographic shellability on WCOMP^{\circ s} to provide three different formulas for the convolutional inverse of the zeta function, also known as the Möbius function. One of these formulas, when s = 1, is at the same time a generalization of the classical number-theoretical theorem that describes the inverse of the Riemann zeta function and also a classical inversion theorem in the theory of symmetric functions. When s = 2, the formula specializes to an interesting inversion theorem already studied by González D'León.