Cores¶
A \(k\)-core is a partition from which no rim hook of size \(k\) can be removed. Alternatively, a \(k\)-core is an integer partition such that the Ferrers diagram for the partition contains no cells with a hook of size (a multiple of) \(k\).
Authors:
- Anne Schilling and Mike Zabrocki (2011): initial version
- Travis Scrimshaw (2012): Added latex output for Core class
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sage.combinat.core.
Core
¶ A \(k\)-core is an integer partition from which no rim hook of size \(k\) can be removed.
EXAMPLES:
sage: c = Core([2,1],4); c [2, 1] sage: c = Core([3,1],4); c Traceback (most recent call last): ... ValueError: [3, 1] is not a 4-core
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sage.combinat.core.
Cores
(k, length=None, **kwargs)¶ A \(k\)-core is a partition from which no rim hook of size \(k\) can be removed. Alternatively, a \(k\)-core is an integer partition such that the Ferrers diagram for the partition contains no cells with a hook of size (a multiple of) \(k\).
The \(k\)-cores generally have two notions of size which are useful for different applications. One is the number of cells in the Ferrers diagram with hook less than \(k\), the other is the total number of cells of the Ferrers diagram. In the implementation in Sage, the first of notion is referred to as the
length
of the \(k\)-core and the second is thesize
of the \(k\)-core. The class of Cores requires that either the size or the length of the elements in the class is specified.EXAMPLES:
We create the set of the \(4\)-cores of length \(6\). Here the length of a \(k\)-core is the size of the corresponding \((k-1)\)-bounded partition, see also
length()
:sage: C = Cores(4, 6); C 4-Cores of length 6 sage: C.list() [[6, 3], [5, 2, 1], [4, 1, 1, 1], [4, 2, 2], [3, 3, 1, 1], [3, 2, 1, 1, 1], [2, 2, 2, 1, 1, 1]] sage: C.cardinality() 7 sage: C.an_element() [6, 3]
We may also list the set of \(4\)-cores of size \(6\), where the size is the number of boxes in the core, see also
size()
:sage: C = Cores(4, size=6); C 4-Cores of size 6 sage: C.list() [[4, 1, 1], [3, 2, 1], [3, 1, 1, 1]] sage: C.cardinality() 3 sage: C.an_element() [4, 1, 1]
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sage.combinat.core.
Cores_length
¶ The class of \(k\)-cores of length \(n\).
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sage.combinat.core.
Cores_size
¶ The class of \(k\)-cores of size \(n\).