Identifier
Identifier
Mp00021: Cores to bounded partition Integer partitions
Images
([2],3) generating graphics... => [2] generating graphics...
([1,1],3) generating graphics... => [1,1] generating graphics...
([3,1],3) generating graphics... => [2,1] generating graphics...
([2,1,1],3) generating graphics... => [1,1,1] generating graphics...
([4,2],3) generating graphics... => [2,2] generating graphics...
([3,1,1],3) generating graphics... => [2,1,1] generating graphics...
([2,2,1,1],3) generating graphics... => [1,1,1,1] generating graphics...
([5,3,1],3) generating graphics... => [2,2,1] generating graphics...
([4,2,1,1],3) generating graphics... => [2,1,1,1] generating graphics...
([3,2,2,1,1],3) generating graphics... => [1,1,1,1,1] generating graphics...
([6,4,2],3) generating graphics... => [2,2,2] generating graphics...
([5,3,1,1],3) generating graphics... => [2,2,1,1] generating graphics...
([4,2,2,1,1],3) generating graphics... => [2,1,1,1,1] generating graphics...
([3,3,2,2,1,1],3) generating graphics... => [1,1,1,1,1,1] generating graphics...
([2],4) generating graphics... => [2] generating graphics...
([1,1],4) generating graphics... => [1,1] generating graphics...
([3],4) generating graphics... => [3] generating graphics...
([2,1],4) generating graphics... => [2,1] generating graphics...
([1,1,1],4) generating graphics... => [1,1,1] generating graphics...
([4,1],4) generating graphics... => [3,1] generating graphics...
([2,2],4) generating graphics... => [2,2] generating graphics...
([3,1,1],4) generating graphics... => [2,1,1] generating graphics...
([2,1,1,1],4) generating graphics... => [1,1,1,1] generating graphics...
([5,2],4) generating graphics... => [3,2] generating graphics...
([4,1,1],4) generating graphics... => [3,1,1] generating graphics...
([3,2,1],4) generating graphics... => [2,2,1] generating graphics...
([3,1,1,1],4) generating graphics... => [2,1,1,1] generating graphics...
([2,2,1,1,1],4) generating graphics... => [1,1,1,1,1] generating graphics...
([6,3],4) generating graphics... => [3,3] generating graphics...
([5,2,1],4) generating graphics... => [3,2,1] generating graphics...
([4,1,1,1],4) generating graphics... => [3,1,1,1] generating graphics...
([4,2,2],4) generating graphics... => [2,2,2] generating graphics...
([3,3,1,1],4) generating graphics... => [2,2,1,1] generating graphics...
([3,2,1,1,1],4) generating graphics... => [2,1,1,1,1] generating graphics...
([2,2,2,1,1,1],4) generating graphics... => [1,1,1,1,1,1] generating graphics...
([2],5) generating graphics... => [2] generating graphics...
([1,1],5) generating graphics... => [1,1] generating graphics...
([3],5) generating graphics... => [3] generating graphics...
([2,1],5) generating graphics... => [2,1] generating graphics...
([1,1,1],5) generating graphics... => [1,1,1] generating graphics...
([4],5) generating graphics... => [4] generating graphics...
([3,1],5) generating graphics... => [3,1] generating graphics...
([2,2],5) generating graphics... => [2,2] generating graphics...
([2,1,1],5) generating graphics... => [2,1,1] generating graphics...
([1,1,1,1],5) generating graphics... => [1,1,1,1] generating graphics...
([5,1],5) generating graphics... => [4,1] generating graphics...
([3,2],5) generating graphics... => [3,2] generating graphics...
([4,1,1],5) generating graphics... => [3,1,1] generating graphics...
([2,2,1],5) generating graphics... => [2,2,1] generating graphics...
([3,1,1,1],5) generating graphics... => [2,1,1,1] generating graphics...
([2,1,1,1,1],5) generating graphics... => [1,1,1,1,1] generating graphics...
([6,2],5) generating graphics... => [4,2] generating graphics...
([5,1,1],5) generating graphics... => [4,1,1] generating graphics...
([3,3],5) generating graphics... => [3,3] generating graphics...
([4,2,1],5) generating graphics... => [3,2,1] generating graphics...
([4,1,1,1],5) generating graphics... => [3,1,1,1] generating graphics...
([2,2,2],5) generating graphics... => [2,2,2] generating graphics...
([3,2,1,1],5) generating graphics... => [2,2,1,1] generating graphics...
([3,1,1,1,1],5) generating graphics... => [2,1,1,1,1] generating graphics...
([2,2,1,1,1,1],5) generating graphics... => [1,1,1,1,1,1] generating graphics...
([2],6) generating graphics... => [2] generating graphics...
([1,1],6) generating graphics... => [1,1] generating graphics...
([3],6) generating graphics... => [3] generating graphics...
([2,1],6) generating graphics... => [2,1] generating graphics...
([1,1,1],6) generating graphics... => [1,1,1] generating graphics...
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([2,2],6) generating graphics... => [2,2] generating graphics...
([2,1,1],6) generating graphics... => [2,1,1] generating graphics...
([1,1,1,1],6) generating graphics... => [1,1,1,1] generating graphics...
([5],6) generating graphics... => [5] generating graphics...
([4,1],6) generating graphics... => [4,1] generating graphics...
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([3,1,1],6) generating graphics... => [3,1,1] generating graphics...
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([2,1,1,1],6) generating graphics... => [2,1,1,1] generating graphics...
([1,1,1,1,1],6) generating graphics... => [1,1,1,1,1] generating graphics...
([6,1],6) generating graphics... => [5,1] generating graphics...
([4,2],6) generating graphics... => [4,2] generating graphics...
([5,1,1],6) generating graphics... => [4,1,1] generating graphics...
([3,3],6) generating graphics... => [3,3] generating graphics...
([3,2,1],6) generating graphics... => [3,2,1] generating graphics...
([4,1,1,1],6) generating graphics... => [3,1,1,1] generating graphics...
([2,2,2],6) generating graphics... => [2,2,2] generating graphics...
([2,2,1,1],6) generating graphics... => [2,2,1,1] generating graphics...
([3,1,1,1,1],6) generating graphics... => [2,1,1,1,1] generating graphics...
([2,1,1,1,1,1],6) generating graphics... => [1,1,1,1,1,1] generating graphics...
([7,2],6) generating graphics... => [5,2] generating graphics...
([6,1,1],6) generating graphics... => [5,1,1] generating graphics...
([4,3],6) generating graphics... => [4,3] generating graphics...
([5,2,1],6) generating graphics... => [4,2,1] generating graphics...
([5,1,1,1],6) generating graphics... => [4,1,1,1] generating graphics...
([3,3,1],6) generating graphics... => [3,3,1] generating graphics...
([3,2,2],6) generating graphics... => [3,2,2] generating graphics...
([4,2,1,1],6) generating graphics... => [3,2,1,1] generating graphics...
([4,1,1,1,1],6) generating graphics... => [3,1,1,1,1] generating graphics...
([2,2,2,1],6) generating graphics... => [2,2,2,1] generating graphics...
([3,2,1,1,1],6) generating graphics... => [2,2,1,1,1] generating graphics...
([3,1,1,1,1,1],6) generating graphics... => [2,1,1,1,1,1] generating graphics...
([2,2,1,1,1,1,1],6) generating graphics... => [1,1,1,1,1,1,1] generating graphics...
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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].
Properties
gradedA map $\phi: A \rightarrow B$ is graded for graded sets $A$ and $B$ if $\operatorname{deg}(a) = \operatorname{deg}(a')$ implies $\operatorname{deg}(\phi(a)) = \operatorname{deg}(\phi(a'))$ for all $a, a' \in A$., bijectiveA map $\phi: A \rightarrow B$ is bijective if it is both injective and surjective. This is if (1) $\phi(a) = \phi(b)$ implies $a=b$, and (2) for any $b \in B$, there is an $a \in A$ such that $\phi(a) = b$.
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
Code
def to_bounded_partition(elt):
    k_boundary = elt.to_partition().k_boundary(elt.k())
    return Partition(k_boundary.row_lengths())

Updated
May 06, 2019 at 14:42 by Christian Stump