- FindStat

Your data matches 175 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000775
(load all 25 compositions to match this statistic)

Mapping

St000775: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The multiplicity of the largest eigenvalue in a graph.

Matching statistic: St001933
(load all 5 compositions to match this statistic)

Mapping

Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 97%●distinct values known / distinct values provided: 50%

Values

([],1)
 => []
 => ? = 1
([],2)
 => []
 => ? = 2
([(0,1)],2)
 => [1]
 => 1
([],3)
 => []
 => ? = 3
([(1,2)],3)
 => [1]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
([],4)
 => []
 => ? = 4
([(2,3)],4)
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [2]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [3]
 => 1
([(0,3),(1,2)],4)
 => [1,1]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [3]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => 1
([],5)
 => []
 => ? = 5
([(3,4)],5)
 => [1]
 => 1
([(2,4),(3,4)],5)
 => [2]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [3]
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => 1
([(1,4),(2,3)],5)
 => [1,1]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [3]
 => 1
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => 1
([],6)
 => []
 => ? = 6
([(4,5)],6)
 => [1]
 => 1
([(3,5),(4,5)],6)
 => [2]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [3]
 => 1

Description

The largest multiplicity of a part in an integer partition.

Matching statistic: St000667
(load all 4 compositions to match this statistic)

Mapping

Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 95%●distinct values known / distinct values provided: 50%

Values

([],1)
 => []
 => []
 => ? = 1
([],2)
 => []
 => []
 => ? = 2
([(0,1)],2)
 => [1]
 => [1]
 => 1
([],3)
 => []
 => []
 => ? = 3
([(1,2)],3)
 => [1]
 => [1]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => [1,1]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => 1
([],4)
 => []
 => []
 => ? = 4
([(2,3)],4)
 => [1]
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [2]
 => [1,1]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => 1
([(0,3),(1,2)],4)
 => [1,1]
 => [2]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1]
 => 1
([],5)
 => []
 => []
 => ? = 5
([(3,4)],5)
 => [1]
 => [1]
 => 1
([(2,4),(3,4)],5)
 => [2]
 => [1,1]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => 1
([(1,4),(2,3)],5)
 => [1,1]
 => [2]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [3]
 => [1,1,1]
 => 1
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [2,1]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => 1
([],6)
 => []
 => []
 => ? ∊ {1,1,1,1,6}
([(4,5)],6)
 => [1]
 => [1]
 => 1
([(3,5),(4,5)],6)
 => [2]
 => [1,1]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => 1
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [13]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,6}
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [13]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,6}
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,6}
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,6}

Description

The greatest common divisor of the parts of the partition.

Matching statistic: St000706
(load all 6 compositions to match this statistic)

Mapping

Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000706: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 93%●distinct values known / distinct values provided: 50%

Values

([],1)
 => []
 => ? = 1
([],2)
 => []
 => ? ∊ {1,2}
([(0,1)],2)
 => [1]
 => ? ∊ {1,2}
([],3)
 => []
 => ? ∊ {1,3}
([(1,2)],3)
 => [1]
 => ? ∊ {1,3}
([(0,2),(1,2)],3)
 => [2]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
([],4)
 => []
 => ? ∊ {1,4}
([(2,3)],4)
 => [1]
 => ? ∊ {1,4}
([(1,3),(2,3)],4)
 => [2]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [3]
 => 1
([(0,3),(1,2)],4)
 => [1,1]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [3]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => 1
([],5)
 => []
 => ? ∊ {1,5}
([(3,4)],5)
 => [1]
 => ? ∊ {1,5}
([(2,4),(3,4)],5)
 => [2]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [3]
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => 1
([(1,4),(2,3)],5)
 => [1,1]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [3]
 => 1
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => 1
([],6)
 => []
 => ? ∊ {1,1,1,1,1,3}
([(4,5)],6)
 => [1]
 => ? ∊ {1,1,1,1,1,3}
([(3,5),(4,5)],6)
 => [2]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [3]
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => 1
([(2,5),(3,4)],6)
 => [1,1]
 => 2
([(2,5),(3,4),(4,5)],6)
 => [3]
 => 1
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => 1
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [13]
 => ? ∊ {1,1,1,1,1,3}
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [13]
 => ? ∊ {1,1,1,1,1,3}
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [14]
 => ? ∊ {1,1,1,1,1,3}
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [15]
 => ? ∊ {1,1,1,1,1,3}

Description

The product of the factorials of the multiplicities of an integer partition.

Matching statistic: St000993
(load all 7 compositions to match this statistic)

Mapping

Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 93%●distinct values known / distinct values provided: 50%

Values

([],1)
 => []
 => ? = 1
([],2)
 => []
 => ? ∊ {1,2}
([(0,1)],2)
 => [1]
 => ? ∊ {1,2}
([],3)
 => []
 => ? ∊ {1,3}
([(1,2)],3)
 => [1]
 => ? ∊ {1,3}
([(0,2),(1,2)],3)
 => [2]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
([],4)
 => []
 => ? ∊ {1,4}
([(2,3)],4)
 => [1]
 => ? ∊ {1,4}
([(1,3),(2,3)],4)
 => [2]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [3]
 => 1
([(0,3),(1,2)],4)
 => [1,1]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [3]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => 1
([],5)
 => []
 => ? ∊ {1,5}
([(3,4)],5)
 => [1]
 => ? ∊ {1,5}
([(2,4),(3,4)],5)
 => [2]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [3]
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => 1
([(1,4),(2,3)],5)
 => [1,1]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [3]
 => 1
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => 1
([],6)
 => []
 => ? ∊ {1,1,1,1,1,6}
([(4,5)],6)
 => [1]
 => ? ∊ {1,1,1,1,1,6}
([(3,5),(4,5)],6)
 => [2]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [3]
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => 1
([(2,5),(3,4)],6)
 => [1,1]
 => 2
([(2,5),(3,4),(4,5)],6)
 => [3]
 => 1
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => 1
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [13]
 => ? ∊ {1,1,1,1,1,6}
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [13]
 => ? ∊ {1,1,1,1,1,6}
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [14]
 => ? ∊ {1,1,1,1,1,6}
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [15]
 => ? ∊ {1,1,1,1,1,6}

Description

The multiplicity of the largest part of an integer partition.

Matching statistic: St001571
(load all 4 compositions to match this statistic)

Mapping

Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001571: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 93%●distinct values known / distinct values provided: 50%

Values

([],1)
 => []
 => []
 => ? = 1
([],2)
 => []
 => []
 => ? = 2
([(0,1)],2)
 => [1]
 => [1]
 => 1
([],3)
 => []
 => []
 => ? = 3
([(1,2)],3)
 => [1]
 => [1]
 => 1
([(0,2),(1,2)],3)
 => [2]
 => [1,1]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => 1
([],4)
 => []
 => []
 => ? = 4
([(2,3)],4)
 => [1]
 => [1]
 => 1
([(1,3),(2,3)],4)
 => [2]
 => [1,1]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => 1
([(0,3),(1,2)],4)
 => [1,1]
 => [2]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1]
 => 1
([],5)
 => []
 => []
 => ? = 5
([(3,4)],5)
 => [1]
 => [1]
 => 1
([(2,4),(3,4)],5)
 => [2]
 => [1,1]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => 1
([(1,4),(2,3)],5)
 => [1,1]
 => [2]
 => 2
([(1,4),(2,3),(3,4)],5)
 => [3]
 => [1,1,1]
 => 1
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [2,1]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => 1
([],6)
 => []
 => []
 => ? ∊ {1,1,1,1,1,1,1,1,1,6}
([(4,5)],6)
 => [1]
 => [1]
 => 1
([(3,5),(4,5)],6)
 => [2]
 => [1,1]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => 1
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,6}
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,6}
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,6}
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [13]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,6}
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,6}
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [12]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,6}
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [13]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,6}
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,6}
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [15]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? ∊ {1,1,1,1,1,1,1,1,1,6}

Description

The Cartan determinant of the integer partition. Let

$p=[p_1,...,p_r]$ be a given integer partition with highest part t. Let

$A=K[x]/(x^t)$ be the finite dimensional algebra over the field

$K$ and

$M$ the direct sum of the indecomposable

$A$ -modules of vector space dimension

$p_i$ for each

$i$ . Then the Cartan determinant of

$p$ is the Cartan determinant of the endomorphism algebra of

$M$ over

$A$ . Explicitly, this is the determinant of the matrix

$\left(\min(\bar p_i, \bar p_j)\right)_{i,j}$ , where

$\bar p$ is the set of distinct parts of the partition.

Matching statistic: St001389
(load all 2 compositions to match this statistic)

Mapping

Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 93%●distinct values known / distinct values provided: 50%

Values

([],1)
 => []
 => []
 => ?
 => ? = 1
([],2)
 => []
 => []
 => ?
 => ? ∊ {1,2}
([(0,1)],2)
 => [1]
 => [1]
 => []
 => ? ∊ {1,2}
([],3)
 => []
 => []
 => ?
 => ? ∊ {1,3}
([(1,2)],3)
 => [1]
 => [1]
 => []
 => ? ∊ {1,3}
([(0,2),(1,2)],3)
 => [2]
 => [1,1]
 => [1]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([],4)
 => []
 => []
 => ?
 => ? ∊ {1,2,4}
([(2,3)],4)
 => [1]
 => [1]
 => []
 => ? ∊ {1,2,4}
([(1,3),(2,3)],4)
 => [2]
 => [1,1]
 => [1]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,3),(1,2)],4)
 => [1,1]
 => [2]
 => []
 => ? ∊ {1,2,4}
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([],5)
 => []
 => []
 => ?
 => ? ∊ {1,2,5}
([(3,4)],5)
 => [1]
 => [1]
 => []
 => ? ∊ {1,2,5}
([(2,4),(3,4)],5)
 => [2]
 => [1,1]
 => [1]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(1,4),(2,3)],5)
 => [1,1]
 => [2]
 => []
 => ? ∊ {1,2,5}
([(1,4),(2,3),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [2,1]
 => [1]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => 1
([],6)
 => []
 => []
 => ?
 => ? ∊ {1,1,2,6}
([(4,5)],6)
 => [1]
 => [1]
 => []
 => ? ∊ {1,1,2,6}
([(3,5),(4,5)],6)
 => [2]
 => [1,1]
 => [1]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(2,5),(3,4)],6)
 => [1,1]
 => [2]
 => []
 => ? ∊ {1,1,2,6}
([(2,5),(3,4),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [2,1]
 => [1]
 => 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {1,1,2,6}

Description

The number of partitions of the same length below the given integer partition. For a partition

$\lambda_1 \geq \dots \lambda_k > 0$ , this number is

$\det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$

Matching statistic: St001780
(load all 6 compositions to match this statistic)

Mapping

Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001780: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 93%●distinct values known / distinct values provided: 33%

Values

([],1)
 => []
 => []
 => ?
 => ? = 1
([],2)
 => []
 => []
 => ?
 => ? ∊ {1,2}
([(0,1)],2)
 => [1]
 => [1]
 => []
 => ? ∊ {1,2}
([],3)
 => []
 => []
 => ?
 => ? ∊ {1,3}
([(1,2)],3)
 => [1]
 => [1]
 => []
 => ? ∊ {1,3}
([(0,2),(1,2)],3)
 => [2]
 => [1,1]
 => [1]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([],4)
 => []
 => []
 => ?
 => ? ∊ {1,2,4}
([(2,3)],4)
 => [1]
 => [1]
 => []
 => ? ∊ {1,2,4}
([(1,3),(2,3)],4)
 => [2]
 => [1,1]
 => [1]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,3),(1,2)],4)
 => [1,1]
 => [2]
 => []
 => ? ∊ {1,2,4}
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([],5)
 => []
 => []
 => ?
 => ? ∊ {1,2,5}
([(3,4)],5)
 => [1]
 => [1]
 => []
 => ? ∊ {1,2,5}
([(2,4),(3,4)],5)
 => [2]
 => [1,1]
 => [1]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(1,4),(2,3)],5)
 => [1,1]
 => [2]
 => []
 => ? ∊ {1,2,5}
([(1,4),(2,3),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [2,1]
 => [1]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => 1
([],6)
 => []
 => []
 => ?
 => ? ∊ {1,2,3,6}
([(4,5)],6)
 => [1]
 => [1]
 => []
 => ? ∊ {1,2,3,6}
([(3,5),(4,5)],6)
 => [2]
 => [1,1]
 => [1]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(2,5),(3,4)],6)
 => [1,1]
 => [2]
 => []
 => ? ∊ {1,2,3,6}
([(2,5),(3,4),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [2,1]
 => [1]
 => 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {1,2,3,6}

Description

The order of promotion on the set of standard tableaux of given shape.

Matching statistic: St001899
(load all 2 compositions to match this statistic)

Mapping

Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001899: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 93%●distinct values known / distinct values provided: 33%

Values

([],1)
 => []
 => []
 => ?
 => ? = 1
([],2)
 => []
 => []
 => ?
 => ? ∊ {1,2}
([(0,1)],2)
 => [1]
 => [1]
 => []
 => ? ∊ {1,2}
([],3)
 => []
 => []
 => ?
 => ? ∊ {1,3}
([(1,2)],3)
 => [1]
 => [1]
 => []
 => ? ∊ {1,3}
([(0,2),(1,2)],3)
 => [2]
 => [1,1]
 => [1]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([],4)
 => []
 => []
 => ?
 => ? ∊ {1,2,4}
([(2,3)],4)
 => [1]
 => [1]
 => []
 => ? ∊ {1,2,4}
([(1,3),(2,3)],4)
 => [2]
 => [1,1]
 => [1]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,3),(1,2)],4)
 => [1,1]
 => [2]
 => []
 => ? ∊ {1,2,4}
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([],5)
 => []
 => []
 => ?
 => ? ∊ {1,2,5}
([(3,4)],5)
 => [1]
 => [1]
 => []
 => ? ∊ {1,2,5}
([(2,4),(3,4)],5)
 => [2]
 => [1,1]
 => [1]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(1,4),(2,3)],5)
 => [1,1]
 => [2]
 => []
 => ? ∊ {1,2,5}
([(1,4),(2,3),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [2,1]
 => [1]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => 1
([],6)
 => []
 => []
 => ?
 => ? ∊ {1,2,3,6}
([(4,5)],6)
 => [1]
 => [1]
 => []
 => ? ∊ {1,2,3,6}
([(3,5),(4,5)],6)
 => [2]
 => [1,1]
 => [1]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(2,5),(3,4)],6)
 => [1,1]
 => [2]
 => []
 => ? ∊ {1,2,3,6}
([(2,5),(3,4),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [2,1]
 => [1]
 => 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {1,2,3,6}

Description

The total number of irreducible representations contained in the higher Lie character for an integer partition.

Matching statistic: St001900
(load all 2 compositions to match this statistic)

Mapping

Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001900: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 93%●distinct values known / distinct values provided: 33%

Values

([],1)
 => []
 => []
 => ?
 => ? = 1
([],2)
 => []
 => []
 => ?
 => ? ∊ {1,2}
([(0,1)],2)
 => [1]
 => [1]
 => []
 => ? ∊ {1,2}
([],3)
 => []
 => []
 => ?
 => ? ∊ {1,3}
([(1,2)],3)
 => [1]
 => [1]
 => []
 => ? ∊ {1,3}
([(0,2),(1,2)],3)
 => [2]
 => [1,1]
 => [1]
 => 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([],4)
 => []
 => []
 => ?
 => ? ∊ {1,2,4}
([(2,3)],4)
 => [1]
 => [1]
 => []
 => ? ∊ {1,2,4}
([(1,3),(2,3)],4)
 => [2]
 => [1,1]
 => [1]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,3),(1,2)],4)
 => [1,1]
 => [2]
 => []
 => ? ∊ {1,2,4}
([(0,3),(1,2),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([],5)
 => []
 => []
 => ?
 => ? ∊ {1,2,5}
([(3,4)],5)
 => [1]
 => [1]
 => []
 => ? ∊ {1,2,5}
([(2,4),(3,4)],5)
 => [2]
 => [1,1]
 => [1]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(1,4),(2,3)],5)
 => [1,1]
 => [2]
 => []
 => ? ∊ {1,2,5}
([(1,4),(2,3),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [2,1]
 => [1]
 => 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [9]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => 1
([],6)
 => []
 => []
 => ?
 => ? ∊ {1,2,3,6}
([(4,5)],6)
 => [1]
 => [1]
 => []
 => ? ∊ {1,2,3,6}
([(3,5),(4,5)],6)
 => [2]
 => [1,1]
 => [1]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(2,5),(3,4)],6)
 => [1,1]
 => [2]
 => []
 => ? ∊ {1,2,3,6}
([(2,5),(3,4),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [2,1]
 => [1]
 => 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => [1,1,1]
 => [1,1]
 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {1,2,3,6}

Description

The number of distinct irreducible representations contained in the higher Lie character for an integer partition.

The following 165 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001924The number of cells in an integer partition whose arm and leg length coincide. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001432The order dimension of the partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000733The row containing the largest entry of a standard tableau. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St000297The number of leading ones in a binary word. St000392The length of the longest run of ones in a binary word. St001372The length of a longest cyclic run of ones of a binary word. St000781The number of proper colouring schemes of a Ferrers diagram. St000877The depth of the binary word interpreted as a path. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001568The smallest positive integer that does not appear twice in the partition. St000655The length of the minimal rise of a Dyck path. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St001593This is the number of standard Young tableaux of the given shifted shape. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000284The Plancherel distribution on integer partitions. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000618The number of self-evacuating tableaux of given shape. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St000762The sum of the positions of the weak records of an integer composition. St000678The number of up steps after the last double rise of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001733The number of weak left to right maxima of a Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001890The maximum magnitude of the Möbius function of a poset. St001256Number of simple reflexive modules that are 2-stable reflexive. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001592The maximal number of simple paths between any two different vertices of a graph. St001732The number of peaks visible from the left. St000026The position of the first return of a Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001191Number of simple modules

$S$ with

$Ext_A^i(S,A)=0$ for all

$i=0,1,...,g-1$ in the corresponding Nakayama algebra

$A$ with global dimension

$g$ . St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001518The number of graphs with the same ordinary spectrum as the given graph. St000456The monochromatic index of a connected graph. St001118The acyclic chromatic index of a graph. St001281The normalized isoperimetric number of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000553The number of blocks of a graph. St000785The number of distinct colouring schemes of a graph. St000916The packing number of a graph. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St000331The number of upper interactions of a Dyck path. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n−1}]$ by adding

$c_0$ to

$c_{n−1}$ . St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001722The number of minimal chains with small intervals between a binary word and the top element. St000003The number of standard Young tableaux of the partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000053The number of valleys of the Dyck path. St000075The orbit size of a standard tableau under promotion. St000182The number of permutations whose cycle type is the given integer partition. St000183The side length of the Durfee square of an integer partition. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000321The number of integer partitions of n that are dominated by an integer partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000345The number of refinements of a partition. St000390The number of runs of ones in a binary word. St000517The Kreweras number of an integer partition. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000628The balance of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000675The number of centered multitunnels of a Dyck path. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000847The number of standard Young tableaux whose descent set is the binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000897The number of different multiplicities of parts of an integer partition. St000913The number of ways to refine the partition into singletons. St000935The number of ordered refinements of an integer partition. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000983The length of the longest alternating subword. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001196The global dimension of

$A$ minus the global dimension of

$eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001313The number of Dyck paths above the lattice path given by a binary word. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001595The number of standard Young tableaux of the skew partition. St001597The Frobenius rank of a skew partition. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001884The number of borders of a binary word. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001330The hat guessing number of a graph. St000287The number of connected components of a graph. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001877Number of indecomposable injective modules with projective dimension 2. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees.