Q-analogues of the Fibo-Stirling numbers
arXiv: Combinatorics(2017)
摘要
Let F_n denote the n^th Fibonacci number relative to the initial conditions F_0=0 and F_1=1. Bach, Paudyal, and Remmel introduced Fibonacci analogues of the Stirling numbers called Fibo-Stirling numbers of the first and second kind. These numbers serve as the connection coefficients between the Fibo-falling factorial basis {(x)_↓_F,n:n ≥ 0} and the Fibo-rising factorial basis {(x)_↑_F,n:n ≥ 0} which are defined by (x)_↓_F,0 = (x)_↑_F,0 = 1 and for k ≥ 1, (x)_↓_F,k = x(x-F_1) ⋯ (x-F_k-1) and (x)_↑_F,k = x(x+F_1) ⋯ (x+F_k-1). We gave a general rook theory model which allowed us to give combinatorial interpretations of the Fibo-Stirling numbers of the first and second kind. There are two natural q-analogues of the falling and rising Fibo-factorial basis. That is, let [x]_q = q^x-1/q-1. Then we let [x]_↓_q,F,0 = [x]_↓_q,F,0 = [x]_↑_q,F,0 = [x]_↑_q,F,0=1 and, for k > 0, we let [x]_↓_q,F,k = [x]_q [x-F_1]_q ⋯ [x-F_k-1]_q, [x]_↓_q,F,k= [x]_q ([x]_q-[F_1]_q) ⋯ ([x]_q-[F_k-1]_q), [x]_↑_q,F,k= [x]_q [x+F_1]_q ⋯ [x+F_k-1]_q, and [x]_↑_q,F,k= [x]_q ([x]_q+[F_1]_q) ⋯ ([x]_q+[F_k-1]_q). In this paper, we show we can modify the rook theory model of Bach, Paudyal, and Remmel to give combinatorial interpretations for the two different types q-analogues of the Fibo-Stirling numbers which arise as the connection coefficients between the two different q-analogues of the Fibonacci falling and rising factorial bases.
更多查看译文
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络