Geodesic
The Bruhat order on the involutions of the symmetric group
Journal of Algebraic Combinatorics, Tome 20 (2004) no. 3, pp. 243-261.

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Summary: In this paper we study the partially ordered set of the involutions of the symmetric group $S _{n}$ with the order induced by the Bruhat order of $S _{n}$. We prove that this is a graded poset, with rank function given by the average of the number of inversions and the number of excedances, and that it is lexicographically shellable, hence Cohen-Macaulay, and Eulerian.
Keywords: symmetric group, Bruhat order, involution, $E$L-shellability, Cohen-Macaulay 
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     author = {Incitti, Federico},
     title = {The {Bruhat} order on the involutions of the symmetric group},
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Incitti, Federico. The Bruhat order on the involutions of the symmetric group. Journal of Algebraic Combinatorics, Tome 20 (2004) no. 3, pp. 243-261. http://geodesic.mathdoc.fr/item/JAC_2004__20_3_a7/