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

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Max B. Kutler

Department of Mathematics

Harvey Mudd College

301 Platt Boulevard

Claremont, CA 91711

USA

C. Ryan Vinroot

Department of Mathematics

College of William and Mary

P. O. Box 8795

Williamsburg, VA 23187

USA

**Abstract:**

The number of elements whose square is the identity in
the symmetric group *S*_{n} 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 *p*th 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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