# Multidimensional Young diagrams

Consider a Young diagram defined as follows:

A Young diagram (also called a Ferrers diagram, particularly when represented using dots) is a finite collection of boxes, or cells, arranged in left-justified rows, with the row lengths in non-increasing order. Listing the number of boxes in each row gives a partition $$\lambda$$ of a non-negative integer $$n$$, the total number of boxes of the diagram.

For example we may write 1+4+5=10:

Question: Are there higher-dimensional versions, using cubes, such that the "faces" of the diagram are each themselves Young diagrams?

Here is an example, with three distinct faces, each representing diagrams: 1+2+3+3, 2+2+3+3, and 0+0+4+4. The faces are the Young diagrams on the faces of the cube in this case. It has 6 faces, and three pairs of (up,right,in), each a Young diagram. In 2d, there is only 1 face (1 diagram). In 3d, one has a cube with six faces, but only three are unique diagrams. One diagram in the 3d case is forced from the other two (up,right,up,right....up and up,up,in,in,up lead to the other necessarily being right,right,in,in,right).

If so, is there a way of writing an integer in terms of the diagram, in the same way as an integer can be represented via one of many Young diagrams (i.e. integer partitions)? This would represent a restricted integer partition, but in a relatively unusual way.

For example the image below would represent the integer partition ((2+2) + (3+3)) + ((2+2) + (3+3)) = 20.

• I don't see the relation between the 3D illustration and the numbers in the question. I don't see what you mean by faces. What would (in dimension 2) you consider to be a face of a Young diagram? – Marc van Leeuwen Jun 29 '20 at 13:06
• In 2d, there is only 1 face (1 diagram). In 3d, one has a cube with six faces, but only three are unique diagrams. One diagram in the 3d case is forced from the other two (up,right,up,right....up and up,up,in,in,up lead to the other necessarily being right,right,in,in,right). – apkg Jun 29 '20 at 13:09