Ferrers diagram: represent partition λ=(λ₁,λ₂,...,λ_k) as rows of dots. Row i has λ_i dots, left-aligned. Partition (4,2,1): ●●●● / ●● / ●. Transpose = conjugate. Young diagram = version with boxes instead of dots.
How to Use
Enter partition parts:
- Diagram: Dot display
- Conjugate: Transposed
- Hooks: Hook lengths
Visual Insights
The Ferrers diagram instantly reveals: number of parts (rows), largest part (columns), conjugate (transpose), Durfee square (largest inscribed square), and self-conjugacy (symmetry about diagonal).
Hook Lengths
Hook at cell (i,j): h(i,j) = λ_i + λ'_j - i - j + 1. The hook length formula: f^λ = n!/Π h(i,j) counts standard Young tableaux. This is the Frame-Robinson-Thrall formula, fundamental in representation theory.
Step-by-Step Instructions
- 1Enter parts.
- 2View diagram.
- 3Toggle conjugate.
- 4See Durfee.
- 5Check hooks.