Hooksets of Young Diagrams

• Hooksets of Young Diagrams are collections of hook lengths from a partition’s Young diagram that capture deep combinatorial and algebraic properties.
• The framework employs Frobenius coordinates to encode partitions, enabling explicit structural theorems and enumeration results for core partitions and symmetric functions.
• This approach connects partition theory with numerical semigroups, integrable systems, and operator theory, providing actionable insights for combinatorial and algebraic research.

A Young diagram is a graphical realization of a partition of a nonnegative integer, and the "hookset" of a Young diagram is the collection of all hook lengths realized by its constituent boxes. The study of hooksets draws together classical partition theory, the rich algebraic structure of numerical semigroups, and modern developments in the theory of symmetric functions, integrable systems, and representations. The hookset not only encodes deep combinatorial properties of the partition but also provides a fundamental coordinate system ("hook variables" or Frobenius coordinates) that governs the algebraic manipulation of Schur functions, cut-and-join operators, and $\tau$-functions. Recent research has established precise correspondences between partitions with prescribed hooksets and algebraic objects such as numerical monoids, developed explicit structural theorems, and extended enumeration results for special classes such as simultaneous $s/t$-cores and complete hooksets (Keith et al., 2010, Mironov et al., 2019).

1. Definitions: Partitions, Young Diagrams, Hooks, and Hooksets

A partition $$\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_k)$$ of a nonnegative integer $n$ is a finite nonincreasing sequence with $\sum_{i=1}^k \lambda_i = n$ and $$\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k \geq 1$$. Its Young diagram is formed by placing $\lambda_j$ boxes in the $j$-th row (using the fourth-quadrant convention), with the cell at $(i, j)$ present if $\lambda_j \geq i$.

The hook at a cell $(i, j)$ is $$h_{ij} = (\lambda_i - j) + |\{ a: \lambda_a \geq j \}|$$. The hookset of $\lambda$ is $Hk(\lambda) = \{ h_{ij} : (i, j) \in \lambda \}$; it consists of all hook lengths realized in the diagram (Keith et al., 2010).

Frobenius (hook) coordinates provide an alternative encoding: given a partition $\lambda$ with $r$ diagonal boxes, the sequences $a_i = \lambda_i - i$, $b_i = \lambda'_i - i$ (where $\lambda'$ is the conjugate partition) for $i = 1, \dots, r$ encode the arm and leg lengths. Every hook length in $\lambda$ is $a_i + b_j + 1$ as $i$, $j$ range (Mironov et al., 2019).

A $k$-core is a partition whose hookset contains no multiples of $k$.

2. The Hookset–Numerical Monoids Correspondence

A central development is the identification of a bijection between hooksets and the complements of numerical monoids of given genus. For any numerical set $S \subset \mathbb{N} \cup \{0\}$ (contains 0, cofinite), its atom monoid $$A(S) = \{ n \in S : n + s \in S \;\forall s \in S \}$$ is a numerical monoid.

Theorem (Hooksets–Monoids correspondence): Among all partitions, hooksets of genus $g$ are exactly the complements of numerical monoids of genus $g$:

$$\phi : \{\text{numerical sets}\} \longleftrightarrow \{\text{partitions}\}$$

so that

$Hk(\lambda) = \overline{A(S)}$

where $\overline{A(S)}$ is the complement of $A(S)$ in nonnegative integers (Keith et al., 2010).

The construction uses labeled NE lattice paths (the partition profile): east (spacer) corresponds to $i \in S$, north (bead) to $i \notin S$. A hook of length $k$ arises when an east step at $i$ and north at $i+k$ imply $k \notin A(S)$, i.e., $k$ is in the hookset if and only if it is not an atom of $S$.

For $k$-cores, partitions correspond to numerical sets with $k \in A(S)$.

3. Structural Theorems and Sufficient Conditions

Key results govern when specific hooksets are realized and the structure of families of such partitions:

• Finite Simultaneous $S$-core property: The set of partitions that are simultaneously $s_i$-core for a set $S = \{s_1, \dots, s_k\}$ is finite if and only if $\gcd(S) = 1$.
• Hook existence criteria: Given a set $\{s_i\}$, if $\lambda$’s profile is strictly above the diagonal at positions distance $s_i$ from its endpoints, then all required hooks $s_i$ appear (Dyck-path criterion). This yields infinite families of partitions containing prescribed hooksets (Keith et al., 2010).
• Strict Dyck profile: If $\lambda$ is a strict Dyck path (profile always above the main diagonal except at endpoints), the hookset is the full initial segment $\{1,2,\ldots,h_{\max}\}$.

Explicit formulae for $k$-core generating functions for complete non-multiples-of-$k$ hooksets, together with abacus/bijection-based enumeration algorithms, are provided (Keith et al., 2010).

4. Hook Variables in Symmetric Functions and Operator Theory

The hook-variable (Frobenius) parametrization is pivotal for Schur functions, their variants, and the associated operators:

• Schur and skew Schur functions: Ordinary, shifted, and skew Schur functions admit determinant formulas in Frobenius coordinates. For ordinary Schur functions,

$$S_\lambda\{p\} = \det_{1 \leq i,j \leq r} S_{(a_i|b_j)}\{p\}$$

where $S_{(a | b)}\{p\}$ is the single-hook Schur function (Mironov et al., 2019).

• Cut-and-join operators: The usual operators $\widehat{W}_\Delta$ diagonalize on $S_\lambda\{p\}$, and expand in hook variables solely over single-hook diagrams once unit cycles are removed from $\Delta$.
• Casimir and Ruijsenaars Hamiltonians: SL$(\infty)$ Casimir operators $C_n$ act diagonally with eigenvalues in hook variables; their generating function yields the $q = t$ Ruijsenaars Hamiltonian, which also admits a single-hook expansion.
• Multicomponent (Matisse) KP $\tau$-functions: Products of single-hook Schur functions in different times generate new integrable $\tau$-functions, with the combinatorics governed by Frobenius data.

5. Quantitative Results, Bounds, and Enumerative Applications

The bijection equips the study of hooksets with enumerative methods from numerical semigroup theory:

• For $S$-core families (where all $s_i \in S$ are contained in the atom monoid), the largest possible hook length (outer hook) is bounded above by the Frobenius number $f$ of the corresponding numerical monoid. Thus, the partition size $n \leq f^2/4$ (Keith et al., 2010).
• For $S = \{s, t\}$ coprime, $f = st - s - t$ gives explicit upper bounds.
• The number of partitions whose hookset is $\{1, 2, \ldots, n\}$ is equinumerous with numerical sets $S$ whose atom monoid is $\{0, n+1, n+2, \ldots\}$.
• An explicit generating function for 3-cores with complete hooksets is established:

$$\sum_{n \geq 0} cc_3(n) q^n = 1 + q + 2\sum_{k \geq 2}(q^{k^2} + q^{k^2 - k})$$

where $cc_3(n)$ is the number of 3-cores of size $n$ whose hookset is all non-multiples of 3 up to the maximal hook.

6. Connections with Numerical Semigroups and Open Problems

There is a deep and structurally rich interplay between hooksets and numerical semigroup theory (Keith et al., 2010):

• The number $n_g$ of numerical monoids of genus $g$ appears conjectured to obey Fibonacci-like recursion $n_g \approx n_{g-1} + n_{g-2}$; this is mirrored in the enumeration of partitions with maximal hookset $\{1, \ldots, g\}$.
• Tools from numerical semigroup theory (Apéry sets, Wilf’s conjecture) may be transferred to hookset combinatorics, suggesting new approaches to open enumeration and uniqueness problems (e.g., the nonuniqueness of a partition for a given hookset).
• Enumeration of the number of partitions with a given hookset, or distinguishing partitions by both hookset and part-multiset, remains an open problem.

This suggests cross-fertilization between partition theory and numerical semigroup theory, potentially yielding new enumeration results and structural insights.

7. Summary and Outlook

Hooksets function both as combinatorial invariants and as the optimal coordinate system for a range of algebraic and analytic structures associated with partitions and Young diagrams. Their correspondence with numerical monoids grants access to structural theorems, sharp enumerative bounds, explicit parametrizations for cores and prescribed hooksets, and operational benefits for symmetric functions, integrable systems, and representation theory (Keith et al., 2010, Mironov et al., 2019). Ongoing research aims to further classify hooksets, prove conjectured enumeration formulae, and extend the interplay with operator theory and matrix models.