The Distributions of the Entries of Young Tableaux

Let T be a standard Young tableau of shape λ⊢k. We show that the probability that a randomly chosen Young tableau of n cells contains T as a subtableau is, in the limit n→∞, equal to f^λ/k!, where f^λ is the number of all tableaux of shape λ. In other words, the probability that a large tableau contains T is equal to the number of tableaux whose shape is that of T, divided by k!. We give several applications, to the probabilities that a set of prescribed entries will appear in a set of prescribed cells of a tableau, and to the probabilities that subtableaux of given shapes will occur. Our argument rests on a notion of quasirandomness of families of permutations, and we give sufficient conditions for this to hold.