# Tableaux

##### Definition: Young Diagram

A Young diagram of size n is a sequence of non- negative integers such that . We denote this by and draw it as a collection of unit boxes where each is the length of a row.

The length is the number of rows in . We denote the set of all Young diagrams by , including the empty diagram .

Below we see the Young diagram [4,3,2] on the left with its transpose on the right.

##### Definition: Young Tableau

A semi-standard Young tableau of shape and rank N is a filling of boxes of a Young diagram with numbers from 1 to N such that the numbers strictly increase along the columns and weakly increase along the rows.

##### Definition: Domino Diagram

A domino diagram is a Young diagram that can be tiled by combinations of 2x1 and 1x2 rectangles (which we call dominoes)

##### Definition: Domino Tableau

A domino tableau with shape and rank is a filling of dominoes of a tiling of a domino diagram with numbers from 1 to N such that the numbers strictly increase along the columns and weakly increase along the rows.

##### Definition: Schur Polynomial

For a given Young Diagram , we defined the corresponding Schur polynomial by

If the content of P has 1s, 2s, ... , ns, then .