Geodesic
On \(q\)-analogs of recursions for the number of involutions and prime order elements in symmetric groups
Journal of integer sequences, Tome 13 (2010) no. 3
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.
Classification : 05A10, 05A15
Keywords: q-analogs, q-binomial coefficients, vector spaces over finite fields, recursions, symmetric group, Eulerian generating functions 
@article{JIS_2010__13_3_a4,
     author = {Kutler,  Max B. and Vinroot,  C.Ryan},
     title = {On \(q\)-analogs of recursions for the number of involutions and prime order elements in symmetric groups},
     journal = {Journal of integer sequences},
     year = {2010},
     volume = {13},
     number = {3},
     zbl = {1228.05023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2010__13_3_a4/}
}
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Kutler,  Max B.; Vinroot,  C.Ryan. On \(q\)-analogs of recursions for the number of involutions and prime order elements in symmetric groups. Journal of integer sequences, Tome 13 (2010) no. 3. http://geodesic.mathdoc.fr/item/JIS_2010__13_3_a4/