Your data matches 50 different statistics following compositions of up to 3 maps.
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Matching statistic: St001570
(load all 4 compositions to match this statistic)
Mapping
Mp00156: Graphs —line graph⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => 1
([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(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)
 => 0
([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(1,4),(2,3),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => 1
([(0,1),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => 2
([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 0
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => 0
([(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(2,5),(3,4),(4,5)],6)
 => ([(0,2),(1,2)],3)
 => 1
([(1,2),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 2
([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],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)
 => 0
([(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 0
([(0,5),(1,5),(2,4),(3,4)],6)
 => ([(0,3),(1,2)],4)
 => 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
Description
The minimal number of edges to add to make a graph Hamiltonian. A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Matching statistic: St001933
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 50%
Values
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? = 0 - 1
([(0,3),(1,3),(2,3)],4)
 => [3]
 => []
 => ? = 0 - 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => []
 => ? = 1 - 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => ? = 0 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 0 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ? = 0 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => ? = 0 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => ? = 0 - 1
([(1,4),(2,4),(3,4)],5)
 => [3]
 => []
 => ? = 0 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => []
 => ? = 0 - 1
([(1,4),(2,3),(3,4)],5)
 => [3]
 => []
 => ? = 1 - 1
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [1]
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => ? = 0 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => []
 => ? = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => []
 => ? = 0 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => ? = 0 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => ? = 0 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => []
 => ? = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 1 = 2 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ? = 0 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 1
([(2,5),(3,5),(4,5)],6)
 => [3]
 => []
 => ? = 0 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 1
([(2,5),(3,4),(4,5)],6)
 => [3]
 => []
 => ? = 1 - 1
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [1]
 => 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => ? = 0 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [4]
 => []
 => ? = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 1 = 2 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4]
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [5]
 => []
 => ? = 1 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [5]
 => []
 => ? = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [5]
 => []
 => ? = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 0 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 0 - 1
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => [1,1]
 => 2 = 3 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
 => [4]
 => []
 => ? = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => [3,1]
 => [1]
 => 1 = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1]
 => [1]
 => 1 = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,2]
 => [2]
 => 1 = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 1 = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [3]
 => 1 = 2 - 1
([(2,3),(4,6),(5,6)],7)
 => [2,1]
 => [1]
 => 1 = 2 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [3,1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => 1 = 2 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => [2,2]
 => [2]
 => 1 = 2 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => [3,2]
 => [2]
 => 1 = 2 - 1
([(1,6),(2,5),(3,4)],7)
 => [1,1,1]
 => [1,1]
 => 2 = 3 - 1
([(1,2),(3,6),(4,5),(5,6)],7)
 => [3,1]
 => [1]
 => 1 = 2 - 1
([(0,3),(1,2),(4,6),(5,6)],7)
 => [2,1,1]
 => [1,1]
 => 2 = 3 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => [3,1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
 => [4,1]
 => [1]
 => 1 = 2 - 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => 1 = 2 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1]
 => [1]
 => 1 = 2 - 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => [3,2]
 => [2]
 => 1 = 2 - 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => [3,2]
 => [2]
 => 1 = 2 - 1
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [5,1]
 => [1]
 => 1 = 2 - 1
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => 1 = 2 - 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => [3,3]
 => [3]
 => 1 = 2 - 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => 1 = 2 - 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => [3,1,1]
 => [1,1]
 => 2 = 3 - 1
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => [5,1]
 => [1]
 => 1 = 2 - 1
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => [3,3]
 => [3]
 => 1 = 2 - 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
 => [3,3]
 => [3]
 => 1 = 2 - 1
Description
The largest multiplicity of a part in an integer partition.
Matching statistic: St000506
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000506: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 50%
Values
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? = 0 - 2
([(0,3),(1,3),(2,3)],4)
 => [3]
 => []
 => ? = 0 - 2
([(0,3),(1,2),(2,3)],4)
 => [3]
 => []
 => ? = 1 - 2
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => ? = 0 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 0 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ? = 0 - 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => ? = 0 - 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => ? = 0 - 2
([(1,4),(2,4),(3,4)],5)
 => [3]
 => []
 => ? = 0 - 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => []
 => ? = 0 - 2
([(1,4),(2,3),(3,4)],5)
 => [3]
 => []
 => ? = 1 - 2
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [1]
 => 0 = 2 - 2
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => ? = 0 - 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => []
 => ? = 1 - 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => []
 => ? = 0 - 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 - 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => ? = 0 - 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 - 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 - 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 - 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => []
 => ? = 1 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 1 - 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ? = 0 - 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(2,5),(3,5),(4,5)],6)
 => [3]
 => []
 => ? = 0 - 2
([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 2
([(2,5),(3,4),(4,5)],6)
 => [3]
 => []
 => ? = 1 - 2
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [1]
 => 0 = 2 - 2
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => ? = 0 - 2
([(1,5),(2,5),(3,4),(4,5)],6)
 => [4]
 => []
 => ? = 1 - 2
([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [5]
 => []
 => ? = 1 - 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,4),(3,4)],6)
 => [2,2]
 => [2]
 => 0 = 2 - 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [5]
 => []
 => ? = 1 - 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [5]
 => []
 => ? = 1 - 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => [1,1]
 => 1 = 3 - 2
([(1,5),(2,4),(3,4),(3,5)],6)
 => [4]
 => []
 => ? = 1 - 2
([(0,1),(2,5),(3,4),(4,5)],6)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,2]
 => [2]
 => 0 = 2 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [3]
 => 0 = 2 - 2
([(2,3),(4,6),(5,6)],7)
 => [2,1]
 => [1]
 => 0 = 2 - 2
([(1,2),(3,6),(4,6),(5,6)],7)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(1,6),(2,6),(3,5),(4,5)],7)
 => [2,2]
 => [2]
 => 0 = 2 - 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => [3,2]
 => [2]
 => 0 = 2 - 2
([(1,6),(2,5),(3,4)],7)
 => [1,1,1]
 => [1,1]
 => 1 = 3 - 2
([(1,2),(3,6),(4,5),(5,6)],7)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(0,3),(1,2),(4,6),(5,6)],7)
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
([(2,3),(4,5),(4,6),(5,6)],7)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => [3,2]
 => [2]
 => 0 = 2 - 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => [3,2]
 => [2]
 => 0 = 2 - 2
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => [4,2]
 => [2]
 => 0 = 2 - 2
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => [3,3]
 => [3]
 => 0 = 2 - 2
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
 => [4,2]
 => [2]
 => 0 = 2 - 2
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => [3,1,1]
 => [1,1]
 => 1 = 3 - 2
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => [3,3]
 => [3]
 => 0 = 2 - 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
 => [3,3]
 => [3]
 => 0 = 2 - 2
Description
The number of standard desarrangement tableaux of shape equal to the given partition. A '''standard desarrangement tableau''' is a standard tableau whose first ascent is even. Here, an ascent of a standard tableau is an entry $i$ such that $i+1$ appears to the right or above $i$ in the tableau (with respect to English tableau notation). This is also the nullity of the random-to-random operator (and the random-to-top) operator acting on the simple module of the symmetric group indexed by the given partition. See also: * [[St000046]]: The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition * [[St000500]]: Eigenvalues of the random-to-random operator acting on the regular representation.
Matching statistic: St001176
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 50%
Values
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? = 0 - 2
([(0,3),(1,3),(2,3)],4)
 => [3]
 => []
 => ? = 0 - 2
([(0,3),(1,2),(2,3)],4)
 => [3]
 => []
 => ? = 1 - 2
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => ? = 0 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 0 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ? = 0 - 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => ? = 0 - 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => ? = 0 - 2
([(1,4),(2,4),(3,4)],5)
 => [3]
 => []
 => ? = 0 - 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => []
 => ? = 0 - 2
([(1,4),(2,3),(3,4)],5)
 => [3]
 => []
 => ? = 1 - 2
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [1]
 => 0 = 2 - 2
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => ? = 0 - 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => []
 => ? = 1 - 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => []
 => ? = 0 - 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 - 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => ? = 0 - 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 - 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 - 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 - 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => []
 => ? = 1 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 1 - 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ? = 0 - 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(2,5),(3,5),(4,5)],6)
 => [3]
 => []
 => ? = 0 - 2
([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 2
([(2,5),(3,4),(4,5)],6)
 => [3]
 => []
 => ? = 1 - 2
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [1]
 => 0 = 2 - 2
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => ? = 0 - 2
([(1,5),(2,5),(3,4),(4,5)],6)
 => [4]
 => []
 => ? = 1 - 2
([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [5]
 => []
 => ? = 1 - 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,4),(3,4)],6)
 => [2,2]
 => [2]
 => 0 = 2 - 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [5]
 => []
 => ? = 1 - 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [5]
 => []
 => ? = 1 - 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => [1,1]
 => 1 = 3 - 2
([(1,5),(2,4),(3,4),(3,5)],6)
 => [4]
 => []
 => ? = 1 - 2
([(0,1),(2,5),(3,4),(4,5)],6)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,2]
 => [2]
 => 0 = 2 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [3]
 => 0 = 2 - 2
([(2,3),(4,6),(5,6)],7)
 => [2,1]
 => [1]
 => 0 = 2 - 2
([(1,2),(3,6),(4,6),(5,6)],7)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(1,6),(2,6),(3,5),(4,5)],7)
 => [2,2]
 => [2]
 => 0 = 2 - 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => [3,2]
 => [2]
 => 0 = 2 - 2
([(1,6),(2,5),(3,4)],7)
 => [1,1,1]
 => [1,1]
 => 1 = 3 - 2
([(1,2),(3,6),(4,5),(5,6)],7)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(0,3),(1,2),(4,6),(5,6)],7)
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
([(2,3),(4,5),(4,6),(5,6)],7)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => [3,2]
 => [2]
 => 0 = 2 - 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => [3,2]
 => [2]
 => 0 = 2 - 2
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => [4,2]
 => [2]
 => 0 = 2 - 2
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => [3,3]
 => [3]
 => 0 = 2 - 2
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
 => [4,2]
 => [2]
 => 0 = 2 - 2
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => [3,1,1]
 => [1,1]
 => 1 = 3 - 2
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => [3,3]
 => [3]
 => 0 = 2 - 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
 => [3,3]
 => [3]
 => 0 = 2 - 2
Description
The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St001440
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001440: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 50%
Values
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? = 0 - 2
([(0,3),(1,3),(2,3)],4)
 => [3]
 => []
 => ? = 0 - 2
([(0,3),(1,2),(2,3)],4)
 => [3]
 => []
 => ? = 1 - 2
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => ? = 0 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 0 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ? = 0 - 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => ? = 0 - 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => ? = 0 - 2
([(1,4),(2,4),(3,4)],5)
 => [3]
 => []
 => ? = 0 - 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => []
 => ? = 0 - 2
([(1,4),(2,3),(3,4)],5)
 => [3]
 => []
 => ? = 1 - 2
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [1]
 => 0 = 2 - 2
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => ? = 0 - 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => []
 => ? = 1 - 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => []
 => ? = 0 - 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 - 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => ? = 0 - 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 - 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 - 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 - 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => []
 => ? = 1 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 1 - 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ? = 0 - 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(2,5),(3,5),(4,5)],6)
 => [3]
 => []
 => ? = 0 - 2
([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 2
([(2,5),(3,4),(4,5)],6)
 => [3]
 => []
 => ? = 1 - 2
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [1]
 => 0 = 2 - 2
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => ? = 0 - 2
([(1,5),(2,5),(3,4),(4,5)],6)
 => [4]
 => []
 => ? = 1 - 2
([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [5]
 => []
 => ? = 1 - 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,4),(3,4)],6)
 => [2,2]
 => [2]
 => 0 = 2 - 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [5]
 => []
 => ? = 1 - 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [5]
 => []
 => ? = 1 - 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => [1,1]
 => 1 = 3 - 2
([(1,5),(2,4),(3,4),(3,5)],6)
 => [4]
 => []
 => ? = 1 - 2
([(0,1),(2,5),(3,4),(4,5)],6)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,2]
 => [2]
 => 0 = 2 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [3]
 => 0 = 2 - 2
([(2,3),(4,6),(5,6)],7)
 => [2,1]
 => [1]
 => 0 = 2 - 2
([(1,2),(3,6),(4,6),(5,6)],7)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(1,6),(2,6),(3,5),(4,5)],7)
 => [2,2]
 => [2]
 => 0 = 2 - 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => [3,2]
 => [2]
 => 0 = 2 - 2
([(1,6),(2,5),(3,4)],7)
 => [1,1,1]
 => [1,1]
 => 1 = 3 - 2
([(1,2),(3,6),(4,5),(5,6)],7)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(0,3),(1,2),(4,6),(5,6)],7)
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
([(2,3),(4,5),(4,6),(5,6)],7)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => [3,2]
 => [2]
 => 0 = 2 - 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => [3,2]
 => [2]
 => 0 = 2 - 2
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => [4,2]
 => [2]
 => 0 = 2 - 2
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => [3,3]
 => [3]
 => 0 = 2 - 2
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
 => [4,2]
 => [2]
 => 0 = 2 - 2
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => [3,1,1]
 => [1,1]
 => 1 = 3 - 2
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => [3,3]
 => [3]
 => 0 = 2 - 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
 => [3,3]
 => [3]
 => 0 = 2 - 2
Description
The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition.
Matching statistic: St001714
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 50%
Values
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? = 0 - 2
([(0,3),(1,3),(2,3)],4)
 => [3]
 => []
 => ? = 0 - 2
([(0,3),(1,2),(2,3)],4)
 => [3]
 => []
 => ? = 1 - 2
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => ? = 0 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 0 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ? = 0 - 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => ? = 0 - 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => ? = 0 - 2
([(1,4),(2,4),(3,4)],5)
 => [3]
 => []
 => ? = 0 - 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => []
 => ? = 0 - 2
([(1,4),(2,3),(3,4)],5)
 => [3]
 => []
 => ? = 1 - 2
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [1]
 => 0 = 2 - 2
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => ? = 0 - 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => []
 => ? = 1 - 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => []
 => ? = 0 - 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 - 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => ? = 0 - 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 - 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 - 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 - 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => []
 => ? = 1 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 1 - 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ? = 0 - 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(2,5),(3,5),(4,5)],6)
 => [3]
 => []
 => ? = 0 - 2
([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 2
([(2,5),(3,4),(4,5)],6)
 => [3]
 => []
 => ? = 1 - 2
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [1]
 => 0 = 2 - 2
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => ? = 0 - 2
([(1,5),(2,5),(3,4),(4,5)],6)
 => [4]
 => []
 => ? = 1 - 2
([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [5]
 => []
 => ? = 1 - 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,4),(3,4)],6)
 => [2,2]
 => [2]
 => 0 = 2 - 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [5]
 => []
 => ? = 1 - 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [5]
 => []
 => ? = 1 - 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => [1,1]
 => 1 = 3 - 2
([(1,5),(2,4),(3,4),(3,5)],6)
 => [4]
 => []
 => ? = 1 - 2
([(0,1),(2,5),(3,4),(4,5)],6)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,2]
 => [2]
 => 0 = 2 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [3]
 => 0 = 2 - 2
([(2,3),(4,6),(5,6)],7)
 => [2,1]
 => [1]
 => 0 = 2 - 2
([(1,2),(3,6),(4,6),(5,6)],7)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(1,6),(2,6),(3,5),(4,5)],7)
 => [2,2]
 => [2]
 => 0 = 2 - 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => [3,2]
 => [2]
 => 0 = 2 - 2
([(1,6),(2,5),(3,4)],7)
 => [1,1,1]
 => [1,1]
 => 1 = 3 - 2
([(1,2),(3,6),(4,5),(5,6)],7)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(0,3),(1,2),(4,6),(5,6)],7)
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
([(2,3),(4,5),(4,6),(5,6)],7)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => [3,2]
 => [2]
 => 0 = 2 - 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => [3,2]
 => [2]
 => 0 = 2 - 2
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => [4,2]
 => [2]
 => 0 = 2 - 2
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => [3,3]
 => [3]
 => 0 = 2 - 2
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
 => [4,2]
 => [2]
 => 0 = 2 - 2
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => [3,1,1]
 => [1,1]
 => 1 = 3 - 2
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => [3,3]
 => [3]
 => 0 = 2 - 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
 => [3,3]
 => [3]
 => 0 = 2 - 2
Description
The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. In particular, partitions with statistic $0$ are wide partitions.
Matching statistic: St001961
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001961: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 50%
Values
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => ? = 0 - 2
([(0,3),(1,3),(2,3)],4)
 => [3]
 => []
 => ? = 0 - 2
([(0,3),(1,2),(2,3)],4)
 => [3]
 => []
 => ? = 1 - 2
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => ? = 0 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => ? = 0 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => ? = 0 - 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => ? = 0 - 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => ? = 0 - 2
([(1,4),(2,4),(3,4)],5)
 => [3]
 => []
 => ? = 0 - 2
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => []
 => ? = 0 - 2
([(1,4),(2,3),(3,4)],5)
 => [3]
 => []
 => ? = 1 - 2
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [1]
 => 0 = 2 - 2
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => ? = 0 - 2
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => []
 => ? = 1 - 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => []
 => ? = 0 - 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 - 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => ? = 0 - 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 - 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 - 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 0 - 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => []
 => ? = 1 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => ? = 1 - 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => ? = 0 - 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => ? = 0 - 2
([(2,5),(3,5),(4,5)],6)
 => [3]
 => []
 => ? = 0 - 2
([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 2
([(2,5),(3,4),(4,5)],6)
 => [3]
 => []
 => ? = 1 - 2
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [1]
 => 0 = 2 - 2
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => ? = 0 - 2
([(1,5),(2,5),(3,4),(4,5)],6)
 => [4]
 => []
 => ? = 1 - 2
([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [5]
 => []
 => ? = 1 - 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,4),(3,4)],6)
 => [2,2]
 => [2]
 => 0 = 2 - 2
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [5]
 => []
 => ? = 1 - 2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [5]
 => []
 => ? = 1 - 2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => ? = 0 - 2
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => [1,1]
 => 1 = 3 - 2
([(1,5),(2,4),(3,4),(3,5)],6)
 => [4]
 => []
 => ? = 1 - 2
([(0,1),(2,5),(3,4),(4,5)],6)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,2]
 => [2]
 => 0 = 2 - 2
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [3]
 => 0 = 2 - 2
([(2,3),(4,6),(5,6)],7)
 => [2,1]
 => [1]
 => 0 = 2 - 2
([(1,2),(3,6),(4,6),(5,6)],7)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(1,6),(2,6),(3,5),(4,5)],7)
 => [2,2]
 => [2]
 => 0 = 2 - 2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => [3,2]
 => [2]
 => 0 = 2 - 2
([(1,6),(2,5),(3,4)],7)
 => [1,1,1]
 => [1,1]
 => 1 = 3 - 2
([(1,2),(3,6),(4,5),(5,6)],7)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(0,3),(1,2),(4,6),(5,6)],7)
 => [2,1,1]
 => [1,1]
 => 1 = 3 - 2
([(2,3),(4,5),(4,6),(5,6)],7)
 => [3,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => [3,2]
 => [2]
 => 0 = 2 - 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => [3,2]
 => [2]
 => 0 = 2 - 2
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => [4,2]
 => [2]
 => 0 = 2 - 2
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => [3,3]
 => [3]
 => 0 = 2 - 2
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
 => [4,2]
 => [2]
 => 0 = 2 - 2
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => 0 = 2 - 2
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => [3,1,1]
 => [1,1]
 => 1 = 3 - 2
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => [5,1]
 => [1]
 => 0 = 2 - 2
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => [3,3]
 => [3]
 => 0 = 2 - 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
 => [3,3]
 => [3]
 => 0 = 2 - 2
Description
The sum of the greatest common divisors of all pairs of parts.
Matching statistic: St000207
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 50%
Values
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => []
 => ? = 0 - 1
([(0,3),(1,3),(2,3)],4)
 => [3]
 => []
 => []
 => ? = 0 - 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => []
 => []
 => ? = 1 - 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => []
 => ? = 0 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => []
 => ? = 0 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => []
 => ? = 0 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(1,4),(2,4),(3,4)],5)
 => [3]
 => []
 => []
 => ? = 0 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => []
 => []
 => ? = 0 - 1
([(1,4),(2,3),(3,4)],5)
 => [3]
 => []
 => []
 => ? = 1 - 1
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => []
 => ? = 0 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => []
 => []
 => ? = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => []
 => []
 => ? = 0 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => []
 => ? = 0 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => []
 => []
 => ? = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(2,5),(3,5),(4,5)],6)
 => [3]
 => []
 => []
 => ? = 0 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(2,5),(3,4),(4,5)],6)
 => [3]
 => []
 => []
 => ? = 1 - 1
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => []
 => ? = 0 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [4]
 => []
 => []
 => ? = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [5]
 => []
 => []
 => ? = 1 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [2,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [5]
 => []
 => []
 => ? = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [5]
 => []
 => []
 => ? = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => [1,1]
 => [2]
 => 2 = 3 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
 => [4]
 => []
 => []
 => ? = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => [3,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [3]
 => [1,1,1]
 => 1 = 2 - 1
([(2,3),(4,6),(5,6)],7)
 => [2,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [3,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => [2,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => [3,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(1,6),(2,5),(3,4)],7)
 => [1,1,1]
 => [1,1]
 => [2]
 => 2 = 3 - 1
([(1,2),(3,6),(4,5),(5,6)],7)
 => [3,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,3),(1,2),(4,6),(5,6)],7)
 => [2,1,1]
 => [1,1]
 => [2]
 => 2 = 3 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => [3,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
 => [4,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => [3,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => [3,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [5,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => [4,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => [3,3]
 => [3]
 => [1,1,1]
 => 1 = 2 - 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
 => [4,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => [3,1,1]
 => [1,1]
 => [2]
 => 2 = 3 - 1
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => [5,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => [3,3]
 => [3]
 => [1,1,1]
 => 1 = 2 - 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
 => [3,3]
 => [3]
 => [1,1,1]
 => 1 = 2 - 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has all vertices in integer lattice points.
Matching statistic: St000208
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 50%
Values
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => []
 => ? = 0 - 1
([(0,3),(1,3),(2,3)],4)
 => [3]
 => []
 => []
 => ? = 0 - 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => []
 => []
 => ? = 1 - 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => []
 => ? = 0 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => []
 => ? = 0 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => []
 => ? = 0 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(1,4),(2,4),(3,4)],5)
 => [3]
 => []
 => []
 => ? = 0 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => []
 => []
 => ? = 0 - 1
([(1,4),(2,3),(3,4)],5)
 => [3]
 => []
 => []
 => ? = 1 - 1
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => []
 => ? = 0 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => []
 => []
 => ? = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => []
 => []
 => ? = 0 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => []
 => ? = 0 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => []
 => []
 => ? = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(2,5),(3,5),(4,5)],6)
 => [3]
 => []
 => []
 => ? = 0 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(2,5),(3,4),(4,5)],6)
 => [3]
 => []
 => []
 => ? = 1 - 1
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => []
 => ? = 0 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [4]
 => []
 => []
 => ? = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [5]
 => []
 => []
 => ? = 1 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [2,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [5]
 => []
 => []
 => ? = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [5]
 => []
 => []
 => ? = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => [1,1]
 => [2]
 => 2 = 3 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
 => [4]
 => []
 => []
 => ? = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => [3,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [3]
 => [1,1,1]
 => 1 = 2 - 1
([(2,3),(4,6),(5,6)],7)
 => [2,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [3,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => [2,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => [3,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(1,6),(2,5),(3,4)],7)
 => [1,1,1]
 => [1,1]
 => [2]
 => 2 = 3 - 1
([(1,2),(3,6),(4,5),(5,6)],7)
 => [3,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,3),(1,2),(4,6),(5,6)],7)
 => [2,1,1]
 => [1,1]
 => [2]
 => 2 = 3 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => [3,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
 => [4,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => [3,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => [3,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [5,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => [4,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => [3,3]
 => [3]
 => [1,1,1]
 => 1 = 2 - 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
 => [4,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => [3,1,1]
 => [1,1]
 => [2]
 => 2 = 3 - 1
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => [5,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => [3,3]
 => [3]
 => [1,1,1]
 => 1 = 2 - 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
 => [3,3]
 => [3]
 => [1,1,1]
 => 1 = 2 - 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that $P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has only integer lattice points as vertices. See also [[St000205]], [[St000206]] and [[St000207]].
Matching statistic: St000667
Mapping
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 50%
Values
([(0,1),(0,2),(1,2)],3)
 => [3]
 => []
 => []
 => ? = 0 - 1
([(0,3),(1,3),(2,3)],4)
 => [3]
 => []
 => []
 => ? = 0 - 1
([(0,3),(1,2),(2,3)],4)
 => [3]
 => []
 => []
 => ? = 1 - 1
([(1,2),(1,3),(2,3)],4)
 => [3]
 => []
 => []
 => ? = 0 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => []
 => []
 => ? = 0 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => []
 => []
 => ? = 0 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(1,4),(2,4),(3,4)],5)
 => [3]
 => []
 => []
 => ? = 0 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [4]
 => []
 => []
 => ? = 0 - 1
([(1,4),(2,3),(3,4)],5)
 => [3]
 => []
 => []
 => ? = 1 - 1
([(0,1),(2,4),(3,4)],5)
 => [2,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => [3]
 => []
 => []
 => ? = 0 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [4]
 => []
 => []
 => ? = 1 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4]
 => []
 => []
 => ? = 0 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4]
 => []
 => []
 => ? = 0 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [4]
 => []
 => []
 => ? = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => []
 => []
 => ? = 1 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(2,5),(3,5),(4,5)],6)
 => [3]
 => []
 => []
 => ? = 0 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [4]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(2,5),(3,4),(4,5)],6)
 => [3]
 => []
 => []
 => ? = 1 - 1
([(1,2),(3,5),(4,5)],6)
 => [2,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
 => [3]
 => []
 => []
 => ? = 0 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
 => [4]
 => []
 => []
 => ? = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => [5]
 => []
 => []
 => ? = 1 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [2,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => [5]
 => []
 => []
 => ? = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [5]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => [5]
 => []
 => []
 => ? = 1 - 1
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [6]
 => []
 => []
 => ? = 0 - 1
([(0,5),(1,4),(2,3)],6)
 => [1,1,1]
 => [1,1]
 => [2]
 => 2 = 3 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
 => [4]
 => []
 => []
 => ? = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => [3,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [3]
 => [1,1,1]
 => 1 = 2 - 1
([(2,3),(4,6),(5,6)],7)
 => [2,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [3,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => [2,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
 => [3,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(1,6),(2,5),(3,4)],7)
 => [1,1,1]
 => [1,1]
 => [2]
 => 2 = 3 - 1
([(1,2),(3,6),(4,5),(5,6)],7)
 => [3,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,3),(1,2),(4,6),(5,6)],7)
 => [2,1,1]
 => [1,1]
 => [2]
 => 2 = 3 - 1
([(2,3),(4,5),(4,6),(5,6)],7)
 => [3,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)
 => [4,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => [3,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => [3,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [5,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)
 => [4,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)
 => [3,3]
 => [3]
 => [1,1,1]
 => 1 = 2 - 1
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
 => [4,2]
 => [2]
 => [1,1]
 => 1 = 2 - 1
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
 => [4,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)
 => [3,1,1]
 => [1,1]
 => [2]
 => 2 = 3 - 1
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => [5,1]
 => [1]
 => [1]
 => 1 = 2 - 1
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)
 => [3,3]
 => [3]
 => [1,1,1]
 => 1 = 2 - 1
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)
 => [3,3]
 => [3]
 => [1,1,1]
 => 1 = 2 - 1
Description
The greatest common divisor of the parts of the partition.
The following 40 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000706The product of the factorials of the multiplicities of an integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000993The multiplicity of the largest part of an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001389The number of partitions of the same length below the given integer partition. St001571The Cartan determinant of the integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001754The number of tolerances of a finite lattice. St000455The second largest eigenvalue of a graph if it is integral. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000260The radius of a connected graph. St001877Number of indecomposable injective modules with projective dimension 2. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001568The smallest positive integer that does not appear twice in the partition. St000567The sum of the products of all pairs of parts. St000929The constant term of the character polynomial of an integer partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000668The least common multiple of the parts of the partition. 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. St000933The number of multipartitions of sizes given by an integer partition. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition.