Q-analogues of convolutions of Fibonacci numbers

Let NR([k]) denote the set of words over the alphabet [k] = {1, . . . , k} with no consecutive repeated letters. Given a word w = w1 . . . wn ∈ NR([k]), or more generally in [k]∗, we say that a pair 〈wi, wj〉 matches the μ pattern if i < j, wi < wj, and there is no i < k < j such that wi ≤ wk ≤ wj. We say that 〈wi, wj〉 is a trivial μ-match if wi + 1 = wj and a nontrivial μ-match if wi + 1 < wj. For each word w in NR([k]), let the weight of w be given by t|w|pntriv(w)qtriv(w), where |w| is the length of w, ntriv(w) is the number of nontrivial μ-matches in w and triv(w) is the number of trivial μ-matches in w. We study the generating functions N (p, q, t) that sum the weights of all words w in NR([3]) starting with the letter i and ending with the letter j. In particular, we show that N(p, 1, t) = ∑ r≥0 tp (1− t− t2)r+1 so that the number of words in NR([3]) starting with 3, ending with 1, and having r nontrivial μ-matches is counted by the convolutions of r + 1 copies of the Fibonacci numbers. It follows that the coefficient of p in N(p, q, t) is a q-analogue of the generating function of the convolution of r + 1 copies of the Fibonacci numbers. The main goals of this paper are to compute the generating functions N (p, q, t) and prove a number of combinatorial properties of their coefficients.