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Báo cáo toán học: "The Enumeration of Sequences with Restrictions on their Partial Sum"

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: The Enumeration of Sequences with Restrictions on their Partial Sums. | The Enumeration of Sequences with Restrictions on their Partial Sums Stephen Suen Department of Mathematics and Statistics University of South Florida ssuen@usf.edu Kevin P. Wagner Department of Mathematics and Statistics University of South Florida kwagner@mail.usf.edu Submitted Mar 26 2010 Accepted Nov 17 2010 Published Nov 26 2010 Mathematics Subject Classification 05A15 05A10 Abstract We examine sequences containing p t s and pt r 1 s where p t and r are integers satisfying p 0 t 1 and pt r 0. We develop a rotation method to enumerate the number of sequences meeting additional requirements related to their partial sums. We also define downcrossings about I and their downcrossing numbers and obtain formulas for the number of sequences for which the sum of the downcrossing numbers equals k for I r 1. We finish with an investigation of the first downcrossing number about I for any I. Keywords. Lattice paths ballot problem rotation method crossings crossing sums generalized binomial series. 1 Introduction We shall assume throughout that p t and r are integers satisfying p 0 t 1 and pt r 0. Let Q Qp r Q r denote the collection of all sequences containing p t s and pt r 1 s. For a sequence u G Q let Uj denote its jth digit and let Sk w denote its kth partial sum. That is k Sk E j 1 wj with So 0 and Spt p r r. One common way to picture the sequences in relation to their partial sums is by considering the sequences as paths j Sj 0 j pt p r with each 1 meaning go right one go up one and each t meaning go right one go down t. We study the number of paths with conditions on their partial sums on their number of crossings crossing THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R160 1 numbers and crossing sums which we shall define later. Our interest in these sequences originated from our investigation of the acceptance urn model involving pt r 1 balls and p t balls. For t 1 see Chen et. al. 1 and Suen and Wagner 12 . These sequences are also related to lattice paths see