A p, q-Analogue of the Generalized Derangement Numbers |
| |
Authors: | Karen S Briggs Jeffrey B Remmel |
| |
Institution: | (1) Department of Mathematics and Computer Science, North Georgia College & State University, Dahlonega, GA 30597, USA;(2) Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA |
| |
Abstract: | In this paper, we study the numbers D
n,k
which are defined as the number of permutations σ of the symmetric group S
n
such that σ has no cycles of length j for j ≤ k. In the case k = 1, D
n,1 is simply the number of derangements of an n-element set. As such, we shall call the numbers D
n,k
generalized derangement numbers. Garsia and Remmel 4] defined some natural q-analogues of D
n,1, denoted by D
n,1(q), which give rise to natural q-analogues of the two classical recursions of the number of derangements. The method of Garsia and Remmel can be easily extended
to give natural p, q-analogues D
n,1(p, q) which satisfy natural p, q-analogues of the two classical recursions for the number of derangements. In 4], Garsia and Remmel also suggested an approach
to define q-analogues of the numbers D
n,k
. In this paper, we show that their ideas can be extended to give a p, q-analogue of the generalized derangements numbers. Again there are two classical recursions for eneralized derangement numbers.
However, the p, q-analogues of the two classical recursions are not as straightforward when k ≥ 2.
Partially supported by NSF grant DMS 0400507. |
| |
Keywords: | AMS Subject Classification" target="_blank">AMS Subject Classification 05A30 05A19 05A15 05A05 05E15 |
本文献已被 SpringerLink 等数据库收录! |
|