information:classification
Tableaux

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 x, x, x..., but now one usually uses a set of numbers for brevity. In their original application to representations of the symmetric group, Young tableaux have 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 n entries is given by the involution numbers

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 n to occur exactly once.

In a standard Young tableau, the integer k {\displaystyle k} is a descent if k + 1 {\displaystyle k+1} appears in a row strictly below k {\displaystyle k}. The sum of the descents is called the major index of the tableau.