Journal of Integer Sequences, Vol. 13 (2010), Article 10.3.6

On q-Analogs of Recursions for the Number of Involutions and Prime Order Elements in Symmetric Groups

Max B. Kutler
Department of Mathematics
Harvey Mudd College
301 Platt Boulevard
Claremont, CA 91711

C. Ryan Vinroot
Department of Mathematics
College of William and Mary
P. O. Box 8795
Williamsburg, VA 23187


The number of elements whose square is the identity in the symmetric group Sn is recursive in n. This recursion may be proved combinatorially, and there is also a nice exponential generating function for this sequence. We study q-analogs of this phenomenon. We begin with sums involving q-binomial coefficients which come up naturally when counting elements in finite classical groups which square to the identity, and we obtain a recursive-like identity for the number of such elements in finite special orthogonal groups. We then study a q-analog for the number of elements in the symmetric group whose pth power is the identity, for some fixed prime p. We find an Eulerian generating function for these numbers, and we prove the q-analog of the recursion for these numbers by giving a combinatorial interpretation in terms of vector spaces over finite fields.

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(Concerned with sequences A000085 A052501.)

Received August 31 2009; revised versions received December 4 2009; March 9 2010. Published in Journal of Integer Sequences, March 12 2010.

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