Geodesic
Square involutions
Journal of integer sequences, Tome 14 (2011) no. 3.

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Summary: A square involution is a square permutation which is also an involution. In this paper we give the enumeration of square involutions, using purely combinatorial methods, by establishing a bijective correspondence with a class of lattice paths. As a corollary to our result, we enumerate various subclasses of square involutions, including the classes of $triangular, decomposable$, and fat involutions. 
@article{JIS_2011__14_3_a6,
     author = {Disanto, Filippo and Frosini, Andrea and Rinaldi, Simone},
     title = {Square involutions},
     journal = {Journal of integer sequences},
     publisher = {mathdoc},
     volume = {14},
     number = {3},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2011__14_3_a6/}
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Disanto, Filippo; Frosini, Andrea; Rinaldi, Simone. Square involutions. Journal of integer sequences, Tome 14 (2011) no. 3. http://geodesic.mathdoc.fr/item/JIS_2011__14_3_a6/