Tags: #definition

conjugate partition

Given some partition $\lambda =({\lambda }_1,\dots ,{\lambda }_k)$, we define the conjugate partition as ${\lambda }^{\prime }=({\lambda }_1^{\prime },\dots )$ where (these are equivalent definitions):

• ${\lambda }_i^{\prime }$ is the number of partitions ${\lambda }_j$ that has value greater than or equal to $i$.
• Looking at the Young diagram, we count rows instead of columns.
• We reflect the diagram across the diagonal

Example:

In the above example, we have the partition of (3, 3, 2, 1, 1). Its conjugate is (5, 3, 2).

See also

References