Periodicity and Parity Theorems for a Statistic on r-Mino Arrangements
نویسندگان
چکیده
If r > 2, the r-Fibonacci numbers F (r) n are defined by F (r) 0 = F (r) 1 = · · · = F (r) r−1 = 1, with F (r) n = F (r) n−1 + F (r) n−r if n > r. The r-Lucas numbers L (r) n are defined by L (r) 1 = L (r) 2 = · · · = L (r) r−1 = 1 and L (r) r = r + 1, with L (r) n = L (r) n−1 + L (r) n−r if n > r + 1. If r = 2, the F (r) n and L (r) n reduce, respectively, to the classical Fibonacci and Lucas numbers (parametrized as in Wilf [12], by F0 = F1 = 1, etc., and L1 = 1, L2 = 3, etc.). Polynomial generalizations of Fn and/or Ln have arisen as generating functions for statistics on binary words [1], lattice paths [5], and linear and circular domino arrangements [8]. Generalizations of F (r) n and/or L (r) n have arisen similarly in connection with statistics on Morse code sequences [4] as well as on linear and circular r-mino arrangements [9]. Cigler [3] introduces and studies a new class of q-Fibonacci polynomials, generalizing the classical sequence, which arise in connection with a certain statistic on Morse code sequences in which the dashes have length 2. The same statistic, which we’ll denote by π, applied more generally to linear r-mino arrangements, leads to the polynomial generalization
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