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1. Definition

A $k$-core is an integer partition $\lambda$ which has no hooks whose size is a multiples of $k$, or, equivalently, $\lambda$ has no rim hooks that are multiples of $k$ [JK81].

2. Properties

$$\prod_{n=1}^\infty \frac{(1 - q^{2n})^2}{(1 - q^n)} = \sum_{n=0}^\infty c_2(n) q^n = 1 + q + q^3 + q^6 + q^{10} + q^{15} + q^{21} + \cdots$$

3. Statistics

We have the following 5 statistics on Cores in the database:

St000158 Cores ⟶ ℤ
The length of a core.
St000190 Cores ⟶ ℤ
The size of a core.
St000191 Cores ⟶ ℤ
The number of strong covers of a core.
St000192 Cores ⟶ ℤ
Number of covers of a core in weak Bruhat order.
St000202 Cores ⟶ ℤ
The number of k-cores contained in the k-core.

4. Maps

We have the following 2 maps from and to Cores in the database:

Mp00021 Cores ⟶ Integer partitions
to bounded partition
Mp00022 Cores ⟶ Integer partitions
to partition

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 $k$.

The $k$-core $\lambda$ is obtained from its corresponding $(k-1)$-bounded partition by finding the partition $\mu$ so that the skew partition $\lambda / \mu$ meets the following criteria:

  1. Every row of $\lambda / \mu$ has the same number of cells as the $(k-1)$-bounded partition

  2. No cell in $\lambda / \mu$ has a hook length exceeding $(k-1)$

  3. Each empty cell above $\lambda / \mu$ has a hook length exceeding $(k-1)$. [LLM12]

For $k$-cores, we can map the corresponding $(k-1)$-bounded partitions to minimal length affine permutation representatives by using the content of the cells of the of the Ferrers diagram modulo $k$. This reduced content is called the residue of a cell. We read across the rows from right to left, starting at the bottom row and working up, and writing $s_i$ when the content of the cell mod $k$ is $i$.

For minimal length representatives of $\tilde{S}_k / S_k$, we apply the left action of $\omega \in \tilde{S}_k / S_k$ to the empty core to obtain a $k$-core. Define the left action by letting $s_i \cdot \lambda$ be the partition:

  1. if there is at least one addable corner of residue $i$, then the result is $\lambda$ with all addable corners of $\lambda$ of residue $i$ added.

  2. if there is at least one removable corner of residue $i$, then the result is $\lambda$ with all removable corners of $\lambda$ of residue $i$ removed.

  3. otherwise the result is $\lambda$. [LLM12]

$$s_2s_4s_3s_1s_0s_4s_2s_1s_0 \cdot \emptyset \to s_2s_4s_3s_1s_0s_4s_2s_1 \cdot \left\lvert\begin{matrix}\hline 0\\ \hline\end{matrix}\right\rvert \\ \to s_2s_4s_3s_1s_0s_4s_2 \cdot \left\lvert \begin{matrix}\hline 0&1\\ \hline\end{matrix}\right\rvert \to s_2s_4s_3s_1s_0s_4 \cdot \left\lvert \begin{matrix}\hline 0&1&2\\ \hline\end{matrix}\right\rvert \\ \to s_2s_4s_3s_1s_0 \cdot \left\lvert \begin{matrix}\hline 0&1&2\\4\\ \hline\end{matrix}\right\rvert \to \cdots \to \left\lvert \begin{matrix}\hline 0&1&2&3&4\\4&0&1&2\\3&4\\2\\ \hline\end{matrix}\right\rvert$$

5. Covers and Ordering

Note we define a $k$-core's length as the number of elements with a hook length of less than $k$, or equivalently the size of the corresponding $(k-1)$-bounded partition or the length of the corresponding affine permutation.

5.1. Weak Covers

We say that the $k$-core $\lambda$ covers the $k$-core $\mu$ if, and only if, the length of $\lambda$ is one more than the length of $\mu$ and there exists some $i$ such that $\lambda=s_i \cdot \mu$. This defines a weak Bruhat order on $k$-cores. [LLM12]

5.2. Strong Covers

We say that the $k$-core $\lambda$ covers the $k$-core $\mu$ if, and only if, the length of $\lambda$ is one more than the length of $\mu$ and $\mu$ is contained in $\lambda$. This defines a strong order on $k$-cores. [LLM12]

6. $(a,b)$-cores

6.1. Definitions

6.2. Properties


7. 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.

8. Sage examples