Young Tableau


The Young tableau (plural, "tableaux") of a Ferrers diagram is obtained by placing the numbers 1, ..., n in the n boxes of the diagram. A "standard" Young tableau is a Young tableau in which the numbers form an increasing sequence along each line and along each column. For example, the standard Young tableaux of size n=3 are given by {{1,2,3}}, {{1,3},{2}}, {{1,2},{3}}, and {{1},{2},{3}}, illustrated above. The bumping algorithm is used to construct a standard Young tableau from a permutation of {1,...,n}, and the number of standard Young tableaux of size 1, 2, 3, ... are 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, ... (OEIS A000085). These numbers can be generated by the recurrence relation


with a(1)=1 and a(2)=2. This is the same as the number of permutation involutions on n elements (Skiena 1990, p. 32).


The number of all possible standard Young tableaux of a given shape can also be considered, and can be calculated with the hook length formula. For example, the illustration above shows the 35 standard tableaux of shape {3,2,1,1}.

There is a correspondence between a permutation and a pair of Young tableaux, known as the Schensted correspondence.

A Young tableau in which numbers are nondecreasing along lines and increasing along columns is called a semistandard Young tableau.

See also

Bumping Algorithm, Durfee Square, Ferrers Diagram, Hook Length Formula, Partition, Partition Function P, Permutation Involution, Random Tableau Schensted Correspondence, Tableau Class

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Young Tableau

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Weisstein, Eric W. "Young Tableau." From MathWorld--A Wolfram Web Resource.

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