Context-free grammars, generating functions and combinatorial arrays

Abstract

Let [Rn,k]n,k≥0 be an array of nonnegative numbers satisfying the recurrence relation Rn,k=(a1n+a2k+a3)Rn−1,k+(b1n+b2k+b3)Rn−1,k−1+(c1n+c2k+c3)Rn−1,k−2 with R0,0=1 and Rn,k=0 unless 0≤k≤n. In this paper, we first prove that the array [Rn,k]n,k≥0 can be generated by some context-free Grammars, which gives a unified proof of many known results. Furthermore, we present criteria for real rootedness of row-generating functions and asymptotical normality of rows of [Rn,k]n,k≥0. Applying the criteria to some arrays related to tree-like tableaux, interior and left peaks, alternating runs, flag descent numbers of group of type B, and so on, we get many results in a unified manner. Additionally, we also obtain the continued fraction expansions for generating functions related to above examples. As results, we prove the strong q-log-convexity of some generating functions.