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: Counting subwords in a partition of a set. | Counting subwords in a partition of a set Toufik Mansour Department of Mathematics University of Haifa 31905 Haifa Israel toufik@ Mark Shattuck Department of Mathematics University of Tennessee Knoxville TN 37996 shattuck@ Sherry . Yan Department of Mathematics Zhejiang Normal University 321004 Jinhua . China huifangyan@ Submitted Sep 22 2009 Accepted Jan 15 2010 Published Jan 22 2010 Mathematics Subject Classification 05A18 05A15 05A05 68R05 Abstract A partition n of the set n 1 2 . n is a collection B1 . Bk of nonempty disjoint subsets of n called blocks whose union equals n . In this paper we find explicit formulas for the generating functions for the number of partitions of n containing exactly k blocks where k is fixed according to the number of occurrences of a subword pattern T for several classes of patterns including all words of length 3. In addition we find simple explicit formulas for the total number of occurrences of the patterns in question within all the partitions of n containing k blocks providing both algebraic and combinatorial proofs. 1 Introduction A partition of n 1 2 . n is a decomposition of n into non-overlapping subsets B1 B2 . Bk called blocks which are listed in increasing order of their least elements 1 k n . We will represent a partition n B1 B2 . Bk in the canonical sequential form n n1n2 nn such that j G Bn. 1 j n. Therefore a sequence n n1n2 nn over the alphabet k represents a partition of n with k blocks if and only if it is a restricted growth function of n onto k see . 11 13 14 for details . For instance The third author was supported by the National Natural Science Foundation of China no. 10901141 . THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R19 1 123214154 is the canonical sequential form of the partition 1 5 7 2 4 3 6 9 8 of 9 . Throughout this paper partitions will be identified with their corresponding canonical sequences. The set of all partitions of n with exactly k .