Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001714
Mapping
St001714: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0
[2]
 => 0
[1,1]
 => 1
[3]
 => 0
[2,1]
 => 0
[1,1,1]
 => 2
[4]
 => 0
[3,1]
 => 0
[2,2]
 => 0
[2,1,1]
 => 2
[1,1,1,1]
 => 3
[5]
 => 0
[4,1]
 => 0
[3,2]
 => 0
[3,1,1]
 => 1
[2,2,1]
 => 1
[2,1,1,1]
 => 4
[1,1,1,1,1]
 => 4
[6]
 => 0
[5,1]
 => 0
[4,2]
 => 0
[4,1,1]
 => 1
[3,3]
 => 0
[3,2,1]
 => 0
[3,1,1,1]
 => 3
[2,2,2]
 => 1
[2,2,1,1]
 => 4
[2,1,1,1,1]
 => 6
[1,1,1,1,1,1]
 => 5
[7]
 => 0
[6,1]
 => 0
[5,2]
 => 0
[5,1,1]
 => 1
[4,3]
 => 0
[4,2,1]
 => 0
[4,1,1,1]
 => 2
[3,3,1]
 => 0
[3,2,2]
 => 1
[3,2,1,1]
 => 3
[3,1,1,1,1]
 => 5
[2,2,2,1]
 => 3
[2,2,1,1,1]
 => 7
[2,1,1,1,1,1]
 => 8
[1,1,1,1,1,1,1]
 => 6
[8]
 => 0
[7,1]
 => 0
[6,2]
 => 0
[6,1,1]
 => 1
[5,3]
 => 0
[5,2,1]
 => 0
Description
The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. In particular, partitions with statistic $0$ are wide partitions.
Matching statistic: St000772
(load all 2 compositions to match this statistic)
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: 8% ●values known / values provided: 11%●distinct values known / distinct values provided: 8%
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)
 => ? ∊ {0,1} + 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,1,3] => ([(1,2)],3)
 => ? ∊ {0,1} + 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {0,2} + 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)
 => ? ∊ {0,2} + 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => ? ∊ {0,2,3} + 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)
 => ? ∊ {0,2,3} + 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)
 => ? ∊ {0,2,3} + 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,2,3,4,6,5] => ([(4,5)],6)
 => ? ∊ {0,1,1,4,4} + 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)
 => ? ∊ {0,1,1,4,4} + 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {0,1,1,4,4} + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,1,1,4,4} + 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)
 => ? ∊ {0,1,1,4,4} + 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)
 => ? ∊ {0,0,0,1,3,4,5,6} + 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)
 => ? ∊ {0,0,0,1,3,4,5,6} + 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)
 => ? ∊ {0,0,0,1,3,4,5,6} + 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => ? ∊ {0,0,0,1,3,4,5,6} + 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)
 => ? ∊ {0,0,0,1,3,4,5,6} + 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => ? ∊ {0,0,0,1,3,4,5,6} + 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)
 => ? ∊ {0,0,0,1,3,4,5,6} + 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)
 => ? ∊ {0,0,0,1,3,4,5,6} + 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)
 => ? ∊ {0,0,1,1,2,3,3,5,6,7,8} + 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)
 => ? ∊ {0,0,1,1,2,3,3,5,6,7,8} + 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)
 => ? ∊ {0,0,1,1,2,3,3,5,6,7,8} + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,1,1,2,3,3,5,6,7,8} + 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)
 => ? ∊ {0,0,1,1,2,3,3,5,6,7,8} + 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)
 => ? ∊ {0,0,1,1,2,3,3,5,6,7,8} + 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)
 => ? ∊ {0,0,1,1,2,3,3,5,6,7,8} + 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)
 => ? ∊ {0,0,1,1,2,3,3,5,6,7,8} + 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)
 => ? ∊ {0,0,1,1,2,3,3,5,6,7,8} + 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)
 => ? ∊ {0,0,1,1,2,3,3,5,6,7,8} + 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)
 => ? ∊ {0,0,1,1,2,3,3,5,6,7,8} + 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)
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10} + 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)
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10} + 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)
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10} + 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)
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10} + 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)
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10} + 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)
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10} + 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,2,3,5,4,6] => ([(4,5)],6)
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10} + 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)
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10} + 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)
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10} + 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10} + 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)
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10} + 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)
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10} + 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)
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10} + 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)
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10} + 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)
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10} + 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)
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10} + 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)
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10} + 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)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,5,5,5,6,8,8,9,11,11,12,13} + 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)
 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,3,3,5,5,5,6,8,8,9,11,11,12,13} + 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 $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $1$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$. The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St001498
(load all 4 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 8%
Values
[1]
 => [1,0]
 => [1,0]
 => [1,0]
 => ? = 0
[2]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => ? ∊ {0,1}
[1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => ? ∊ {0,1}
[3]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? ∊ {0,0,2}
[2,1]
 => [1,0,1,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? ∊ {0,0,2}
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? ∊ {0,0,2}
[4]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,2,3}
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,2,3}
[2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? ∊ {0,0,0,2,3}
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,2,3}
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,2,3}
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,1,1,4,4}
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,1,1,4,4}
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,1,1,4,4}
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,1,1,4,4}
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,1,1,4,4}
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,1,1,4,4}
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,1,1,4,4}
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,3,4,5,6}
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,3,4,5,6}
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,3,4,5,6}
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,3,4,5,6}
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,3,4,5,6}
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,3,4,5,6}
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,3,4,5,6}
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 0
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,3,4,5,6}
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,3,4,5,6}
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,3,4,5,6}
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,2,3,3,5,6,7,8}
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,2,3,3,5,6,7,8}
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,2,3,3,5,6,7,8}
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,2,3,3,5,6,7,8}
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,2,3,3,5,6,7,8}
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,2,3,3,5,6,7,8}
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,2,3,3,5,6,7,8}
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,2,3,3,5,6,7,8}
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 0
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,2,3,3,5,6,7,8}
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,2,3,3,5,6,7,8}
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 0
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,2,3,3,5,6,7,8}
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,2,3,3,5,6,7,8}
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,1,2,3,3,5,6,7,8}
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10}
[7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10}
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10}
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10}
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10}
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10}
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10}
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10}
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10}
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 0
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 0
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 0
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 0
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => 0
[3,3,3]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 1
[3,3,2,1]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 0
[3,2,2,2]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 0
[3,2,2,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => 0
[2,2,2,2,1]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 0
[2,2,2,1,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => 0
[5,3,2]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[4,4,2]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 1
[4,3,2,1]
 => [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => 0
[4,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => 0
[3,3,3,1]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 1
[3,3,2,2]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 0
[3,3,2,1,1]
 => [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => 0
[3,2,2,2,1]
 => [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => 0
[2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 0
[2,2,2,2,1,1]
 => [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => 0
[5,4,2]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[5,3,3]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => 1
[4,4,3]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => 1
[4,4,2,1]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => 0
[4,3,3,1]
 => [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => 1
[4,3,2,2]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => 0
[3,3,3,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 0
[3,3,3,1,1]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => 1
[3,3,2,2,1]
 => [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => 0
[3,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => 0
[2,2,2,2,2,1]
 => [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => 0
[5,5,2]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[5,4,3]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => 1
[4,4,4]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 2
[4,4,3,1]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => 1
[4,4,2,2]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => 0
[4,3,3,2]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => 0
[3,3,3,3]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 1
[3,3,3,2,1]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => 0
[3,3,2,2,2]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => 0
[2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => 0
[5,5,3]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => 1
Description
The normalised height of a Nakayama algebra with magnitude 1. We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
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: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 4%
Values
[1]
 => [1,0,1,0]
 => [[1,1],[]]
 => ([],1)
 => ? = 0
[2]
 => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,1}
[1,1]
 => [1,0,1,1,0,0]
 => [[2,1],[]]
 => ([],1)
 => ? ∊ {0,1}
[3]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,2}
[2,1]
 => [1,0,1,0,1,0]
 => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,2}
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => ([],1)
 => ? ∊ {0,0,2}
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,2,3}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,2,3}
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,2,3}
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,2,3}
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,2,3}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[3,3,3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,4,4}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,4,4}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,4,4}
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,4,4}
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,4,4}
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,4,4}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,4,4}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[4,4,4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,3,4,5,6}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[4,4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,3,4,5,6}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,3,4,5,6}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,3,4,5,6}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,3,4,5,6}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,3,4,5,6}
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,3,4,5,6}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,3,4,5,6}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,3,4,5,6}
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[4,4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,3,4,5,6}
[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)
 => ? ∊ {0,0,0,0,0,1,1,3,4,5,6}
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [[4,4,4,4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[3,3,3,3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[3,3,3,3],[2,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [[3,3,3,2],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[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)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[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)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[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)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[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)
 => ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10}
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [[5,5,5,5],[4]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10}
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [[4,4,4,3],[3]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10}
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [[4,4,4,4],[3,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10}
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [[4,4,3],[3]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10}
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [[2,2,2,2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10}
[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
[5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[5,3,3,2]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[5,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[4,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[4,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[5,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 0
[5,4,4,2,1]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => [[6,4],[3]]
 => ([(0,2),(2,1)],3)
 => 0
[5,4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [[6,3],[2]]
 => ([(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: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 4%
Values
[1]
 => [1,0,1,0]
 => [[1,1],[]]
 => ([],1)
 => ? = 0
[2]
 => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => ([],1)
 => ? ∊ {0,1}
[1,1]
 => [1,0,1,1,0,0]
 => [[2,1],[]]
 => ([],1)
 => ? ∊ {0,1}
[3]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,2}
[2,1]
 => [1,0,1,0,1,0]
 => [[1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,2}
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => ([],1)
 => ? ∊ {0,0,2}
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,2,3}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,2,3}
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,2,3}
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,2,3}
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,2,3}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [[3,3,3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,4,4}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,4,4}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,4,4}
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,4,4}
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,4,4}
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,4,4}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,1,1,4,4}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [[4,4,4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,3,4,5,6}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [[4,4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,3,4,5,6}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,3,4,5,6}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,3,4,5,6}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,3,4,5,6}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,3,4,5,6}
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,3,4,5,6}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,3,4,5,6}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,1,1,3,4,5,6}
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [[4,4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,1,1,3,4,5,6}
[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)
 => ? ∊ {0,0,0,0,0,1,1,3,4,5,6}
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [[4,4,4,4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [[3,3,3,3,3],[2]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [[3,3,3,3],[2,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [[3,3,3,2],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[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)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2,1],[]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[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)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[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)
 => ? ∊ {0,0,0,0,0,0,1,1,2,3,3,5,6,7,8}
[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)
 => ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10}
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [[5,5,5,5],[4]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10}
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [[4,4,4,3],[3]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10}
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [[4,4,4,4],[3,1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10}
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [[4,4,3],[3]]
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10}
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [[2,2,2,2,2],[1]]
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,1,2,2,2,2,3,4,7,7,7,7,10,10}
[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
[5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [[3,3,3,2],[2,2,1]]
 => ([(0,2),(2,1)],3)
 => 0
[5,3,3,2]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [[3,3,2,2],[2,1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[5,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2,1],[2,1]]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1
[4,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2,1],[1,1]]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[4,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1,1],[1]]
 => ([(0,2),(2,1)],3)
 => 0
[5,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1,1],[1,1]]
 => ([(0,2),(2,1)],3)
 => 0
[5,4,4,2,1]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => [[6,4],[3]]
 => ([(0,2),(2,1)],3)
 => 0
[5,4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [[6,3],[2]]
 => ([(0,2),(2,1)],3)
 => 0
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St000771
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 10%
Values
[1]
 => [1,0]
 => [1] => ([],1)
 => 1 = 0 + 1
[2]
 => [1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[1,1]
 => [1,1,0,0]
 => [1,2] => ([],2)
 => ? = 1 + 1
[3]
 => [1,0,1,0,1,0]
 => [2,1,3] => ([(1,2)],3)
 => ? ∊ {0,2} + 1
[2,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 1 = 0 + 1
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,3,2] => ([(1,2)],3)
 => ? ∊ {0,2} + 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? ∊ {0,0,2,3} + 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 1 = 0 + 1
[2,2]
 => [1,1,1,0,0,0]
 => [1,2,3] => ([],3)
 => ? ∊ {0,0,2,3} + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,2,3} + 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => ([(2,3)],4)
 => ? ∊ {0,0,2,3} + 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ? ∊ {0,0,1,4,4} + 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,1,4,4} + 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1 = 0 + 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,1,4,4} + 1
[2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? ∊ {0,0,1,4,4} + 1
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? ∊ {0,0,1,4,4} + 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
 => ? ∊ {0,0,0,1,1,3,4,5,6} + 1
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,1,1,3,4,5,6} + 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1 = 0 + 1
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 1 = 0 + 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => ([(2,3)],4)
 => ? ∊ {0,0,0,1,1,3,4,5,6} + 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,3,4,5,6} + 1
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,4,1,5,3,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? ∊ {0,0,0,1,1,3,4,5,6} + 1
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([],4)
 => ? ∊ {0,0,0,1,1,3,4,5,6} + 1
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ? ∊ {0,0,0,1,1,3,4,5,6} + 1
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,3,1,5,4,6] => ([(1,2),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,1,1,3,4,5,6} + 1
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
 => ? ∊ {0,0,0,1,1,3,4,5,6} + 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,6,5,7] => ([(1,6),(2,5),(3,4)],7)
 => ? ∊ {0,0,0,1,1,3,3,5,6,7,8} + 1
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,6,5,7] => ([(1,2),(3,6),(4,5),(5,6)],7)
 => ? ∊ {0,0,0,1,1,3,3,5,6,7,8} + 1
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 1 = 0 + 1
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,6,3,5,7] => ([(1,6),(2,5),(3,4),(3,5),(4,6)],7)
 => ? ∊ {0,0,0,1,1,3,3,5,6,7,8} + 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,3,3,5,6,7,8} + 1
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1 = 0 + 1
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,4,1,6,3,7,5] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 1 = 0 + 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,3,3,5,6,7,8} + 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3 = 2 + 1
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [2,3,1,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
 => ? ∊ {0,0,0,1,1,3,3,5,6,7,8} + 1
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,4,1,5,3,7,6] => ([(0,1),(2,5),(3,4),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,1,1,3,3,5,6,7,8} + 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,1,1,3,3,5,6,7,8} + 1
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
 => ? ∊ {0,0,0,1,1,3,3,5,6,7,8} + 1
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,1,5,4,7,6] => ([(0,3),(1,2),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,1,1,3,3,5,6,7,8} + 1
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4,7,6] => ([(1,6),(2,5),(3,4)],7)
 => ? ∊ {0,0,0,1,1,3,3,5,6,7,8} + 1
[8]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,6,5,8,7] => ([(0,7),(1,6),(2,5),(3,4)],8)
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,3,4,7,7,7,7,10,10} + 1
[7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,6,5,8,7] => ([(0,3),(1,2),(4,7),(5,6),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,3,4,7,7,7,7,10,10} + 1
[6,2]
 => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,4,6,1,3,5,7] => ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,3,4,7,7,7,7,10,10} + 1
[6,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,6,3,5,8,7] => ([(0,1),(2,5),(3,4),(4,6),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,3,4,7,7,7,7,10,10} + 1
[5,3]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => 1 = 0 + 1
[5,2,1]
 => [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [2,4,1,6,7,3,5] => ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
 => 1 = 0 + 1
[5,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,4,1,6,3,8,5,7] => ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,3,4,7,7,7,7,10,10} + 1
[4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,3,4,7,7,7,7,10,10} + 1
[4,3,1]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [2,3,1,4,6,5] => ([(1,2),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,3,4,7,7,7,7,10,10} + 1
[4,2,2]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 2 = 1 + 1
[4,2,1,1]
 => [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [2,4,1,5,3,6,7] => ([(2,6),(3,5),(4,5),(4,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,3,4,7,7,7,7,10,10} + 1
[4,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,4,1,6,3,7,5,8] => ([(1,7),(2,6),(3,4),(3,5),(4,6),(5,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,3,4,7,7,7,7,10,10} + 1
[3,3,2]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,3,4,7,7,7,7,10,10} + 1
[3,3,1,1]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,4,2,6,3,5] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,3,4,7,7,7,7,10,10} + 1
[3,2,2,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,3,4,7,7,7,7,10,10} + 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
 => [2,3,1,6,4,7,5] => ([(0,6),(1,5),(2,4),(3,4),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,3,4,7,7,7,7,10,10} + 1
[3,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,4,1,5,3,7,6,8] => ([(1,2),(3,6),(4,5),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,3,4,7,7,7,7,10,10} + 1
[2,2,2,2]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,3,4,7,7,7,7,10,10} + 1
[2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,3,4,7,7,7,7,10,10} + 1
[2,2,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => [1,4,2,5,3,7,6] => ([(1,2),(3,6),(4,5),(5,6)],7)
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,3,4,7,7,7,7,10,10} + 1
[2,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,1,5,4,7,6,8] => ([(1,4),(2,3),(5,7),(6,7)],8)
 => ? ∊ {0,0,0,0,0,0,0,2,2,2,2,3,4,7,7,7,7,10,10} + 1
[6,3]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,4,6,1,7,3,5] => ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => 1 = 0 + 1
[5,2,2]
 => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,4,6,7,1,3,5] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => 1 = 0 + 1
[4,3,2]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 1 + 1
[4,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [2,4,1,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 2 = 1 + 1
[4,3,3]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 4 = 3 + 1
[4,2,2,2]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,4,5,6,1,7,3] => ([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 2 = 1 + 1
[5,3,3]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,4,5,6,7,1,3] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 3 = 2 + 1
[4,3,2,2]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,3,4,6,1,7,5] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 2 = 1 + 1
[5,4,3]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [2,3,4,6,7,1,5] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => 2 = 1 + 1
[4,3,3,3]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 5 = 4 + 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 $$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$ Its eigenvalues are $0,4,4,6$, so the statistic is $2$. The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore statistic $1$.
Matching statistic: St001880
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 6%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => ([],2)
 => ? = 0
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => ([(1,2)],3)
 => ? ∊ {0,1}
[1,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => ? ∊ {0,1}
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ? ∊ {0,0,2}
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ([],3)
 => ? ∊ {0,0,2}
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => ? ∊ {0,0,2}
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ? ∊ {0,0,0,2,3}
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => ? ∊ {0,0,0,2,3}
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => ? ∊ {0,0,0,2,3}
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => ? ∊ {0,0,0,2,3}
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? ∊ {0,0,0,2,3}
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ? ∊ {0,0,0,1,1,4,4}
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => ? ∊ {0,0,0,1,1,4,4}
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => ? ∊ {0,0,0,1,1,4,4}
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => ([(1,3),(2,3)],4)
 => ? ∊ {0,0,0,1,1,4,4}
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {0,0,0,1,1,4,4}
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ([(0,4),(4,1),(4,2),(4,3)],5)
 => ? ∊ {0,0,0,1,1,4,4}
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ? ∊ {0,0,0,1,1,4,4}
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,6),(1,5),(1,6),(2,4),(2,5),(5,3),(6,3),(6,4)],7)
 => ? ∊ {0,0,0,0,0,1,3,4,5,6}
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6)
 => ? ∊ {0,0,0,0,0,1,3,4,5,6}
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ? ∊ {0,0,0,0,0,1,3,4,5,6}
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,3,4,5,6}
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,3,4,5,6}
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ([],4)
 => ? ∊ {0,0,0,0,0,1,3,4,5,6}
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,1,3,4,5,6}
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? ∊ {0,0,0,0,0,1,3,4,5,6}
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
 => ? ∊ {0,0,0,0,0,1,3,4,5,6}
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
 => ? ∊ {0,0,0,0,0,1,3,4,5,6}
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,7),(1,6),(1,7),(2,5),(2,6),(5,4),(6,3),(6,4),(7,3),(7,5)],8)
 => ? ∊ {0,0,0,0,0,0,1,1,3,3,5,6,7,8}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(6,3),(6,4)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,3,3,5,6,7,8}
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => ([(0,2),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3)],6)
 => ? ∊ {0,0,0,0,0,0,1,1,3,3,5,6,7,8}
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(5,3)],6)
 => ? ∊ {0,0,0,0,0,0,1,1,3,3,5,6,7,8}
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ([(1,4),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,3,3,5,6,7,8}
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,3,3,5,6,7,8}
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,3,3,5,6,7,8}
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,3,3,5,6,7,8}
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(4,1)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,3,3,5,6,7,8}
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,3,3,5,6,7,8}
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => ([(0,4),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => ? ∊ {0,0,0,0,0,0,1,1,3,3,5,6,7,8}
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ? ∊ {0,0,0,0,0,0,1,1,3,3,5,6,7,8}
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
 => 2
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
 => ? ∊ {0,0,0,0,0,0,1,1,3,3,5,6,7,8}
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
 => ? ∊ {0,0,0,0,0,0,1,1,3,3,5,6,7,8}
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,8),(1,7),(1,8),(2,6),(2,7),(5,4),(6,3),(6,4),(7,3),(7,5),(8,5),(8,6)],9)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,4,7,7,7,7,10,10}
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,7),(1,6),(1,7),(2,5),(2,6),(6,3),(6,4),(7,3),(7,4),(7,5)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,4,7,7,7,7,10,10}
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(6,3)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,4,7,7,7,7,10,10}
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(5,3),(6,3)],7)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,4,7,7,7,7,10,10}
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(3,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,4,7,7,7,7,10,10}
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,3),(3,4),(3,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,4,7,7,7,7,10,10}
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4),(3,5)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,4,7,7,7,7,10,10}
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,2,2,2,4,7,7,7,7,10,10}
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
 => 2
[2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => ([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
 => 3
[3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 2
[2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
 => 3
[3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,1)],6)
 => 1
[3,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => ([(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(4,3),(5,6)],7)
 => 3
[2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
 => 3
[3,3,2,2,1]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[3,3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
 => 3
[3,3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
 => ([(0,5),(1,6),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => 2
[3,3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => ([(0,3),(0,4),(0,5),(1,6),(3,6),(4,6),(5,1),(6,2)],7)
 => 2
[3,3,2,2,2,1]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ([(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3),(5,4)],7)
 => 2
[4,4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,1),(4,5),(5,6)],7)
 => 1
[4,4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
 => 2
[4,4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
 => ([(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2),(5,3)],7)
 => 2
[3,3,3,2,2,1]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(0,5),(2,6),(3,6),(4,6),(5,6),(6,1)],7)
 => 2
[4,4,3,3,1]
 => [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => ([(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1),(5,2)],7)
 => 1
[4,4,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,6),(4,5),(6,5)],7)
 => 2
[4,4,3,3,2]
 => [1,1,0,0,1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
 => ([(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,1)],7)
 => 1
[4,4,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 1
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.