Editing Young tableau (section)

=== Tableaux ===
[[Image:Young tableaux for 541 partition.svg|thumb|right|150px|A standard Young tableau of shape (5, 4, 1): the numbers 1-10 in the boxes increase in every row and every column.]]

A '''Young tableau''' is obtained by filling in the boxes of the Young diagram with symbols taken from some ''alphabet'', which is usually required to be a [[totally ordered set]]. Originally that alphabet was a set of indexed variables {{mvar|''x''<sub>1</sub>}}, {{mvar|''x''<sub>2</sub>}}, {{mvar|''x''<sub>3</sub>}}..., but now one usually uses a set of numbers for brevity. In their original application to [[representations of the symmetric group]], Young tableaux have {{mvar|''n''}} distinct entries, arbitrarily assigned to boxes of the diagram.  A tableau is called '''standard''' if the entries in each row and each column are increasing. The number of distinct standard Young tableaux on {{mvar|''n''}} entries is given by the [[involution number]]s
:1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, ... {{OEIS|A000085}}.[[File:Standard Young Tableaux.png|thumb|All standard Young tableaux with at most 5 boxes]]

In other applications, it is natural to allow the same number to appear more than once (or not at all) in a tableau.  A tableau is called '''semistandard''', or ''column strict'', if the entries weakly increase along each row and strictly increase down each column.  Recording the number of times each number appears in a tableau gives a sequence known as the '''weight''' of the tableau.   Thus the standard Young tableaux are precisely the semistandard tableaux of weight (1,1,...,1), which requires every integer up to {{mvar|''n''}} to occur exactly once.

In a standard Young tableau, the integer <math>k</math> is a '''descent''' if <math>k+1</math> appears in a row strictly below <math>k</math>. The sum of the descents is called the '''major index''' of the tableau.<ref name="ste89"/>