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Definition & Example
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- A $k$-core is an [integer partition](/IntegerPartitions) $\lambda$ which has no hooks whose size are multiples of $k$.
- Equivalently, $\lambda$ has no rim hooks that are multiples of $k$ <>.
- The **length** of a $k$-core is the number of boxes in its diagram with hook length less than $k$.
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- $k$-cores are graphically represented by the associated [Ferrers](http://en.wikipedia.org/wiki/Norman_Macleod_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](http://oeis.org/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 <>.
Properties
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- 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 $k$ [](http://arxiv.org/abs/1301.3569)<>.
- There is a bijection between $k$-cores and *affine Grassmannian elements*.
$(a,b)$-cores {#ab-cores}
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- 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 <>.
See [](http://arxiv.org/abs/1308.0572)<> for open questions and conjectures concerning $(a,b)$-cores relating to generalized Catalan numbers and $q,t$-Catalan numbers.
References
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Sage examples
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{{{#!sagecell
for n in [2,3,4,5]:
for k in [1 .. n]:
print n,k, Cores(n,k).cardinality()
for c in Cores(5,4):
print c
}}}
Technical information for database usage
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- A $k$-core is uniquely represented as a pair `(X,n)` with `X` being the partition and `n` 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 $$(2,3),(3,3),(4,3),(5,3),(6,3),(2,4),(3,4),(4,4),(5,4),(6,4),(2,5),(3,5),(4,5),(5,5),(6,5),(2,6),(3,6),(4,6),(5,6),(6,6),(7,6).$$