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Definition & Example

A $k$core is an integer partition $\lambda$ which has no hooks whose size are 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 its diagram with hook length less than $k$.
the 7 Cores of size (5, 6)  
([5],6)  ([4,1],6)  ([3,2],6)  ([3,1,1],6)  ([2,2,1],6)  ([2,1,1,1],6)  ([1,1,1,1,1],6) 

$k$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].
Properties

There is a bijective correspondence between $k$cores and $(k1)$bounded partitions. The $(k1)$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.
$(a,b)$cores {#abcores}

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
$$\frac{1}{a+b} \binom{a+b}{a,b} = \frac{(a+b1)!}{a!b!},$$see [And02].
See [AHJ14] for open questions and conjectures concerning $(a,b)$cores relating to generalized Catalan numbers and $q,t$Catalan numbers.
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. 237243
 [JK81] James, Gordon; Kerber, Adalbert, The representation theory of the symmetric group, AddisonWesley Publishing Co., Reading, Mass. (1981)
 [LLM12] Lam, T., Lapointe, L., Morse, J., et al., kSchur 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. 11651169
Sage examples
Technical information for database usage

A $k$core is uniquely represented as a pair
(X,n)
withX
being the partition andn
its length. 
Cores are graded by pairs
(n,k)
such that $k$cores are graded by their lengths. 
The database contains all cores with parameters
$$(2,3),(3,3),(4,3),(5,3),(6,3),(2,4),(3,4),(4,4),(5,4),(6,4),(2,5),(3,5),(4,5),(5,5),(6,5),(2,6),(3,6),(4,6),(5,6),(6,6),(7,6).$$