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 → ∞, equal to ν(π(T ∗))/k!, where π(T ∗) is the shape (= Ferrers diagram) of T ∗ and ν(π) is the number of all tableaux of shape π. In other words, the probability that a large tableau contains T ∗ is equal to the number of tableaux whose s...

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/k!, where f 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 sever...

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The (shifted) plactic monoid. The celebrated Robinson-Schensted-Knuth correspondence (14) is a bijection between words in a linearly ordered alphabet X = {1 < 2 < 3 < · · · } and pairs of Young tableaux with entries in X . More precisely, each word corresponds to a pair consisting of a semistandard insertion tableau and a standard recording tableau. The words producing a given insertion tableau...

Bijective proofs of the hook formulas for the number of ordinary standard Young tableaux and for the number of shifted standard Young tableaux are given. They are formulated in a uniform manner, and in fact prove q-analogues of the ordinary and shifted hook formulas. The proofs proceed by combining the ordinary, respectively shifted, Hillman–Grassl algorithm and Stanley's (P, ω)-partition theor...

Based on Schützenberger’s evacuation and a modification of jeu de taquin, we give a bijective proof of an identity connecting the generating function of reverse semistandard Young tableaux with bounded entries with the generating function of all semistandard Young tableaux. This solves Exercise 7.102 b of Richard Stanley’s book ‘Enumerative Combinatorics 2’.

The RSK correspondence generalises the Robinson-Schensted correspondence by replacing permutation matrices by matrices with entries in N, and standard Young tableaux by semistandard ones. For r ∈ N>0, the Robinson-Schensted correspondence can be trivially extended, using the r-quotient map, to one between r-coloured permutations and pairs of standard r-ribbon tableaux built on a fixed r-core (t...