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1. Definition & Example
A $k$-core is an integer partition $\lambda$ which has no hooks whose size is a multiples of $k$.
Equivalently, $\lambda$ has no rim hooks that are multiples of $k$ [JK81].
The length of a $k$-core is the number of boxes in in diagram with hook length less than $k$.
The five $4$-cores of length $5$ |
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([5,2],4) |
([4,1,1],4) |
([3,2,1],4) |
([3,1,1,1],4) |
([2,2,1,1,1],4) |
$k$-cores are graphically represented by the associated Ferrers diagram (or Young diagram) as a collection of boxes.
For fixed $k \geq 1$, the number $c_k(n)$ of $k$-cores of size $n$ is A175595. The generating function for $c_k(n)$ is $$\sum_{n=0}^\infty c_k(n) q^n = \prod_{n=1}^\infty \frac{(1 - q^{kn})^k}{(1 - q^n)},$$ see [Ono95].
2. FindStat representation and coverage
A $k$-core is uniquely represented as a tuple (X,n) with X being the list of its parts and n its length.
$k$-cores are graded by tuples (n,k) with length n.
- The database contains all cores with parameters
$$(2,3),(3,3),(4,3),(5,3),(6,3),(2,4),(3,4),(4,4),(5,4),(6,4),(2,5),(3,5),(4,5),(5,5),(6,5),(2,6),(3,6),(4,6),(5,6),(6,6),(7,6).$$
3. Additional information
3.1. Properties
There is a bijective correspondence between $k$-cores and $(k-1)$-bounded partitions. The $(k-1)$-bounded partition corresponding to the $k$-core $\lambda$ is obtained by removing all the cells of $\lambda$ whose hook length is greater than or equal to $k$ [LLM12].
There is a bijection between $k$-cores and affine Grassmannian elements.
3.2. $(a,b)$-cores
A partition $\lambda$ is an $(a,b)$-core $\lambda$ if, and only if, it is both an $a$-core and a $b$-core.
The total number of $(a,b)$-core partitions is finite if, and only if, $a$ and $b$ are coprime, in which case the number is $$\frac{1}{a+b} \binom{a+b}{a,b} = \frac{(a+b-1)!}{a!b!},$$ see [And02].
There multiple open questions and conjectures concerning simultaneous $(a,b)$-cores relating to generalized Catalan numbers and $q,t$-Catalan numbers. See Results and conjectures on simultaneous core partitions at arXiv:1308.0572. [AHJ14]
4. References
- [AHJ14] Armstrong, D., Hanusa, C. R. H., Jones, B. C., Results and conjectures on simultaneous core partitions, arXiv:1308.0572 (2014).
[And02] Anderson, Jaclyn, Partitions which are simultaneously $t_1$- and $t_2$-core, Disc. Math., 248, (2002), pp. 237-243.
[JK81] James, Gordon; Kerber, Adalbert, The representation theory of the symmetric group, Addison-Wesley Publishing Co., Reading, Mass. (1981).
[LLM12] Lam, T., Lapointe, L., Morse, J., et al., k-Schur functions and affine Schubert calculus, arXiv:1301.3569 (2012).
[Ono95] Ono, Ken, A Note on the number of $t$-Core Partitions, Rocky Mountain Journal of Mathematics 25 (1995), no. 3, pp. 1165-1169.
5. Sage examples