S-infinity representations and combinatorial identities

For various probability measures on the space of the infinite standard Youn g tableaux we study the probability that in a random tableau, the (i, j)(th ) entry equals a given number n. Beside the combinatorics of finite standar d tableaux, the main tools here are from the Vershik-Kerov character theory of S-infinity. The analysis of these probabilities leads to many explicit combinatorial identities, some of which are related to hypergeometric serie s.