tailieunhanh - Báo cáo toán học: "Generating function for K-restricted jagged partitions"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Generating function for K-restricted jagged partitions. | Generating function for K-restricted jagged partitions . Fortin P. Jacob and P. Mathieu Departement de physique de génie physique et d optique Universite Laval Quebec Canada G1K 7P4. Submitted Oct 9 2004 Accepted Feb 11 2005 Published Feb 21 2005 Mathematics Subject Classifications 05A15 05A17 05A19 Abstract We present a natural extension of Andrews multiple sums counting partitions of the form Al Xm with Xi Xi k-1 2. The multiple sum that we construct is the generating function for the so-called K-restricted jagged partitions. A jagged partition is a sequence of non-negative integers n1 n2 nm with nm 1 subject to the weakly decreasing conditions ni ni 1 1 and ni ni 2. The K-restriction refers to the following additional conditions ni ni K-1 1 or ni ni 1 1 ni K-2 1 ni K-1. The corresponding generalization of the Rogers-Ramunjan identities is displayed together with a novel combinatorial interpretation. 1 Introduction In 1981 Andrews 2 showed that the generating function for partitions with prescribed number of parts subject to the following difference 2 condition j j k-1 2 1 and containing at most i 1 parts equal to 1 is Fk i z q mi X mk-i 0 qN1 N2-i Li zN q mi q mfc-i 2 Present address Department of Physics and Astronomy Rutgers the State University of New Jersey Piscataway NJ 08854-8019 jffor27@ Present address Department of Mathematical Sciences University of Durham Durham DH1 3L UK lpmathieu@. This work is supported by NSERC. THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 R12 1 with Nj mj mk-1 Lj Nj Nk-1 N L1 3 Lk Lk 1 0 and a n n-1 a q n n 1 atf . 4 i 0 This is a one-parameter deformation of the multiple q-series of the analytic Andrews-Gordon identity 1 3 . In this work we present the derivation of the generating function for jagged partitions of length m which are sequences of non-negative integers n n2 nm satisfying n Uj 1 - 1 Uj Uj 2 nm 1 5 and further subject to the following K-restrictions .