- FindStat

Identifier

Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St001901: Integer partitions ⟶ ℤ

Values

([],1) => [1] => [1] => 1

([],2) => [1,1] => [2] => 1

([(0,1)],2) => [2] => [1,1] => 1

([],3) => [1,1,1] => [2,1] => 1

([(1,2)],3) => [2,1] => [1,1,1] => 1

([(0,1),(0,2)],3) => [2,1] => [1,1,1] => 1

([(0,2),(2,1)],3) => [3] => [3] => 1

([(0,2),(1,2)],3) => [2,1] => [1,1,1] => 1

([],4) => [1,1,1,1] => [4] => 1

([(2,3)],4) => [2,1,1] => [2,1,1] => 1

([(1,2),(1,3)],4) => [2,1,1] => [2,1,1] => 1

([(0,1),(0,2),(0,3)],4) => [2,1,1] => [2,1,1] => 1

([(0,2),(0,3),(3,1)],4) => [3,1] => [3,1] => 1

([(0,1),(0,2),(1,3),(2,3)],4) => [3,1] => [3,1] => 1

([(1,2),(2,3)],4) => [3,1] => [3,1] => 1

([(0,3),(3,1),(3,2)],4) => [3,1] => [3,1] => 1

([(1,3),(2,3)],4) => [2,1,1] => [2,1,1] => 1

([(0,3),(1,3),(3,2)],4) => [3,1] => [3,1] => 1

([(0,3),(1,3),(2,3)],4) => [2,1,1] => [2,1,1] => 1

([(0,3),(1,2)],4) => [2,2] => [1,1,1,1] => 1

([(0,3),(1,2),(1,3)],4) => [2,2] => [1,1,1,1] => 1

([(0,2),(0,3),(1,2),(1,3)],4) => [2,2] => [1,1,1,1] => 1

([(0,3),(2,1),(3,2)],4) => [4] => [2,2] => 1

([(0,3),(1,2),(2,3)],4) => [3,1] => [3,1] => 1

([],5) => [1,1,1,1,1] => [4,1] => 2

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

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

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

([(0,1),(0,2),(0,3),(0,4)],5) => [2,1,1,1] => [2,1,1,1] => 1

([(0,2),(0,3),(0,4),(4,1)],5) => [3,1,1] => [3,2] => 1

([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => [3,1,1] => [3,2] => 1

([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => [3,1,1] => [3,2] => 1

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

([(0,3),(0,4),(4,1),(4,2)],5) => [3,1,1] => [3,2] => 1

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

([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => [4,1] => [2,2,1] => 1

([(0,3),(0,4),(3,2),(4,1)],5) => [3,2] => [3,1,1] => 1

([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => [3,2] => [3,1,1] => 1

([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => [3,2] => [3,1,1] => 1

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

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

([(0,4),(4,1),(4,2),(4,3)],5) => [3,1,1] => [3,2] => 1

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

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

([(0,4),(1,4),(4,2),(4,3)],5) => [3,2] => [3,1,1] => 1

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

([(0,4),(1,4),(2,4),(4,3)],5) => [3,1,1] => [3,2] => 1

([(0,4),(1,4),(2,4),(3,4)],5) => [2,1,1,1] => [2,1,1,1] => 1

([(0,4),(1,4),(2,3)],5) => [2,2,1] => [1,1,1,1,1] => 1

([(0,4),(1,3),(2,3),(2,4)],5) => [2,2,1] => [1,1,1,1,1] => 1

([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => [2,2,1] => [1,1,1,1,1] => 1

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => [2,2,1] => [1,1,1,1,1] => 1

([(0,4),(1,4),(2,3),(4,2)],5) => [4,1] => [2,2,1] => 1

([(0,4),(1,3),(2,3),(3,4)],5) => [3,1,1] => [3,2] => 1

([(0,4),(1,4),(2,3),(2,4)],5) => [2,2,1] => [1,1,1,1,1] => 1

([(0,4),(1,4),(2,3),(3,4)],5) => [3,1,1] => [3,2] => 1

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

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

([(0,4),(1,2),(1,4),(2,3)],5) => [3,2] => [3,1,1] => 1

([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => [3,2] => [3,1,1] => 1

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

([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => [3,2] => [3,1,1] => 1

([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => [3,2] => [3,1,1] => 1

([(0,4),(1,2),(1,4),(4,3)],5) => [3,2] => [3,1,1] => 1

([(0,4),(1,2),(1,3)],5) => [2,2,1] => [1,1,1,1,1] => 1

([(0,4),(1,2),(1,3),(1,4)],5) => [2,2,1] => [1,1,1,1,1] => 1

([(0,2),(0,4),(3,1),(4,3)],5) => [4,1] => [2,2,1] => 1

([(0,4),(1,2),(1,3),(3,4)],5) => [3,1,1] => [3,2] => 1

([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => [4,1] => [2,2,1] => 1

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => [3,1,1] => [3,2] => 1

([(0,3),(0,4),(1,2),(1,4)],5) => [2,2,1] => [1,1,1,1,1] => 1

([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => [2,2,1] => [1,1,1,1,1] => 1

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => [2,2,1] => [1,1,1,1,1] => 1

([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => [3,2] => [3,1,1] => 1

([(0,3),(1,2),(1,4),(3,4)],5) => [3,2] => [3,1,1] => 1

([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => [3,2] => [3,1,1] => 1

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

([(0,3),(3,4),(4,1),(4,2)],5) => [4,1] => [2,2,1] => 1

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

([(0,4),(1,2),(2,4),(4,3)],5) => [4,1] => [2,2,1] => 1

([(0,3),(1,4),(4,2)],5) => [3,2] => [3,1,1] => 1

([(0,4),(3,2),(4,1),(4,3)],5) => [4,1] => [2,2,1] => 1

([(0,4),(1,2),(2,3),(2,4)],5) => [3,2] => [3,1,1] => 1

([(0,4),(2,3),(3,1),(4,2)],5) => [5] => [5] => 1

([(0,3),(1,2),(2,4),(3,4)],5) => [3,2] => [3,1,1] => 1

([(0,4),(1,2),(2,3),(3,4)],5) => [4,1] => [2,2,1] => 1

([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => [4,1] => [2,2,1] => 1

([],6) => [1,1,1,1,1,1] => [4,2] => 2

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

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

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

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

([(0,1),(0,2),(0,3),(0,4),(0,5)],6) => [2,1,1,1,1] => [4,1,1] => 2

([(0,2),(0,3),(0,4),(0,5),(5,1)],6) => [3,1,1,1] => [3,2,1] => 3

([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6) => [3,1,1,1] => [3,2,1] => 3

([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6) => [3,1,1,1] => [3,2,1] => 3

([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => [3,1,1,1] => [3,2,1] => 3

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

([(0,3),(0,4),(0,5),(5,1),(5,2)],6) => [3,1,1,1] => [3,2,1] => 3

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

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

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Description

The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition.

Map

Greene-Kleitman invariant

Description

The Greene-Kleitman invariant of a poset.
This is the partition $(c_1 - c_0, c_2 - c_1, c_3 - c_2, \ldots)$, where $c_k$ is the maximum cardinality of a union of $k$ chains of the poset. Equivalently, this is the conjugate of the partition $(a_1 - a_0, a_2 - a_1, a_3 - a_2, \ldots)$, where $a_k$ is the maximum cardinality of a union of $k$ antichains of the poset.

Map

Glaisher-Franklin

Description

The Glaisher-Franklin bijection on integer partitions.
This map sends the set of even part sizes, each divided by two, to the set of repeated part sizes, see [1, 3.3.1].