  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 this 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.
