Young Diagram Calculator
Size:
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Shape:
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Number of permutations with this shape:
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Product of hook lengths:
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Number of standard tableaux with this shape:
0

Number of semistandard tableaux with this shape and maximum entry :
1


Show hook lengths
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Young Diagrams
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.

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