Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: Dumont’s statistic on words. | Dumont s statistic on words Mark Skandera Department of Mathematics University of Michigan Ann Arbor MI mskan@ Submitted August 4 2000 Accepted January 15 2001. MR Subject Classifications 06A07 68R15 Abstract We define Dumont s statistic on the symmetric group Sn to be the function dmc Sn N which maps a permutation Ơ to the number of distinct nonzero letters in code ơ . Dumont showed that this statistic is Eulerian. Naturally extending Dumont s statistic to the rearrangement classes of arbitrary words we create a generalized statistic which is again Eulerian. As a consequence we show that for each distributive lattice J P which is a product of chains there is a poset Q such that the f-vector of Q is the -vector of J P . This strengthens for products of chains a result of Stanley concerning the flag -vectors of Cohen-Macaulay complexes. We conjecture that the result holds for all finite distributive lattices. 1 Introduction Let Sn be the symmetric group on n letters and let us write each permutation in Sn in one line notation 1 n. We call position i a descent in if i i 1 and an excedance in if i i. Counting descents and excedances we define two permutation statistics des Sn N and exc Sn N by des i I i i i exc i I i i . It is well known that the number of permutations in Sn with k descents equals the number of permutations in Sn with k excedances. This number is often denoted A n k 1 and the generating function AnO X A n k 1 xk 1 X X k 0 K2Sn ft2Sn THE ELECTRONIC JOURNAL OF COMBINATORICS 8 2001 R11 1 is called the nth Eulerian polynomial. Any permutation statistic stat Sn N satisfying An x X . 2S or equivalently k 2 Sn I stat K kg k 2 Sn I des K kg for k 0 . n 1 is called Eulerian. A third Eulerian statistic essentially defined by Dumont 6 counts the number of distinct nonzero letters in the code of a permutation. We define code K to be the word Cl cn where ci j i 1 Kj Kig. Denoting Dumont s statistic by dmc we have dmc K 0 I appears in code k g. .
