Partial Jucys–Murphy elements and star factorizations

In this paper, we look at the number of factorizations of a given permutation into star transpositions. In particular, we give a natural explanation of a hidden symmetry, answering a question of I.P. Goulden and D.M. Jackson. We also have a new proof of their explicit formula. Another result is the normalized class expansion of some central elements of the symmetric group algebra introduced by P. Biane.To obtain these results, we use natural analogs of Jucys–Murphy elements in the algebra of partial permutations of V. Ivanov and S. Kerov. We investigate their properties and use a formula of A. Lascoux and J.Y. Thibon to give the expansion of their power sums on the natural basis of the invariant subalgebra.