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Map identifier: Mp00021

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Map name: to bounded partition

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Domain: Cores

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Codomain: Integer partitions

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Description: The (k-1)-bounded partition of a k-core.

Starting with a $k$-core, deleting all cells of hook length greater than or equal to $k$ yields a $(k-1)$-bounded partition [1, Theorem 7], see also [2, Section 1.2].

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References: [1]   Lapointe, L., Morse, J. Tableaux on $k+1$-cores, reduced words for affine permutations, and $k$-Schur expansions [[MathSciNet:2167475]] [[arXiv:math/0402320]]
[2]   Lam, T., Lapointe, L., Morse, J., Schilling, A., Shimozono, M., Zabrocki, M. $k$-Schur functions and affine Schubert calculus [[MathSciNet:3379711]] [[arXiv:1301.3569]]

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Code:

def mapping(elt):
    k_boundary = elt.to_partition().k_boundary(elt.k())
    return Partition(k_boundary.row_lengths())


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Map images:

([2],3)             => [2]
([1,1],3)           => [1,1]
([2],4)             => [2]
([1,1],4)           => [1,1]
([2],5)             => [2]
([1,1],5)           => [1,1]
([2],6)             => [2]
([1,1],6)           => [1,1]
([3,1],3)           => [2,1]
([2,1,1],3)         => [1,1,1]
([3],4)             => [3]
([2,1],4)           => [2,1]
([1,1,1],4)         => [1,1,1]
([3],5)             => [3]
([2,1],5)           => [2,1]
([1,1,1],5)         => [1,1,1]
([3],6)             => [3]
([2,1],6)           => [2,1]
([1,1,1],6)         => [1,1,1]
([4,2],3)           => [2,2]
([3,1,1],3)         => [2,1,1]
([2,2,1,1],3)       => [1,1,1,1]
([4,1],4)           => [3,1]
([2,2],4)           => [2,2]
([3,1,1],4)         => [2,1,1]
([2,1,1,1],4)       => [1,1,1,1]
([4],5)             => [4]
([3,1],5)           => [3,1]
([2,2],5)           => [2,2]
([2,1,1],5)         => [2,1,1]
([1,1,1,1],5)       => [1,1,1,1]
([4],6)             => [4]
([3,1],6)           => [3,1]
([2,2],6)           => [2,2]
([2,1,1],6)         => [2,1,1]
([1,1,1,1],6)       => [1,1,1,1]
([5,3,1],3)         => [2,2,1]
([4,2,1,1],3)       => [2,1,1,1]
([3,2,2,1,1],3)     => [1,1,1,1,1]
([5,2],4)           => [3,2]
([4,1,1],4)         => [3,1,1]
([3,2,1],4)         => [2,2,1]
([3,1,1,1],4)       => [2,1,1,1]
([2,2,1,1,1],4)     => [1,1,1,1,1]
([5,1],5)           => [4,1]
([3,2],5)           => [3,2]
([4,1,1],5)         => [3,1,1]
([2,2,1],5)         => [2,2,1]
([3,1,1,1],5)       => [2,1,1,1]
([2,1,1,1,1],5)     => [1,1,1,1,1]
([5],6)             => [5]
([4,1],6)           => [4,1]
([3,2],6)           => [3,2]
([3,1,1],6)         => [3,1,1]
([2,2,1],6)         => [2,2,1]
([2,1,1,1],6)       => [2,1,1,1]
([1,1,1,1,1],6)     => [1,1,1,1,1]
([6,4,2],3)         => [2,2,2]
([5,3,1,1],3)       => [2,2,1,1]
([4,2,2,1,1],3)     => [2,1,1,1,1]
([3,3,2,2,1,1],3)   => [1,1,1,1,1,1]
([6,3],4)           => [3,3]
([5,2,1],4)         => [3,2,1]
([4,1,1,1],4)       => [3,1,1,1]
([4,2,2],4)         => [2,2,2]
([3,3,1,1],4)       => [2,2,1,1]
([3,2,1,1,1],4)     => [2,1,1,1,1]
([2,2,2,1,1,1],4)   => [1,1,1,1,1,1]
([6,2],5)           => [4,2]
([5,1,1],5)         => [4,1,1]
([3,3],5)           => [3,3]
([4,2,1],5)         => [3,2,1]
([4,1,1,1],5)       => [3,1,1,1]
([2,2,2],5)         => [2,2,2]
([3,2,1,1],5)       => [2,2,1,1]
([3,1,1,1,1],5)     => [2,1,1,1,1]
([2,2,1,1,1,1],5)   => [1,1,1,1,1,1]
([6,1],6)           => [5,1]
([4,2],6)           => [4,2]
([5,1,1],6)         => [4,1,1]
([3,3],6)           => [3,3]
([3,2,1],6)         => [3,2,1]
([4,1,1,1],6)       => [3,1,1,1]
([2,2,2],6)         => [2,2,2]
([2,2,1,1],6)       => [2,2,1,1]
([3,1,1,1,1],6)     => [2,1,1,1,1]
([2,1,1,1,1,1],6)   => [1,1,1,1,1,1]
([7,2],6)           => [5,2]
([6,1,1],6)         => [5,1,1]
([4,3],6)           => [4,3]
([5,2,1],6)         => [4,2,1]
([5,1,1,1],6)       => [4,1,1,1]
([3,3,1],6)         => [3,3,1]
([3,2,2],6)         => [3,2,2]
([4,2,1,1],6)       => [3,2,1,1]
([4,1,1,1,1],6)     => [3,1,1,1,1]
([2,2,2,1],6)       => [2,2,2,1]
([3,2,1,1,1],6)     => [2,2,1,1,1]
([3,1,1,1,1,1],6)   => [2,1,1,1,1,1]
([2,2,1,1,1,1,1],6) => [1,1,1,1,1,1,1]

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Created: Jan 19, 2020 at 07:28 by FindStatCrew

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Last Updated: Jan 19, 2020 at 07:28 by Martin Rubey