A Young diagram is a visualization of an integer partition. Each row represents one part of the partition, and the size of the part determines the number of boxes in that row. The diagram is drawn so that the largest part is at the top, with remaining rows (weakly) decreasing in length. As a result, Young diagrams have a "top-left justified" appearance.

Every Young diagram is associated with an integer partition of size \(n\), meaning that the sum of the parts is equal to the integer \(n\). This applet allows you to construct a Young diagram with up to 13 parts (of up to 13).

Young diagrams are named for Alfred Young, a British mathematician whose research areas included representation theory, combinatorics, statistics, and their applications to physics and chemistry.

• The hook length \(\mbox{hook}(b)\) of a box \(b\) in a Young diagram is the total number of boxes to the right and below that box, plus 1 for the box itself.
• The shape of the partition is a weakly decreasing list of the parts of the partition. The \(i\)th term of this list is equal to the number of boxes in the \(i\)th row of the Young diagram.
• Every permutation on \(n\) letters can be written in disjoint cycle notation. The list of the lengths of these cycles is called the shape of the permutation. Suppose a partition \(\lambda\) is a partition of \(n\) with \(a_1\) parts of length 1, \(a_2\) parts of length 2, \(a_3\) parts of length 3, etc. The number of permutations on \(n\) letters with \(a_1\) cycles of length 1, \(a_2\) cycles of length 2, \(a_3\) cycles of length 3, etc, is given by \[ z_\lambda = \frac{n!}{1^{a_1}a_1!2^{a_2}a_2!3^{a_3}a_3! \cdots} \]
• A standard Young tableaux is a Young diagram whose \(n\) boxes are filled with the numbers 1 through \(n\) so that the numbers are strictly increasing from left to right in each row, and top to bottom in each column. The number of standard tableaux of a particular shape \(\lambda\) can be found with the hook length formula \[ \frac{n!}{\prod_{b \in Y} \mbox{hook}(b)}\] where the product is taken over all boxes \(b\) in the diagram \(Y\). This is equal to the dimension of the irreducible representation of S\(_n\) corresponding to the partition \(\lambda\).
• A semistandard Young tableaux is a Young diagram whose \(n\) boxes are filled with the numbers 1 through \(n\) so that the numbers are weakly increasing from left to right in each row, and strictly decreasing from top to bottom in each column. These are also called "column strict tableaux". The number of semistandard tableaux of a particular shape \(\lambda\) and maximum entry \(k\) can be found with the hook length formula \[ \prod_{(i,j)\in Y} \frac{k+j-i}{\mbox{hook}(i,j)}\] where \((i,j)\) refers to the box in row \(i\) and column \(j\) in the diagram \(Y\). Note that \(k\) must be at least as large as the number of rows in the diagram in order to build a semistandard tableau. This is equal to the dimension of the irreducible representation of GL\(_k\) corresponding to the partition \(\lambda\) with at most \(k\) parts.

Click on the grid to add or remove a box to the Young diagram, as well as all boxes to its below or to the right of the box. This maintains the required form of a Young diagram.

The size and shape of the partition will be updated, along with the numbers of permutations, standard tableaux, and semistandard tableaux of that shape.

You can choose to display or hide the hook lengths of each box in the diagram, as well as dynamically shading each box by its hook length.

This applet was created using JavaScript and the P5 graphics library. It was last updated in February 2022.