The distributions of the entries of Young tableaux

Brendan D. McKay, Jennifer Morse, Herbert S. Wilf

Published 2000-08-22, updated 2001-06-11Version 3

Let T^* be a standard Young tableau of k cells. We show that the probability that a Young tableau of n cells contains T^* as a subtableau is, in the limit n -> \infty, equal to \nu(\pi(T^*))/k!, where \pi(T^*) is the shape (= Ferrers diagram) of T^* and \nu(\pi) is the number of all tableaux of shape \pi. 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. We then extend these results by finding an explicit formula for the limiting probability that a Young tableau has a given set of entries in a given set of positions. The result is that the limiting probability that a Young tableau has a prescribed set of entries k_1,k_2,..., k_m in a prescribed set of m cells is equal to the sum of the measures of all tableaux on K cells (K=\max{\{k_i\}}) that have the given entries in the given positions, where the measure of a tableau of K cells is the number of tableaux of its shape divided by K!. In the proof we also develop conditions that ensure the quasirandomness of certain families of permutations.