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The main aim of this article is to compute all the moments of the number of$p^\ell$-torsion elements in some type of nite abelian groups. The averagesinvolved in these moments are those de ned for the Cohen-Lenstra heuristics forclass groups and their adaptation for Tate-Shafarevich groups. In particular,we prove that the heuristic model for Tate-Shafarevich groups is compatiblewith the recent conjecture of Poonen and Rains about the moments of the ordersof $p$-Selmer groups of elliptic curves. For our purpose, we are led to de necertain polynomials indexed by integer partitions and to study them in acombinatorial way. Moreover, from our probabilistic model, we derivecombinatorial identities, some of which appearing to be new, the others beingrelated to the theory of symmetric functions. In some sense, our methodtherefore gives for these identities a somehow natural algebraic context.

$p^\ell$-Torsion Points In Finite Abelian Groups And Combinatorial Identities