Geodesic
The algebra of conjugacy classes in symmetric groups, and partial permutations
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part III, Tome 256 (1999), pp. 95-120

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We prove a convolution formula for the conjugacy classes in symmetric groups conjectured by the second author. A combinatorial interpretation of coefficients is provided. As a main tool we introduce new semigroup of partial permutations. We describe its structure, representations, and characters. We also discuss filtrations on the subalgebra of invariants in the semigroup algebra. 
@article{ZNSL_1999_256_a7,
     author = {V. N. Ivanov and S. V. Kerov},
     title = {The algebra of conjugacy classes in symmetric groups, and partial permutations},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {95--120},
     publisher = {mathdoc},
     volume = {256},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1999_256_a7/}
}
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V. N. Ivanov; S. V. Kerov. The algebra of conjugacy classes in symmetric groups, and partial permutations. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part III, Tome 256 (1999), pp. 95-120. http://geodesic.mathdoc.fr/item/ZNSL_1999_256_a7/