Your data matches 2 different statistics following compositions of up to 3 maps.
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Matching statistic: St001283
Mapping
St001283: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1
[2]
 => 0
[1,1]
 => 1
[3]
 => 0
[2,1]
 => 0
[1,1,1]
 => 1
[4]
 => 0
[3,1]
 => 0
[2,2]
 => 0
[2,1,1]
 => 1
[1,1,1,1]
 => 2
[5]
 => 0
[4,1]
 => 0
[3,2]
 => 0
[3,1,1]
 => 0
[2,2,1]
 => 0
[2,1,1,1]
 => 0
[1,1,1,1,1]
 => 1
[6]
 => 0
[5,1]
 => 0
[4,2]
 => 0
[4,1,1]
 => 0
[3,3]
 => 0
[3,2,1]
 => 0
[3,1,1,1]
 => 1
[2,2,2]
 => 0
[2,2,1,1]
 => 1
[2,1,1,1,1]
 => 2
[1,1,1,1,1,1]
 => 1
[7]
 => 0
[6,1]
 => 0
[5,2]
 => 0
[5,1,1]
 => 0
[4,3]
 => 0
[4,2,1]
 => 0
[4,1,1,1]
 => 0
[3,3,1]
 => 0
[3,2,2]
 => 0
[3,2,1,1]
 => 0
[3,1,1,1,1]
 => 0
[2,2,2,1]
 => 0
[2,2,1,1,1]
 => 0
[2,1,1,1,1,1]
 => 0
[1,1,1,1,1,1,1]
 => 1
[8]
 => 0
[7,1]
 => 0
[6,2]
 => 0
[6,1,1]
 => 0
[5,3]
 => 0
[5,2,1]
 => 0
Description
The number of finite solvable groups that are realised by the given partition over the complex numbers. A finite group $G$ is ''realised'' by the partition $(a_1,\dots,a_m)$ if its group algebra over the complex numbers is isomorphic to the direct product of $a_i\times a_i$ matrix rings over the complex numbers. The smallest partition which does not realise a solvable group, but does realise a finite group, is $(5,4,3,3,1)$.
Matching statistic: St000264
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 15%●distinct values known / distinct values provided: 14%
Values
[1]
 => [1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => ? = 1 + 3
[2]
 => [1,1,0,0,1,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => ? = 0 + 3
[1,1]
 => [1,0,1,1,0,0]
 => [1,2] => ([(1,2)],3)
 => ? = 1 + 3
[3]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 0 + 3
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 0 + 3
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => ([(2,3)],4)
 => ? = 1 + 3
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 0 + 3
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? = 0 + 3
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => ([(1,3),(2,3)],4)
 => ? = 0 + 3
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3] => ([(2,3)],4)
 => ? = 1 + 3
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ? = 2 + 3
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 0 + 3
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 0 + 3
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ? = 0 + 3
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,5] => ([(4,5)],6)
 => ? = 1 + 3
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 0 + 3
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 0 + 3
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 0 + 3
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 0 + 3
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 3
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 0 + 3
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => ? = 1 + 3
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0 + 3
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ? = 1 + 3
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,5] => ([(4,5)],6)
 => ? = 2 + 3
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,6] => ([(5,6)],7)
 => ? = 1 + 3
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 3
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 0 + 3
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 0 + 3
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 0 + 3
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 0 + 3
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 0 + 3
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 0 + 3
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 0 + 3
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 0 + 3
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => ? = 0 + 3
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,5] => ([(4,5)],6)
 => ? = 0 + 3
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 0 + 3
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,5] => ([(4,5)],6)
 => ? = 0 + 3
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,6] => ([(5,6)],7)
 => ? = 0 + 3
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,7] => ([(6,7)],8)
 => ? = 1 + 3
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => ? = 0 + 3
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 0 + 3
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 0 + 3
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 0 + 3
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 0 + 3
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 0 + 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 0 + 3
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 0 + 3
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 0 + 3
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 0 + 3
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 0 + 3
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,5] => ([(4,5)],6)
 => ? = 1 + 3
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 0 + 3
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 0 + 3
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3 = 0 + 3
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,5] => ([(4,5)],6)
 => ? = 0 + 3
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,6] => ([(5,6)],7)
 => ? = 0 + 3
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 0 + 3
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,5] => ([(4,5)],6)
 => ? = 1 + 3
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,6] => ([(5,6)],7)
 => ? = 2 + 3
[5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[5,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 0 + 3
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 0 + 3
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 0 + 3
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 0 + 3
[2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[5,4,1]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[5,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [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)
 => 3 = 0 + 3
[3,2,2,2,1]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[6,5]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 0 + 3
[6,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 0 + 3
[5,4,2]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[5,4,1,1]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[5,3,3]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[5,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[5,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[4,4,3]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[4,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[4,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[3,3,3,2]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[3,3,3,1,1]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[3,3,2,2,1]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[2,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => 3 = 0 + 3
[6,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 0 + 3
[6,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 0 + 3
[5,4,3]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [3,1,1,1] => ([(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)
 => 3 = 0 + 3
[5,4,2,1]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[5,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[5,3,3,1]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[5,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[5,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
[4,4,3,1]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 0 + 3
Description
The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.