Explicit formulas are given for the entry-wise marginal probabilities in uniformly random permutations of prescribed hook, two-row, or rectangular Robinson-Schensted shape, together with a limit theorem on the expected number of fixed points.
Stanley,Enumerative combinatorics
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Explicit marginal distributions for permutations with prescribed Robinson-Schensted shape
Explicit formulas are given for the entry-wise marginal probabilities in uniformly random permutations of prescribed hook, two-row, or rectangular Robinson-Schensted shape, together with a limit theorem on the expected number of fixed points.