Processing math: 100%

Your data matches 10 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000513
Mapping
St000513: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0
[2]
 => 1
[1,1]
 => 1
[3]
 => 0
[2,1]
 => 1
[1,1,1]
 => 3
[4]
 => 0
[3,1]
 => 0
[2,2]
 => 2
[2,1,1]
 => 2
[1,1,1,1]
 => 6
[5]
 => 0
[4,1]
 => 0
[3,2]
 => 1
[3,1,1]
 => 1
[2,2,1]
 => 2
[2,1,1,1]
 => 4
[1,1,1,1,1]
 => 10
[6]
 => 0
[5,1]
 => 0
[4,2]
 => 1
[4,1,1]
 => 1
[3,3]
 => 0
[3,2,1]
 => 1
[3,1,1,1]
 => 3
[2,2,2]
 => 3
[2,2,1,1]
 => 3
[2,1,1,1,1]
 => 7
[1,1,1,1,1,1]
 => 15
[7]
 => 0
[6,1]
 => 0
[5,2]
 => 1
[5,1,1]
 => 1
[4,3]
 => 0
[4,2,1]
 => 1
[4,1,1,1]
 => 3
[3,3,1]
 => 0
[3,2,2]
 => 2
[3,2,1,1]
 => 2
[3,1,1,1,1]
 => 6
[2,2,2,1]
 => 3
[2,2,1,1,1]
 => 5
[2,1,1,1,1,1]
 => 11
[1,1,1,1,1,1,1]
 => 21
[8]
 => 0
[7,1]
 => 0
[6,2]
 => 1
[6,1,1]
 => 1
[5,3]
 => 0
[5,2,1]
 => 1
Description
The number of invariant subsets of size 2 when acting with a permutation of given cycle type.
Matching statistic: St001629
(load all 3 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 7% ●values known / values provided: 21%●distinct values known / distinct values provided: 7%
Values
[1]
 => [1,0,1,0]
 => [1,1] => [2] => ? = 0
[2]
 => [1,1,0,0,1,0]
 => [2,1] => [1,1] => ? ∊ {1,1}
[1,1]
 => [1,0,1,1,0,0]
 => [1,2] => [1,1] => ? ∊ {1,1}
[3]
 => [1,1,1,0,0,0,1,0]
 => [3,1] => [1,1] => ? ∊ {0,3}
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1] => [3] => 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,3] => [1,1] => ? ∊ {0,3}
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,1] => [1,1] => ? ∊ {0,0,2,2,6}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1] => [1,1] => ? ∊ {0,0,2,2,6}
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,2] => [2] => ? ∊ {0,0,2,2,6}
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3] => [1,1] => ? ∊ {0,0,2,2,6}
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,1] => ? ∊ {0,0,2,2,6}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,1] => [1,1] => ? ∊ {1,2,4,10}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1] => [1,1] => ? ∊ {1,2,4,10}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,2] => 0
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,2,1] => [1,1,1] => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,2] => [2,1] => 0
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,4] => [1,1] => ? ∊ {1,2,4,10}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,5] => [1,1] => ? ∊ {1,2,4,10}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [6,1] => [1,1] => ? ∊ {0,0,0,1,1,3,3,3,7,15}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [5,1] => [1,1] => ? ∊ {0,0,0,1,1,3,3,3,7,15}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1] => [1,1] => ? ∊ {0,0,0,1,1,3,3,3,7,15}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,1] => [1,1] => ? ∊ {0,0,0,1,1,3,3,3,7,15}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2] => [1,1] => ? ∊ {0,0,0,1,1,3,3,3,7,15}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [4] => 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4] => [1,1] => ? ∊ {0,0,0,1,1,3,3,3,7,15}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1] => ? ∊ {0,0,0,1,1,3,3,3,7,15}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,4] => [1,1] => ? ∊ {0,0,0,1,1,3,3,3,7,15}
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,5] => [1,1] => ? ∊ {0,0,0,1,1,3,3,3,7,15}
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,6] => [1,1] => ? ∊ {0,0,0,1,1,3,3,3,7,15}
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [7,1] => [1,1] => ? ∊ {0,0,1,1,2,2,3,3,5,6,11,21}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [6,1] => [1,1] => ? ∊ {0,0,1,1,2,2,3,3,5,6,11,21}
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,1] => [1,1] => ? ∊ {0,0,1,1,2,2,3,3,5,6,11,21}
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [5,1] => [1,1] => ? ∊ {0,0,1,1,2,2,3,3,5,6,11,21}
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => [1,2] => 0
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [4,1] => [1,1] => ? ∊ {0,0,1,1,2,2,3,3,5,6,11,21}
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [1,1,1] => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,2] => [1,1] => ? ∊ {0,0,1,1,2,2,3,3,5,6,11,21}
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,3] => [1,1] => ? ∊ {0,0,1,1,2,2,3,3,5,6,11,21}
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,4] => [1,1] => ? ∊ {0,0,1,1,2,2,3,3,5,6,11,21}
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,5] => [1,1] => ? ∊ {0,0,1,1,2,2,3,3,5,6,11,21}
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [2,1] => 0
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,5] => [1,1] => ? ∊ {0,0,1,1,2,2,3,3,5,6,11,21}
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,6] => [1,1] => ? ∊ {0,0,1,1,2,2,3,3,5,6,11,21}
[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] => [1,1] => ? ∊ {0,0,1,1,2,2,3,3,5,6,11,21}
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [8,1] => [1,1] => ? ∊ {0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [7,1] => [1,1] => ? ∊ {0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [6,1] => [1,1] => ? ∊ {0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [6,1] => [1,1] => ? ∊ {0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [5,1] => [1,1] => ? ∊ {0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [5,1] => [1,1] => ? ∊ {0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [5,1] => [1,1] => ? ∊ {0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,2] => [1,1] => ? ∊ {0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => [1,2] => 0
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,1] => 0
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => [1,1,1] => 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,5] => [1,1] => ? ∊ {0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,1,1] => 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,2] => 0
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [2,1] => 0
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,5] => [1,1] => ? ∊ {0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,6] => [1,1] => ? ∊ {0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4] => [1,1] => ? ∊ {0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,5] => [1,1] => ? ∊ {0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,6] => [1,1] => ? ∊ {0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [4,1,1] => [1,2] => 0
[5,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,4,1] => [1,1,1] => 1
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [1,3] => 0
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [1,1,2] => 1
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,1,1] => 1
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [3,1] => 0
[2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,4] => [2,1] => 0
[5,4,1]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [4,1,1] => [1,2] => 0
[5,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,4,1] => [1,1,1] => 1
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [5] => 1
[3,2,2,2,1]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,4] => [2,1] => 0
[6,5]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [5,1,1] => [1,2] => 0
[6,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,5,1] => [1,1,1] => 1
[5,4,2]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [4,1,1] => [1,2] => 0
[5,4,1,1]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [4,1,1] => [1,2] => 0
[5,3,3]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [3,2,1] => [1,1,1] => 1
[5,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,4,1] => [1,1,1] => 1
[5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,3,1] => [1,1,1] => 1
[5,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,4,1] => [1,1,1] => 1
[4,4,3]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [3,1,2] => [1,1,1] => 1
[4,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,3,2] => [1,1,1] => 1
[4,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,4] => [2,1] => 0
[3,3,3,2]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,1,3] => [1,1,1] => 1
[3,3,3,1,1]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,2,3] => [1,1,1] => 1
[3,3,2,2,1]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,4] => [2,1] => 0
[2,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,5] => [2,1] => 0
[6,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [5,1,1] => [1,2] => 0
[6,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,5,1] => [1,1,1] => 1
[5,4,3]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [3,1,1,1] => [1,3] => 0
[5,4,2,1]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [4,1,1] => [1,2] => 0
[5,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,3,1,1] => [1,1,2] => 1
[5,3,3,1]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [3,2,1] => [1,1,1] => 1
[5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,3,1] => [1,1,1] => 1
[5,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,4,1] => [1,1,1] => 1
[5,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,3,1] => [2,1,1] => 1
[4,4,3,1]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [3,1,2] => [1,1,1] => 1
Description
The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
Matching statistic: St000456
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 17% ●values known / values provided: 20%●distinct values known / distinct values provided: 17%
Values
[1]
 => [1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[2]
 => [1,1,0,0,1,0]
 => [1,3,2] => ([(1,2)],3)
 => ? ∊ {1,1} + 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,1,3] => ([(1,2)],3)
 => ? ∊ {1,1} + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {1,3} + 1
[2,1]
 => [1,0,1,0,1,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {1,3} + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => ? ∊ {2,2,6} + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {2,2,6} + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ? ∊ {2,2,6} + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,2,3,4,6,5] => ([(4,5)],6)
 => ? ∊ {1,1,2,4,10} + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,2,4,10} + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {1,1,2,4,10} + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,2,4,10} + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => ([(4,5)],6)
 => ? ∊ {1,1,2,4,10} + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,7,6] => ([(5,6)],7)
 => ? ∊ {1,1,1,3,3,3,7,15} + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1 = 0 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,7,15} + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,7,15} + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,7,15} + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 0 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,7,15} + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,7,15} + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,3,3,3,7,15} + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1 = 0 + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => ([(5,6)],7)
 => ? ∊ {1,1,1,3,3,3,7,15} + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,6,8,7] => ([(6,7)],8)
 => ? ∊ {0,0,1,2,2,3,3,5,6,11,21} + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,7,5] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 1 = 0 + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,1,2,2,3,3,5,6,11,21} + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [4,1,2,3,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,2,2,3,3,5,6,11,21} + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,1,2,2,3,3,5,6,11,21} + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {0,0,1,2,2,3,3,5,6,11,21} + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,1,2,2,3,3,5,6,11,21} + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,1,2,2,3,3,5,6,11,21} + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 2 = 1 + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [2,1,6,3,4,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,2,2,3,3,5,6,11,21} + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,1,2,2,3,3,5,6,11,21} + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,1,2,2,3,3,5,6,11,21} + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [2,7,1,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 1 = 0 + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7,8] => ([(6,7)],8)
 => ? ∊ {0,0,1,2,2,3,3,5,6,11,21} + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,6,7,9,8] => ([(7,8)],9)
 => ? ∊ {0,0,1,1,1,1,1,2,3,4,4,4,6,8,10,16,28} + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,8,6] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? ∊ {0,0,1,1,1,1,1,2,3,4,4,4,6,8,10,16,28} + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,6,2,3,4,7,5] => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? ∊ {0,0,1,1,1,1,1,2,3,4,4,4,6,8,10,16,28} + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [5,1,2,3,4,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => ? ∊ {0,0,1,1,1,1,1,2,3,4,4,4,6,8,10,16,28} + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,2,3,4,4,4,6,8,10,16,28} + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [4,5,1,2,6,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [3,1,2,4,6,5] => ([(1,2),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,2,3,4,4,4,6,8,10,16,28} + 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,2,3,5,4,6] => ([(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,2,3,4,4,4,6,8,10,16,28} + 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {0,0,1,1,1,1,1,2,3,4,4,4,6,8,10,16,28} + 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,2,3,4,4,4,6,8,10,16,28} + 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,1,1,1,1,1,2,3,4,4,4,6,8,10,16,28} + 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ? ∊ {0,0,1,1,1,1,1,2,3,4,4,4,6,8,10,16,28} + 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [2,1,7,3,4,5,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => ? ∊ {0,0,1,1,1,1,1,2,3,4,4,4,6,8,10,16,28} + 1
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,3,2,4,5,6] => ([(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,2,3,4,4,4,6,8,10,16,28} + 1
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,1,1,1,1,1,2,3,4,4,4,6,8,10,16,28} + 1
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [2,6,1,3,4,5,7] => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? ∊ {0,0,1,1,1,1,1,2,3,4,4,4,6,8,10,16,28} + 1
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,8,1,3,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? ∊ {0,0,1,1,1,1,1,2,3,4,4,4,6,8,10,16,28} + 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7,8,9] => ([(7,8)],9)
 => ? ∊ {0,0,1,1,1,1,1,2,3,4,4,4,6,8,10,16,28} + 1
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,6,7,8,10,9] => ([(8,9)],10)
 => ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,3,3,4,4,6,6,7,10,12,15,22,36} + 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,9,7] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,8)],9)
 => ? ∊ {0,0,0,0,0,0,0,1,1,2,2,2,3,3,4,4,6,6,7,10,12,15,22,36} + 1
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [5,6,1,2,3,7,4] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 4 = 3 + 1
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,6,7,1,3,4,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 4 = 3 + 1
[6,3,1]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [5,1,6,2,3,7,4] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 3 = 2 + 1
[6,2,1,1]
 => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [4,6,1,2,3,7,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
 => 3 = 2 + 1
[5,4,1]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1 = 0 + 1
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 1 = 0 + 1
[5,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 0 + 1
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 1 = 0 + 1
[4,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [2,6,1,7,3,4,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
 => 3 = 2 + 1
[3,2,2,2,1]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1 = 0 + 1
[3,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,5,7,1,3,4,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 3 = 2 + 1
[6,4,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => [5,1,2,6,3,7,4] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => 2 = 1 + 1
[6,3,1,1]
 => [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
 => [4,1,6,2,3,7,5] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => 2 = 1 + 1
[6,2,1,1,1]
 => [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [3,6,1,2,4,7,5] => ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 3 = 2 + 1
[5,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [2,6,1,3,7,4,5] => ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 0 + 1
[4,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 3 = 2 + 1
[4,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [2,5,1,7,3,4,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => 2 = 1 + 1
[3,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,4,7,1,3,5,6] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => 2 = 1 + 1
[6,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [5,1,2,3,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 1 = 0 + 1
[6,4,1,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
 => [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => 1 = 0 + 1
[6,3,2,1]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => [4,5,6,1,2,7,3] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
 => 5 = 4 + 1
[6,3,1,1,1]
 => [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
 => [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => 1 = 0 + 1
[6,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [2,6,1,3,4,7,5] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 1 = 0 + 1
[5,4,2,1]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[5,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => 2 = 1 + 1
Description
The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St000771
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 20%●distinct values known / distinct values provided: 10%
Values
[1]
 => [1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[2]
 => [1,1,0,0,1,0]
 => [1,3,2] => ([(1,2)],3)
 => ? ∊ {1,1} + 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,1,3] => ([(1,2)],3)
 => ? ∊ {1,1} + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {1,3} + 1
[2,1]
 => [1,0,1,0,1,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {1,3} + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => ? ∊ {2,2,6} + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {2,2,6} + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ? ∊ {2,2,6} + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,2,3,4,6,5] => ([(4,5)],6)
 => ? ∊ {1,1,2,4,10} + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,2,4,10} + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {1,1,2,4,10} + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,2,4,10} + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => ([(4,5)],6)
 => ? ∊ {1,1,2,4,10} + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,7,6] => ([(5,6)],7)
 => ? ∊ {0,0,0,3,3,3,7,15} + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 1 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,3,3,3,7,15} + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,3,3,3,7,15} + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => ? ∊ {0,0,0,3,3,3,7,15} + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,3,3,3,7,15} + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => ? ∊ {0,0,0,3,3,3,7,15} + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,3,3,3,7,15} + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => ([(5,6)],7)
 => ? ∊ {0,0,0,3,3,3,7,15} + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,6,8,7] => ([(6,7)],8)
 => ? ∊ {0,0,1,1,1,3,3,5,6,11,21} + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,7,5] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 3 = 2 + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,1,1,1,3,3,5,6,11,21} + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [4,1,2,3,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,3,3,5,6,11,21} + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,1,1,1,3,3,5,6,11,21} + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {0,0,1,1,1,3,3,5,6,11,21} + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,1,1,1,3,3,5,6,11,21} + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,1,1,1,3,3,5,6,11,21} + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [2,1,6,3,4,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,3,3,5,6,11,21} + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,1,1,1,3,3,5,6,11,21} + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,1,1,1,3,3,5,6,11,21} + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [2,7,1,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 3 = 2 + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7,8] => ([(6,7)],8)
 => ? ∊ {0,0,1,1,1,3,3,5,6,11,21} + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,6,7,9,8] => ([(7,8)],9)
 => ? ∊ {0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,8,6] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? ∊ {0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,6,2,3,4,7,5] => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? ∊ {0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [5,1,2,3,4,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => ? ∊ {0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [4,5,1,2,6,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [3,1,2,4,6,5] => ([(1,2),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,2,3,5,4,6] => ([(4,5)],6)
 => ? ∊ {0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6)
 => ? ∊ {0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ? ∊ {0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [2,1,7,3,4,5,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => ? ∊ {0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,3,2,4,5,6] => ([(4,5)],6)
 => ? ∊ {0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [2,6,1,3,4,5,7] => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? ∊ {0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,8,1,3,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? ∊ {0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7,8,9] => ([(7,8)],9)
 => ? ∊ {0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,6,7,8,10,9] => ([(8,9)],10)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,4,4,6,6,7,10,12,15,22,36} + 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,9,7] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,8)],9)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,3,3,3,3,4,4,6,6,7,10,12,15,22,36} + 1
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [5,6,1,2,3,7,4] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 2 = 1 + 1
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
[4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
[3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,6,7,1,3,4,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 2 = 1 + 1
[6,3,1]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [5,1,6,2,3,7,4] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 1 = 0 + 1
[6,2,1,1]
 => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [4,6,1,2,3,7,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
 => 2 = 1 + 1
[5,4,1]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 1 = 0 + 1
[5,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 0 + 1
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 1 = 0 + 1
[4,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [2,6,1,7,3,4,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
 => 2 = 1 + 1
[3,2,2,2,1]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[3,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,5,7,1,3,4,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 1 = 0 + 1
[6,4,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => [5,1,2,6,3,7,4] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => 1 = 0 + 1
[6,3,1,1]
 => [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
 => [4,1,6,2,3,7,5] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => 1 = 0 + 1
[6,2,1,1,1]
 => [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [3,6,1,2,4,7,5] => ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
[5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2 = 1 + 1
[5,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [2,6,1,3,7,4,5] => ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
[4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 0 + 1
[4,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2 = 1 + 1
[4,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [2,5,1,7,3,4,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => 1 = 0 + 1
[3,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,4,7,1,3,5,6] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => 1 = 0 + 1
[6,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [5,1,2,3,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 2 = 1 + 1
[6,4,1,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
 => [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => 1 = 0 + 1
[6,3,2,1]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => [4,5,6,1,2,7,3] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[6,3,1,1,1]
 => [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
 => [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => 1 = 0 + 1
[6,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [2,6,1,3,4,7,5] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 1 = 0 + 1
[5,4,2,1]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[5,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => 1 = 0 + 1
Description
The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums 0, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian (4−1−2−1−14−1−2−2−14−1−1−2−14). Its eigenvalues are 0,4,4,6, so the statistic is 2. The path on four vertices has eigenvalues 0,4.7…,6,9.2… and therefore statistic 1.
Matching statistic: St000772
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 20%●distinct values known / distinct values provided: 10%
Values
[1]
 => [1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[2]
 => [1,1,0,0,1,0]
 => [1,3,2] => ([(1,2)],3)
 => ? ∊ {1,1} + 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,1,3] => ([(1,2)],3)
 => ? ∊ {1,1} + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {1,3} + 1
[2,1]
 => [1,0,1,0,1,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => ([(2,3)],4)
 => ? ∊ {1,3} + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => ? ∊ {2,2,6} + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {2,2,6} + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ? ∊ {2,2,6} + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,2,3,4,6,5] => ([(4,5)],6)
 => ? ∊ {1,1,2,4,10} + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,2,4,10} + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {1,1,2,4,10} + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {1,1,2,4,10} + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,1,3,4,5,6] => ([(4,5)],6)
 => ? ∊ {1,1,2,4,10} + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,7,6] => ([(5,6)],7)
 => ? ∊ {0,1,1,3,3,3,7,15} + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1 = 0 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,1,1,3,3,3,7,15} + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,3,3,3,7,15} + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => ? ∊ {0,1,1,3,3,3,7,15} + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,1,1,3,3,3,7,15} + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => ? ∊ {0,1,1,3,3,3,7,15} + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,1,1,3,3,3,7,15} + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1 = 0 + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => ([(5,6)],7)
 => ? ∊ {0,1,1,3,3,3,7,15} + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,6,8,7] => ([(6,7)],8)
 => ? ∊ {1,1,1,2,2,3,3,5,6,11,21} + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [6,1,2,3,4,7,5] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 1 = 0 + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {1,1,1,2,2,3,3,5,6,11,21} + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [4,1,2,3,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,2,2,3,3,5,6,11,21} + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,3,3,5,6,11,21} + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,2,2,3,3,5,6,11,21} + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,2,2,3,3,5,6,11,21} + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {1,1,1,2,2,3,3,5,6,11,21} + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [2,1,6,3,4,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,2,2,3,3,5,6,11,21} + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,2,2,3,3,5,6,11,21} + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ? ∊ {1,1,1,2,2,3,3,5,6,11,21} + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [2,7,1,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 1 = 0 + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7,8] => ([(6,7)],8)
 => ? ∊ {1,1,1,2,2,3,3,5,6,11,21} + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,6,7,9,8] => ([(7,8)],9)
 => ? ∊ {1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [7,1,2,3,4,5,8,6] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? ∊ {1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,6,2,3,4,7,5] => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? ∊ {1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [5,1,2,3,4,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
 => ? ∊ {1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [4,5,1,2,6,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [3,1,2,4,6,5] => ([(1,2),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,2,3,5,4,6] => ([(4,5)],6)
 => ? ∊ {1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6)
 => ? ∊ {1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ? ∊ {1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ? ∊ {1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1 = 0 + 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [2,1,7,3,4,5,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => ? ∊ {1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,3,2,4,5,6] => ([(4,5)],6)
 => ? ∊ {1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
 => ? ∊ {1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [2,6,1,3,4,5,7] => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ? ∊ {1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [2,8,1,3,4,5,6,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ? ∊ {1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7,8,9] => ([(7,8)],9)
 => ? ∊ {1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28} + 1
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,6,7,8,10,9] => ([(8,9)],10)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,3,3,3,3,4,4,6,6,7,10,12,15,22,36} + 1
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,9,7] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,8)],9)
 => ? ∊ {0,1,1,1,1,1,1,2,2,2,3,3,3,3,4,4,6,6,7,10,12,15,22,36} + 1
[6,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [5,6,1,2,3,7,4] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 1 = 0 + 1
[5,3,1]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[5,2,1,1]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => [3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
[4,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
[3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[3,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,6,7,1,3,4,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 1 = 0 + 1
[6,3,1]
 => [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
 => [5,1,6,2,3,7,4] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 1 = 0 + 1
[6,2,1,1]
 => [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
 => [4,6,1,2,3,7,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
 => 1 = 0 + 1
[5,4,1]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1 = 0 + 1
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 1 = 0 + 1
[5,2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 0 + 1
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 1 = 0 + 1
[4,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
 => [2,6,1,7,3,4,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
 => 1 = 0 + 1
[3,2,2,2,1]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1 = 0 + 1
[3,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,5,7,1,3,4,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 1 = 0 + 1
[6,4,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
 => [5,1,2,6,3,7,4] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => 1 = 0 + 1
[6,3,1,1]
 => [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
 => [4,1,6,2,3,7,5] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => 1 = 0 + 1
[6,2,1,1,1]
 => [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [3,6,1,2,4,7,5] => ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
[5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 1 = 0 + 1
[5,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [2,6,1,3,7,4,5] => ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
[4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1 = 0 + 1
[4,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 1 = 0 + 1
[4,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [2,5,1,7,3,4,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => 1 = 0 + 1
[3,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,4,7,1,3,5,6] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => 1 = 0 + 1
[6,5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [5,1,2,3,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 1 = 0 + 1
[6,4,1,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
 => [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => 1 = 0 + 1
[6,3,2,1]
 => [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
 => [4,5,6,1,2,7,3] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
[6,3,1,1,1]
 => [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
 => [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => 1 = 0 + 1
[6,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => [2,6,1,3,4,7,5] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 1 = 0 + 1
[5,4,2,1]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[5,3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => 1 = 0 + 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. The distance Laplacian of a graph is the (symmetric) matrix with row and column sums 0, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue. For example, the cycle on four vertices has distance Laplacian (4−1−2−1−14−1−2−2−14−1−1−2−14). Its eigenvalues are 0,4,4,6, so the statistic is 1. The path on four vertices has eigenvalues 0,4.7…,6,9.2… and therefore also statistic 1. The graphs with statistic n−1, n−2 and n−3 have been characterised, see [1].
Matching statistic: St001630
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 7% ●values known / values provided: 9%●distinct values known / distinct values provided: 7%
Values
[1]
 => [1,0,1,0]
 => [[1,1],[]]
 => ([],1)
 => ? = 0
[2]
 => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => ([],1)
 => ? ∊ {1,1}
[1,1]
 => [1,0,1,1,0,0]
 => [[2,1],[]]
 => ([],1)
 => ? ∊ {1,1}
[3]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => ([],1)
 => ? ∊ {0,1,3}
[2,1]
 => [1,0,1,0,1,0]
 => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,1,3}
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => ([],1)
 => ? ∊ {0,1,3}
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,2,2,6}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,2,2,6}
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,2,2,6}
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => ([],1)
 => ? ∊ {0,0,2,2,6}
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => ([],1)
 => ? ∊ {0,0,2,2,6}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[3,3,3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,1,1,2,4,10}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,1,1,2,4,10}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,1,1,2,4,10}
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,2,4,10}
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,2,4,10}
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,2,4,10}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,2,4,10}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[4,4,4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[4,4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[4,4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [[4,4,4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [[4,4,4,4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[3,3,3,3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[3,3,3,3],[2,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [[3,3,3,2],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [[3,3,3,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [[3,3,3,3,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[4,4,4,4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [[5,5,5,5,5],[4]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [[5,5,5,5],[4]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [[4,4,4,3],[3]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [[4,4,4,4],[3,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [[4,4,3],[3]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [[2,2,2,2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [[3,3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[3,3,2,2],[1,1]]
 => ([(0,2),(2,1)],3)
 => 1
[3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [[4,4,4,3],[3,1]]
 => ([(0,2),(2,1)],3)
 => 1
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [[4,4,3],[3,1]]
 => ([(0,2),(2,1)],3)
 => 1
[4,4,2]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [[4,3,2],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[4,4,1,1]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [[4,3,3],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [[4,3,2],[1,1]]
 => ([(0,2),(2,1)],3)
 => 1
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [[4,3,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [[4,4,2],[2,1]]
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[3,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [[4,4,3,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[5,3,3]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [[3,3,2,2],[2,1]]
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[5,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2,1],[2]]
 => ([(0,2),(2,1)],3)
 => 1
[5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [[3,3,3,2],[2,1,1]]
 => ([(0,2),(2,1)],3)
 => 1
[5,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3,1],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[4,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 1
[3,3,3,1,1]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2,1],[1,1]]
 => ([(0,3),(2,1),(3,2)],4)
 => 2
[5,3,3,1]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 1
[5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[4,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[4,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [[4,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001876
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 7% ●values known / values provided: 9%●distinct values known / distinct values provided: 7%
Values
[1]
 => [1,0,1,0]
 => [[1,1],[]]
 => ([],1)
 => ? = 0
[2]
 => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => ([],1)
 => ? ∊ {1,1}
[1,1]
 => [1,0,1,1,0,0]
 => [[2,1],[]]
 => ([],1)
 => ? ∊ {1,1}
[3]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => ([],1)
 => ? ∊ {0,1,3}
[2,1]
 => [1,0,1,0,1,0]
 => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,1,3}
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => ([],1)
 => ? ∊ {0,1,3}
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,2,2,6}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,2,2,6}
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,2,2,6}
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => ([],1)
 => ? ∊ {0,0,2,2,6}
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => ([],1)
 => ? ∊ {0,0,2,2,6}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[3,3,3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,1,1,2,4,10}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,1,1,2,4,10}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,1,1,2,4,10}
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,2,4,10}
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,2,4,10}
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,2,4,10}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,2,4,10}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[4,4,4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[4,4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[4,4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [[4,4,4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [[4,4,4,4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[3,3,3,3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[3,3,3,3],[2,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [[3,3,3,2],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [[3,3,3,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [[3,3,3,3,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[4,4,4,4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [[5,5,5,5,5],[4]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [[5,5,5,5],[4]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [[4,4,4,3],[3]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [[4,4,4,4],[3,1]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [[4,4,3],[3]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [[2,2,2,2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [[3,3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[3,3,2,2],[1,1]]
 => ([(0,2),(2,1)],3)
 => 0
[3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [[4,4,4,3],[3,1]]
 => ([(0,2),(2,1)],3)
 => 0
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [[4,4,3],[3,1]]
 => ([(0,2),(2,1)],3)
 => 0
[4,4,2]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [[4,3,2],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[4,4,1,1]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [[4,3,3],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [[4,3,2],[1,1]]
 => ([(0,2),(2,1)],3)
 => 0
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [[4,3,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [[4,4,2],[2,1]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[3,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [[4,4,3,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[5,3,3]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [[3,3,2,2],[2,1]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[5,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2,1],[2]]
 => ([(0,2),(2,1)],3)
 => 0
[5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [[3,3,3,2],[2,1,1]]
 => ([(0,2),(2,1)],3)
 => 0
[5,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3,1],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[4,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 0
[3,3,3,1,1]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2,1],[1,1]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[5,3,3,1]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
[5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[4,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[4,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [[4,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St001877
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 7% ●values known / values provided: 9%●distinct values known / distinct values provided: 7%
Values
[1]
 => [1,0,1,0]
 => [[1,1],[]]
 => ([],1)
 => ? = 0
[2]
 => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => ([],1)
 => ? ∊ {1,1}
[1,1]
 => [1,0,1,1,0,0]
 => [[2,1],[]]
 => ([],1)
 => ? ∊ {1,1}
[3]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => ([],1)
 => ? ∊ {0,1,3}
[2,1]
 => [1,0,1,0,1,0]
 => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,1,3}
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => ([],1)
 => ? ∊ {0,1,3}
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,2,2,6}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,2,2,6}
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,2,2,6}
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => ([],1)
 => ? ∊ {0,0,2,2,6}
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => ([],1)
 => ? ∊ {0,0,2,2,6}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[3,3,3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,1,1,2,4,10}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,1,1,2,4,10}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,1,1,2,4,10}
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,2,4,10}
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,2,4,10}
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,2,4,10}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,2,4,10}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[4,4,4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[4,4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[4,4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [[4,4,4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [[4,4,4,4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[3,3,3,3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[3,3,3,3],[2,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [[3,3,3,2],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [[3,3,3,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [[3,3,3,3,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[4,4,4,4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [[5,5,5,5,5],[4]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [[5,5,5,5],[4]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [[4,4,4,3],[3]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [[4,4,4,4],[3,1]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [[4,4,3],[3]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [[2,2,2,2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [[3,3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[3,3,2,2],[1,1]]
 => ([(0,2),(2,1)],3)
 => 0
[3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [[4,4,4,3],[3,1]]
 => ([(0,2),(2,1)],3)
 => 0
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [[4,4,3],[3,1]]
 => ([(0,2),(2,1)],3)
 => 0
[4,4,2]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [[4,3,2],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[4,4,1,1]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [[4,3,3],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [[4,3,2],[1,1]]
 => ([(0,2),(2,1)],3)
 => 0
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [[4,3,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [[4,4,2],[2,1]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[3,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [[4,4,3,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[5,3,3]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [[3,3,2,2],[2,1]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[5,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2,1],[2]]
 => ([(0,2),(2,1)],3)
 => 0
[5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [[3,3,3,2],[2,1,1]]
 => ([(0,2),(2,1)],3)
 => 0
[5,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3,1],[2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[4,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 0
[3,3,3,1,1]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2,1],[1,1]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[5,3,3,1]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 0
[5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[4,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[4,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [[4,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St001878
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 7% ●values known / values provided: 9%●distinct values known / distinct values provided: 7%
Values
[1]
 => [1,0,1,0]
 => [[1,1],[]]
 => ([],1)
 => ? = 0
[2]
 => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => ([],1)
 => ? ∊ {1,1}
[1,1]
 => [1,0,1,1,0,0]
 => [[2,1],[]]
 => ([],1)
 => ? ∊ {1,1}
[3]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => ([],1)
 => ? ∊ {0,1,3}
[2,1]
 => [1,0,1,0,1,0]
 => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,1,3}
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => ([],1)
 => ? ∊ {0,1,3}
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,2,2,6}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,2,2,6}
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,2,2,6}
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => ([],1)
 => ? ∊ {0,0,2,2,6}
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => ([],1)
 => ? ∊ {0,0,2,2,6}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[3,3,3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,1,1,2,4,10}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,1,1,2,4,10}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,1,1,2,4,10}
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,1,1,2,4,10}
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,2,4,10}
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,2,4,10}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3,1],[]]
 => ([],1)
 => ? ∊ {0,0,1,1,2,4,10}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[4,4,4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[4,4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[4,4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [[4,4,4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,3,3,3,7,15}
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [[4,4,4,4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[3,3,3,3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[3,3,3,3],[2,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [[3,3,3,2],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [[3,3,3,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [[3,3,3,3,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [[4,4,4,4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,3,5,6,11,21}
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [[5,5,5,5,5],[4]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [[5,5,5,5],[4]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [[4,4,4,3],[3]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [[4,4,4,4],[3,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [[4,4,3],[3]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [[2,2,2,2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,4,4,4,6,8,10,16,28}
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [[3,3,3,2],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[3,3,3]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [[3,3,2,2],[1,1]]
 => ([(0,2),(2,1)],3)
 => 1
[3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[6,2,2]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [[4,4,4,3],[3,1]]
 => ([(0,2),(2,1)],3)
 => 1
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [[4,4,3],[3,1]]
 => ([(0,2),(2,1)],3)
 => 1
[4,4,2]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [[4,3,2],[2]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[4,4,1,1]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [[4,3,3],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [[4,3,2],[1,1]]
 => ([(0,2),(2,1)],3)
 => 1
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [[4,3,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [[4,4,2],[2,1]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[3,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [[4,4,3,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[5,3,3]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [[3,3,2,2],[2,1]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[5,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2,1],[2]]
 => ([(0,2),(2,1)],3)
 => 1
[5,2,2,2]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [[3,3,3,2],[2,1,1]]
 => ([(0,2),(2,1)],3)
 => 1
[5,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3,1],[2,1]]
 => ([(0,2),(2,1)],3)
 => 1
[4,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
[4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [[5,3],[2]]
 => ([(0,2),(2,1)],3)
 => 1
[3,3,3,1,1]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2,1],[1,1]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[5,3,3,1]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [[4,4,3],[3,2]]
 => ([(0,2),(2,1)],3)
 => 1
[5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [[4,4,2],[3,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[4,4,2,2]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [[4,3,2],[2,1]]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[4,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [[4,3,1],[2]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[4,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [[4,2,1],[1]]
 => ([(0,2),(2,1)],3)
 => 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001232
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 17%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 0 + 1
[2]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ? = 1 + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? ∊ {1,3} + 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? ∊ {1,3} + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,2,6} + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,2,6} + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? ∊ {0,0,2,6} + 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,2,6} + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,1,4,10} + 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ? ∊ {0,0,1,4,10} + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ? ∊ {0,0,1,4,10} + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {0,0,1,4,10} + 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,1,4,10} + 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,1,1,1,3,3,3,7,15} + 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,1,1,1,3,3,3,7,15} + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? ∊ {0,0,1,1,1,3,3,3,7,15} + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => ? ∊ {0,0,1,1,1,3,3,3,7,15} + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ? ∊ {0,0,1,1,1,3,3,3,7,15} + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ? ∊ {0,0,1,1,1,3,3,3,7,15} + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => ? ∊ {0,0,1,1,1,3,3,3,7,15} + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ? ∊ {0,0,1,1,1,3,3,3,7,15} + 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,1,1,1,3,3,3,7,15} + 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,1,1,1,3,3,3,7,15} + 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,5,6,11,21} + 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,5,6,11,21} + 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,5,6,11,21} + 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,5,6,11,21} + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,5,6,11,21} + 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,5,6,11,21} + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,5,6,11,21} + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,5,6,11,21} + 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,5,6,11,21} + 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,5,6,11,21} + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,5,6,11,21} + 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,5,6,11,21} + 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,5,6,11,21} + 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,1,1,1,2,2,3,5,6,11,21} + 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,2,4,4,4,6,8,10,16,28} + 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,2,4,4,4,6,8,10,16,28} + 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,2,4,4,4,6,8,10,16,28} + 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,2,4,4,4,6,8,10,16,28} + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,2,4,4,4,6,8,10,16,28} + 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,2,4,4,4,6,8,10,16,28} + 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,2,4,4,4,6,8,10,16,28} + 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,2,4,4,4,6,8,10,16,28} + 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,2,4,4,4,6,8,10,16,28} + 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,2,4,4,4,6,8,10,16,28} + 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,2,4,4,4,6,8,10,16,28} + 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,2,4,4,4,6,8,10,16,28} + 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,2,4,4,4,6,8,10,16,28} + 1
[3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,1,2,4,4,4,6,8,10,16,28} + 1
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 5 = 4 + 1
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[5,3,2,1]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 5 = 4 + 1
[3,3,2,2,1]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[5,4,2,1]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[4,4,3,1]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 5 = 4 + 1
[4,3,3,2]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => 4 = 3 + 1
[4,3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.