Random SYT via Hook Walk     permutations

The hook-walk algorithm (Greene-Nijenhuis-Wilf) generates uniformly random Standard Young Tableaux (SYT) of any given shape. This is a fundamental tool in algebraic combinatorics with applications to representation theory, symmetric functions, and random matrix theory.

How it works:

1. Start with an empty Young diagram of the given shape
2. For each number k from N down to 1:
   • Pick a random starting cell uniformly from all empty cells
   • Perform a random walk within the hook: move right or down with probabilities proportional to arm and leg lengths
   • Stop when reaching a corner cell (arm = leg = 0)
   • Place k at that corner and remove it from the diagram

Properties:

• Uniform sampling: Each SYT of the given shape has equal probability
• Efficient: O(N√N) time complexity for N boxes
• Scalable: Handles shapes up to 100,000 boxes using WASM

Shape input methods:

• Draw Shape: Click cells on the interactive grid to draw Young diagrams by hand
• Manual notation: Enter row lengths like 5,5,5 or 100^50
• Plancherel measure: Sample random partitions by discretizing the Vershik-Kerov limit shape Ω(x) = (2/π)[x√(1-x²) + arcsin(x)]

Visualization:

• Small tableaux (≤200 boxes): Individual numbers displayed in cells
• Large tableaux (>200 boxes): Heat map showing value distribution by deciles