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The Eulerian generating function of q-derangements

DSpace at IIT Bombay

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Title The Eulerian generating function of q-derangements
 
Creator SRINIVASAN, MK
 
Subject finite vector-spaces
eulerian generating functions
q-analog of derangements
eigenvalue-free matrices over a finite field
 
Description Let F-q denote the finite field with q elements. For nonnegative integers n, k, let d(q) (n, k) denote the number of n x n F-q-matrices having k as the sum of the dimensions of the eigenspaces (of the eigenvalues lying in F-q). Let dq (n) = d(q)(n,0), i.e., d(q) (n) denotes the number of n x n F-q-matrices having no eigenvalues in Fq. The Eulerian generating function of dq (n) has been well studied in the last 20 years [Kung, The cycle structure of a linear transformation over a finite field, Linear Algebra Appl. 36 (1981) 141-155, Neumann and Praeger, Derangements and eigenvalue-free elements in finite classical groups, J. London Math. Soc. (2) 58 (1998) 564-586 and Stong, Some asymptotic results on finite vector spaces, Adv. Appl. Math. 9(2) (1988) 167-199]. The main tools have been the rational canonical form, nilpotent matrices, and a q-series identity of Euler. In this paper we take an elementary approach to this problem, based on Mobius inversion, and find the following bivariate generating function: [GRAPHICS] (c) 2006
 
Publisher ELSEVIER SCIENCE BV
 
Date 2011-07-27T01:28:52Z
2011-12-26T12:53:02Z
2011-12-27T05:40:06Z
2011-07-27T01:28:52Z
2011-12-26T12:53:02Z
2011-12-27T05:40:06Z
2006
 
Type Article
 
Identifier DISCRETE MATHEMATICS, 306(17), 2134-2140
0012-365X
http://dx.doi.org/10.1016/j.disc.2006.04.007
http://dspace.library.iitb.ac.in/xmlui/handle/10054/7075
http://hdl.handle.net/10054/7075
 
Language en