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Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Parity Theorems for Statistics on Domino Arrangements. | Parity Theorems for Statistics on Domino Arrangements Mark A. Shattuck Mathematics Department University of Tennessee Knoxville TN 37996-1300 shattuck@math.utk.edu Carl G. Wagner Mathematics Department University of Tennessee Knoxville TN 37996-1300 wagner@math.utk.edu Submitted Oct 12 2004 Accepted Jan 21 2005 Published Jun 14 2005 MR Subject Classifications 11B39 05A15 Abstract We study special values of Carlitz s -Fibonacci and q-Lucas polynomials Fn q t and Ln q t . Brief algebraic and detailed combinatorial treatments are presented the latter based on the fact that these polynomials are bivariate generating functions for a pair of statistics defined respectively on linear and circular domino arrangements. 1 Introduction In what follows N and P denote respectively the nonnegative and the positive integers. If q is an indeterminate then nq 1 q qn 1 if n 2 P 0 q 1 nq 1q2q nq if n 2 P and _ if0 6 k 6 n o.n k q .n-k q I1.1 0 if k 0 or 0 6 n k. A useful variation of 1.1 is the well known formula 10 p.29 Cn X qOdo id1 - fcdfc Xp k n - k t qt 1.2 q do di dk n k t 0 díÉN where p k n k t denotes the number of partitions of the integer t with at most n k parts each no larger than k. This paper elucidates certain features of the q-Fibonacci polynomials Fj q t y qk n h tk n 2 N 1.3 k k n 2 x q THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 N10 1 and the q-Lucas polynomials Ln q t JL.Lnk 0s k . n 2 v Jq v k tk. q ne P. 1.4 Note that Fn 1 1 Fn where F0 F1 1 and Fn Fn-1 Fn-2 n 2 this parameterization of the Fibonacci numbers also employed by Wilf 12 results here in a notation with mnemonic features superior to that of the classical parameterization and Ln 1 1 Ln where Li 1 L2 3 and Ln Ln-1 Ln-2 n 3. Our aim here is to present both algebraic and combinatorial treatments of Fn 1 1 Fn 1 t Ln 1 1 and Ln 1 t . Our algebraic proofs make frequent use of the identity 11 pp. 201-202 k X kqx 1 x 1 qX . 1 qkX k 2 N. 1.5 Our combinatorial proofs use the fact that Fn q t and Ln q t are .
