TAILIEUCHUNG - Báo cáo toán học: " λ-factorials of n"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: λ-factorials of n. | A-factorials of n Yidong Sunk and Jujuan Zhuangk Department of Mathematics Dalian Maritime University 116026 Dalian . China lsydmath@ 1 jjzhuang1979@ Submitted Aug 26 2010 Accepted Nov 27 2010 Published Dec 10 2010 Mathematics Subject Classification 05A05 05A19 05A40 05C05. Abstract Recently by the Riordan identity related to tree enumerations n X ki k 1 n 1 n-k n 1 n 1 U n Sun and Xu have derived another analogous one n è k k 0 7 Dk i n 1 n-k kn 1 where Dk is the number of permutations with no fixed points on 1 2 . k . In the paper we utilize the A-factorials of n defined by Eriksen Freij and Wdstlund to give a unified generalization of these two identities. We provide for it a combinatorial proof by the functional digraph theory and two algebraic proofs. Using the umbral representation of our generalized identity and Abel s binomial formula we deduce several properties for A-factorials of n and establish interesting relations between the generating functions of general and exponential types for any sequence of numbers or polynomials. Keywords Derangement A-factorial of n Charlier polynomial Bell polynomial Hermite polynomial. 1 Introduction Let Sn denote the set of permutations of n 1 2 . n . A fixed point of a permutation n e Sn is an element i G n such that n i i. Denote by fix n the number of fixed THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R169 1 points of n. Recently Eriksen Freij and Wdstlund 6 defined the polynomials called the A-factorials of n by setting fn A o A 1. VS They utilized the polynomials fn A to give closed formulas for the number of derangements permutations with no fixed points with descents in prescribed positions and derived several nice properties for fn A such as fn A A Ê Y fk A A -k k 0 fn A È fk k A 1 n-k k 0 fn A nfn-1 A A 1 n. -f- MA nfn-i A . dA Clearly we have fn 0 Dn 17 A000166 and fn 1 n where Dn is the number of derangements in Sn. The relation indicates that fn A n 0 1 . form

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