On the Random Young Diagrams and Their Cores

Anr-core of a Young diagramλis a residual subdiagram obtained after consecutive removals of the feasibler-long border strips, “rim hooks.” The removal process on the diagramλand the resultingr-core are the essential elements in the Murnaghan–Nakayama formula forχλ, the character of the associated irreducible representation ofSn(n=|λ|), on the conjugacy class {r[n/r]} (n≡0 mod r). A complete characterization ofr-cores is given, which extends a well known result forr=2. Under an assumption that the partitionλis chosen uniformly at random out of all partitions ofn, it is shown that typically ther-core size is of ordern1/2, while the height and the width are of ordern1/4. Fornchosen uniformly at random between 1 andN, the core boundary scaled byN1/4is proved to converge, in distribution, to a random concave curve which consists ofr−1 line segments.