Free rises, restricted partitions, and q-Fibonacci polynomials

Given a word w = w₁w₂... w_n, we will call an ordered pair (i, j) for which 1 ≤ i < j ≤ n and w_i < w_j a free rise. In this paper, we consider two classes of set partitions having cardinality given by the m-Fibonacci numbers and enumerate them according to the number of free rises, where partitions are represented, canonically, as restricted growth functions. In addition, we identify several classes of partitions avoiding two patterns of length four and having cardinality given by the Fibonacci numbers of even index. We enumerate two of these classes jointly according to the number of blocks and to the number of free rises, where we now require that the larger letter in a free rise correspond to the left-most occurrence of a letter of its kind. The resulting x, q-polynomials seem to be new and several identities are derived generalizing those of the even-indexed Fibonacci numbers.