tailieunhanh - Báo cáo toán học: "Combinatorial statistics on type-B analogues of noncrossing partitions and restricted permutations"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Combinatorial statistics on type-B analogues of noncrossing partitions and restricted permutations. | Combinatorial statistics on type-B analogues of noncrossing partitions and restricted permutations Rodica Simion Department of Mathematics The George Washington University Washington DC 20052 Submitted September 5 1999 Accepted January 5 2000 Abstract We define type-B analogues of combinatorial statistics previously studied on noncrossing partitions and show that analogous equidistribution and symmetry properties hold in the case of type-B noncrossing partitions. We also identify pattern-avoiding classes of elements in the hyperoctahedral group which parallel known classes of restricted permutations with respect to their relations to noncrossing partitions. Key words restricted permutations pattern-avoidance noncrossing partitions signed permutations permutation statistics partition statistics ợ-analogue AMS Classification 05 This paper was dedicated by the author to George Andrews on the occasion of his 60th birthday. We are very sad to report that Rodica Simion died on January 7 2000 just two days after the acceptance of this paper. We have made the few minor changes requested by the referee and are honored to present her paper here. The Editors Partially supported by the National Science Foundation award DMS-9970957. 1 THE ELECTRONIC .JOURNAL OF COMBINATORICS 7 2000 R9 2 1 Introduction The goal of this paper is to give type-B analogues of enumerative results concerning combinatorial statistics defined on type-A noncrossing partitions and on certain classes of permutations characterized by pattern-avoidance. To this end we need B-analogues of the combinatorial objects in question. As type-B noncrossing partitions we use those studied by Reiner 20 . In the hyperoctahedral group the natural B-analogue of the symmetric group we identify classes of restricted signed permutations with enumerative properties analogous to those of the 132- and 321-avoiding permutations in the symmetric group. We also propose definitions for four partition statistics lsB lbB rsB and rbB