tailieunhanh - Báo cáo toán học: "Permutations avoiding two patterns of length three"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Permutations avoiding two patterns of length three | Permutations avoiding two patterns of length three Vincent R. Vatter Department of Mathematics Rutgers University Piscataway NJ 08854 vatter@ Submitted Nov 25 2002 Accepted Jan 9 2003 Published Jan 22 2003 MR Subject Classifications 05A15 68R15 Keywords Restricted permutation forbidden subsequence generating tree Abstract We study permutations that avoid two distinct patterns of length three and any additional set of patterns. We begin by showing how to enumerate these permutations using generating trees generalizing the work of Mansour 13 . We then find sufficient conditions for when the number of such permutations is given by a polynomial and answer a question of Egge 6 . Afterwards we show how to use these computations to count permutations that avoid two distinct patterns of length three and contain other patterns a prescribed number of times. 1 Introduction Let q q1q2 . .qk be a permutation in the symmetric group Sk. We call k the length of q and write q k. The reduction of a word w of distinct integers of length k red w is the k-permutation obtained by replacing the smallest number element of w by 1 the second smallest element by 2 and so on. We say that the permutation p p1p2. .pn E Sn contains a q pattern if there is a subsequence pi pi2. .pik of p that reduces to q that is red pixpi2. .pik q. Otherwise we say that p is q-avoiding. For example 3142 contains a 132 pattern because red 142 132 whereas 3124 is 132-avoiding. Let the set Sn q consist of all n-permutations that avoid q. If Q is a set of permutations we define Sn Q Sn q q Q This work has been partially supported by an NSF VIGRE grant to the Rutgers University Department of Mathematics. THE ELECTRONIC JOURNAL OF COMBINATORICS 9 2 2003 R6 1 so Sn Q consists of all n-permutations that avoid every member of Q. We also define S Q Sn Q n 1 and set Sn Q Sn Q - Note that if q1 q2 G Q and q2 contains q1 then the q2 restriction is superfluous since every q1-avoiding permutation is also .