- FindStat

Identifier

Mp00021: Cores —to bounded partition⟶ Integer partitions
St000159: Integer partitions ⟶ ℤ (values match St000318The number of addable cells of the Ferrers diagram of an integer partition., St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition.)

Values

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

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Description

The number of distinct parts of the integer partition.
This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.

Map

to bounded partition

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].