The Algebra of Conjugacy Classes in Symmetric Groups and Partial Permutations

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.