A core partition with modulus , or -core partition, is a partition none of whose hook lengths is divisible by the positive integer . The hook length of a box in its Ferrers diagram is one plus the number of boxes to its right in the same row and below it in the same column. A simultaneous -core partition is both an -core and a -core.

For example, has hook lengths 3, 1, and 1, so it is a 2-core but not a 3-core. For relatively prime positive integers , the number of simultaneous core partitions, including the empty partition, is

(Anderson 2002). For , , the two are the empty partition and .

A corner is a box whose deletion still leaves a Ferrers diagram. Its count is the number of distinct parts. Cho et al. (2026) give formulas and bijections for self-conjugate partitions that are simultaneously -core, refined by their number of corners.

Aydemir (2026) gave formulas for the largest size and length of a simultaneous -core with distinct parts for every relatively prime pair . Size is the sum of the parts, whereas length is the number of parts. For the maximum length, suppose and run the Euclidean algorithm on until the first remainder 1. If is the preceding remainder, back-substitution gives and with , , and , with when (Aydemir 2026). Put and . The maximum length is when is odd and , and otherwise, where is the floor function. For example, the maximum length of a distinct-part -core is 5 and its maximum size is 24.

Removing rim hooks of length from a partition until none remain leaves its unique -core . The -quotient is an ordered -tuple of partitions obtained from the residue classes modulo in the abacus representation. Their sizes satisfy

where denotes the sum of the parts of (Garvan et al. 1990). Jacon (2026) gives a corresponding size formula for charged multipartitions with additional terms depending on the charges.

See also

Ferrers Diagram, Hook Length, Hook Length Formula, Partition, Self-Conjugate Partition

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References

Anderson, J. "Partitions Which Are Simultaneously - and -Core." Disc. Math. 248, 237-243, 2002. https://doi.org/10.1016/S0012-365X(01)00343-0.Aydemir, M. C. "Maximum Length and Size of Simultaneous Core Partitions with Distinct Parts." 26 Oct 2026. https://arxiv.org/abs/2610.35322.Cho, H.; Lee, H.-H.; Lee, K.; Nam, H.; and Sohn, J. "Corners of Self-Conjugate -Core Partitions." Elec. J. Combin. 33, P3.53, 2026. https://doi.org/10.37236/14987.Garvan, F.; Kim, D.; and Stanton, D. "Cranks and -Cores." Invent. Math. 101, 1-17, 1990. https://doi.org/10.1007/BF01231493.Jacon, N. "Size of Multipartitions through Level-Rank Duality." Elec. J. Combin. 33, P4.34, 2026. https://doi.org/10.37236/14190.

Cite this as:

Weisstein, Eric W. "Core Partition." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CorePartition.html

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