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Identifier

Mp00021: Cores —to bounded partition⟶ Integer partitions

Images

=>
            
              Cc0013;cc-rep-0Cc0002;cc-rep-1
            

            ([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]
          

click to show experimental identities (only identities of compositions of up to three maps are shown)

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].

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

Properties

graded, bijective

Sage code

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

Weight

26

Created

Jan 19, 2020 at 07:28 by FindStatCrew

Updated

Jan 19, 2020 at 07:28 by Martin Rubey