Hook Length Formula¶
Count standard Young tableaux of a partition shape by dividing n! by the product of its cell hook lengths.
Core Idea¶
The hook length formula counts the standard Young tableaux of an ordinary partition shape λ with n cells. Each tableau uses 1 through n once, increasing along rows and columns. For every cell, count the cell itself and those directly right and below it; that is its hook length h(u). The count is f^λ = n! / ∏ h(u) over all cells.[^ref-29ea248b2da3]
Scope of Application¶
The rule applies across partition shapes, not just one diagram. For shape (3,1), hooks 4, 2, 1, 1 give 3 tableaux; for (2,2), hooks 3, 2, 2, 1 give 2. The same count is the dimension of the symmetric-group irreducible representation indexed by the shape.[^ref-29ea248b2da3]
Clarity¶
Hook lengths depend on the diagram, not on numbers written in one tableau. The product counts all valid standard fillings. Its factors should not be interpreted as independent cell probabilities, and the unmodified formula does not count skew or semistandard tableaux.[ref-29ea248b2da3][ref-b18a2e9db307]
Manages Complexity¶
Instead of checking up to n! permutations against row and column constraints, calculate n local hook lengths and one quotient. The exact reduction retains the shape dependence while avoiding explicit enumeration.[^ref-29ea248b2da3]
Abstract Reasoning¶
Confirm a straight partition shape and standard filling rule. Compute each right-and-below cell hook, multiply their lengths, and divide n!. If the object to be counted has a different shape or labeling rule, stop and use a theorem for that setting rather than reuse the quotient by analogy.[^ref-29ea248b2da3]
Knowledge Transfer¶
The exact theorem transfers among ordinary partition shapes and, through a proven correspondence, to indexed representation dimensions. It is a specialized Formal Theorem, not every hook-named identity. Plancherel probability uses \((f^\lambda)^2/n!\), not \(f^\lambda\) alone; rooted-tree hooks and semistandard or skew variants require separate statements.[ref-b18a2e9db307][ref-018b6621759e]
[^ref-29ea248b2da3]: Stanley, Topics in Algebraic Combinatorics, Theorem 8.1. [^ref-b18a2e9db307]: Greene, Nijenhuis and Wilf, original hook-formula proof. [^ref-018b6621759e]: Gessel and Seo, “A Refinement of Cayley's Formula for Trees,” Electronic Journal of Combinatorics 11(2) (2006), #R27, §2, defines rooted-tree vertex hooks by descendant count and derives the increasing-labeling quotient.
Relationships to Other Abstractions¶
Current abstraction Hook Length Formula Domain-specific
Parents (1) — more general patterns this builds on
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Hook Length Formula is a kind of Formal theorem Domain-specific
The hook-length formula is a formal theorem.
Hierarchy paths (2) — routes to 2 parentless roots
- Hook Length Formula → Formal theorem → Formal System → Formalization → Representation → Abstraction
- Hook Length Formula → Formal theorem → Formal System → Formalization → Transformation → Function (Mapping)
Neighborhood in Abstraction Space¶
Hook Length Formula sits in a sparse region of the domain-specific corpus (89th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Graph Structures & Combinatorial Objects (44 abstractions)
Nearest neighbors
- Kostka number — 0.82
- Polygon — 0.80
- Finite subdivision rule — 0.80
- Euler Characteristic — 0.80
- Wilf Equivalence — 0.80
Computed from structural-signature embeddings · 2026-10-08