Connection Coefficients Between Generalized Rising and Falling Factorial Bases

Abstract

Let ${\mathcal{S} = (s_1, s_2, \ldots)}$ be any sequence of nonnegative integers and let ${{S_{k} = \sum_{i=1}^k} {s_i}}$.We then define the falling (rising) factorials relative to ${\mathcal{S}}$ by setting ${(x)\downarrow_{k, \mathcal{S}} = (x- S_{1})(x-S_2) \cdots (x-S_k)}$ and ${(x)\uparrow_{k, \mathcal{S}}= (x+S_1)(x+S_{2}) \cdots (x+S_{k})}$ if ${k \geq 1}$ with ${(x)\downarrow_{0,\mathcal{S}}= (x)\uparrow_{0, \mathcal{S}}= 1}$. It follows that ${\{(x)\downarrow_{k,\mathcal{S}}\}_{k > 0}}$ and ${\{(x)\uparrow_{k,\mathcal{S}}\}_{k > 0}}$ are bases for the polynomial ring ${\mathbb{Q}[x]}$. We use a rook theory model due to Miceli and Remmel to give combinatorial interpretations for the connection coefficients between any two of the bases ${\{(x)\downarrow_{k,\mathcal{S}}\}_{k \geq 0}, \{(x)\uparrow_{k,\mathcal{S}}\}_{k \geq 0}, \{(x)\downarrow_{k,\mathcal{T}}\}_{k \geq 0}}$, and ${\{(x)\uparrow_{k, \mathcal{T}}\}_{k \geq 0}}$ for any two sequences of nonnegative integers ${\mathcal{S} =(s_1, s_2, \ldots)}$ and ${\mathcal{T} = (t_1, t_2, \ldots)}$. We also give two different q-analogues of such coefficients. Moreover, we use this rook model to give an alternative combinatorial interpretation of such coefficients in terms of certain pairs of colored permutations and set partitions with restricted insertion patterns.