  We introduce an analytical iterative method, the density matrix recursion
method, to generate arbitrary reduced density matrices of superpositions of
short-range dimer coverings on periodic or non-periodic quantum spin-1/2 ladder
lattices, with an arbitrary number of legs. The method can be used to calculate
bipartite as well as multipartite physical properties, including bipartite and
multi-partite entanglement. We apply this technique to distinguish between
even- and odd-legged ladders. Specifically, we show that while genuine
multi-partite entanglement decreases with increasing system size for the
even-legged ladder states, it does the opposite for odd-legged ones.
