Discrete Mathematics 99 (1992) 63-68 North-Holland 63 Partition statistics on permutations Paul H. Edelman” zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC School of Mathematics, University of Minnesota, Minneapolis, MN 554.55, USA Rodica Simian* * Department of Mathematics, The George W ashington University, W ashington, D. C. 20052 USA Dennis White* School of Mathematics, Universiry of Minnesota, Minneapolis, MN 55455, USA Received 5 July 1989 Revised 1 December 1989 Abstract Edelman, P.H., R. Simion and D. White, Partition statistics on permutations, Discrete Mathematics 99 (1992) 63-68. We describe some properties of a new statistic on permutations. This statistic is closely related to a well-known statistic on set partitions. In [7] four statistics on set partitions were described, each having the q-Stirling numbers [2] as their distribution generating function. In this paper we will show an interesting relationship between one of these statistics, Is, defined below, and permutations. Specifically, we compute the distribution polynomial c qW, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA 0.Z.Y” as well as E& PFlp(& o)PO), where P is, in turn, the weak and strong order on the symmetric group. In each case the polynomial has an interesting factorization. Our formulae (l), (3), and (4) are instances of the following phenomenon: A poset P and a combinatorial * This work was partially supported by NSF grant DMS-8700995. **Some of this work was carried out while visiting the Institute for Mathematics and its Applications and was partially supported by NSF grant CCR-8707539. 0012-365X/92/$05.00 0 1992- Elsevier Science Publishers B.V. All rights reserved