Identifier
Identifier
Mp00021: to bounded partition
Images
([2],3) => [2]
([1,1],3) => [1,1]
([3,1],3) => [2,1]
([2,1,1],3) => [1,1,1]
([4,2],3) => [2,2]
([3,1,1],3) => [2,1,1]
([2,2,1,1],3) => [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]
([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]
([2],4) => [2]
([1,1],4) => [1,1]
([3],4) => [3]
([2,1],4) => [2,1]
([1,1,1],4) => [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]
([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]
([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]
([2],5) => [2]
([1,1],5) => [1,1]
([3],5) => [3]
([2,1],5) => [2,1]
([1,1,1],5) => [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]
([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]
([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]
([2],6) => [2]
([1,1],6) => [1,1]
([3],6) => [3]
([2,1],6) => [2,1]
([1,1,1],6) => [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],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]
([6,1],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]
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