Partition conjugate: transpose the Ferrers diagram. (4,2,1) → (3,2,1,1). The conjugate's parts equal row lengths of the original. Largest part becomes number of parts and vice versa. Self-conjugate ⟺ Ferrers diagram is symmetric.
How to Use
Enter partition parts:
- Conjugate: Transposed partition
- Self: Check symmetry
- Durfee: Largest square
Self-Conjugate
Self-conjugate partitions: λ=λ'. Example: (3,2,1) is self-conjugate. There's a bijection: self-conjugate partitions of n ↔ partitions of n into distinct odd parts. The Durfee square mediates this correspondence.
Durfee Square
The Durfee square is the largest k×k square fitting in the Ferrers diagram. For partition λ: d(λ) = max{k: λ_k ≥ k}. It splits the diagram into the square, a partition to the right, and one below.
Step-by-Step Instructions
- 1Enter parts.
- 2Compute conjugate.
- 3Check self-conjugate.
- 4Find Durfee square.
- 5See visual.