TAILIEUCHUNG - Báo cáo toán học: " Exterior Pairs and Up Step Statistics on Dyck Paths"

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: Exterior Pairs and Up Step Statistics on Dyck Paths. | Exterior Pairs and Up Step Statistics on Dyck Paths Sen-Peng Eu Tung-Shan FU Department of Applied Mathematics Mathematics Faculty National University of Kaohsiung National Pingtung Institute of Commerce Kaohsiung 811 Taiwan ROC Pingtung 900 Taiwan ROC speu@ tsfu@ Submitted Jan 29 2010 Accepted April 8 2011 Published Apr 21 2011 Mathematics Subject Classifications 05A15 05A19 Abstract Let Cn be the set of Dyck paths of length n. In this paper by a new automorphism of ordered trees we prove that the statistic number of exterior pairs introduced by A. Denise and R. Simion on the set Cn is equidistributed with the statistic number of up steps at height h with h 0 mod 3 . Moreover for m 3 we prove that the two statistics number of up steps at height h with h 0 mod m and number of up steps at height h with h m 1 mod m on the set Cn are almost equidistributed . Both results are proved combinatorially. Keywords Dyck path exterior pair ordered tree planted tree continued fraction 1 Introduction Let Cn denote the set of lattice paths called Dyck paths of length n in the plane Z X Z from the origin to the point 2n 0 using up step 1 1 and down step 1 1 that never pass below the x-axis. Let U and D denote an up step and a down step respectively. In 3 Denise and Simion introduced and investigated the two statistics pyramid weight and number of exterior pairs on the set Cn. A pyramid in a Dyck path is a section of the form UhDh a succession of h up steps followed immediately by h down steps where h is called the height of the pyramid. The pyramid is maximal if it is not contained in a higher pyramid. The pyramid weight of a Dyck path is the sum of the heights of its maximal pyramids. An exterior pair in a Dyck path is a pair consisting of an up step and its matching down step which do not belong to any pyramid. For example the path shown in Figure 1 contains three maximal pyramids with a total weight of 4 and two exterior pairs. Partially supported by .

Đã phát hiện trình chặn quảng cáo AdBlock
Trang web này phụ thuộc vào doanh thu từ số lần hiển thị quảng cáo để tồn tại. Vui lòng tắt trình chặn quảng cáo của bạn hoặc tạm dừng tính năng chặn quảng cáo cho trang web này.