Geodesic
Involutions of the Symmetric Group and Congruence B-Orbits (Extended Abstract)
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) (2010).

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We study the poset of Borel congruence classes of symmetric matrices ordered by containment of closures. We give a combinatorial description of this poset and calculate its rank function. We discuss the relation between this poset and the Bruhat poset of involutions of the symmetric group. Also we present the poset of Borel congruence classes of anti-symmetric matrices ordered by containment of closures. We show that there exists a bijection between the set of these classes and the set of involutions of the symmetric group. We give two formulas for the rank function of this poset. 
@article{DMTCS_2010_special_259_a72,
     author = {Bagno, Eli and Cherniavsky, Yonah},
     title = {Involutions of the {Symmetric} {Group} and {Congruence} {B-Orbits} {(Extended} {Abstract)}},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)},
     year = {2010},
     doi = {10.46298/dmtcs.2877},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2877/}
}
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%V DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)
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Bagno, Eli; Cherniavsky, Yonah. Involutions of the Symmetric Group and Congruence B-Orbits (Extended Abstract). Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010) (2010). doi : 10.46298/dmtcs.2877. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2877/

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