Your data matches 31 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000741
(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
St000741: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 2
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 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
[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
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[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
[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
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 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
[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
[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
[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
[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
[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
[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
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 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
[5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
 => 1
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,4,2,3,6,5] => ([(1,2),(3,5),(4,5)],6)
 => 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => 1
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => 1
[3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [2,1,5,3,4,6] => ([(1,2),(3,5),(4,5)],6)
 => 1
[2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
 => 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
[4,4,2]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,4,2,5,3,6] => ([(2,5),(3,4),(4,5)],6)
 => 1
[4,4,1,1]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4,6] => ([(1,2),(3,5),(4,5)],6)
 => 1
[4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,2,4,6,3,5] => ([(2,5),(3,4),(4,5)],6)
 => 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)
 => 2
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,3,2,6,4,5] => ([(1,2),(3,5),(4,5)],6)
 => 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
[3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [3,1,4,2,5,6] => ([(2,5),(3,4),(4,5)],6)
 => 1
[3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4,6] => ([(2,5),(3,4),(4,5)],6)
 => 1
[5,4,1,1]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [3,1,2,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
 => 1
[5,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [2,1,5,3,6,4] => ([(0,1),(2,5),(3,4),(4,5)],6)
 => 1
[5,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
 => 1
[4,4,3]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
 => 1
[4,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
 => 1
[3,3,3,2]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
 => 1
[5,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [2,1,3,5,6,4] => ([(1,2),(3,5),(4,5)],6)
 => 1
[5,3,3,1]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [3,1,4,2,6,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
 => 1
[5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,3,5,2,6,4] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 1
[5,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [2,3,1,4,6,5] => ([(1,2),(3,5),(4,5)],6)
 => 1
[4,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [2,4,1,5,3,6] => ([(1,5),(2,4),(3,4),(3,5)],6)
 => 1
[4,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,6,3,5] => ([(0,1),(2,5),(3,4),(4,5)],6)
 => 1
[5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
 => 1
[5,3,3,2]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
 => 1
Description
The Colin de Verdière graph invariant.
Matching statistic: St000546
(load all 2 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000546: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,2] => 0 = 1 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => [3,1,2] => 1 = 2 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,4,2,3] => 0 = 1 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [3,1,2,4] => 0 = 1 - 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => 0 = 1 - 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [3,1,4,2] => 0 = 1 - 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => 0 = 1 - 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 0 = 1 - 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => 1 = 2 - 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 0 = 1 - 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => 0 = 1 - 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => 0 = 1 - 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,4,2,5,3] => 0 = 1 - 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => 0 = 1 - 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => 0 = 1 - 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,6,4,5] => 0 = 1 - 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,2,4,5,3,6] => 0 = 1 - 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => 0 = 1 - 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,3,4,2,5,6] => 0 = 1 - 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => 0 = 1 - 1
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [3,1,2,5,6,4] => 0 = 1 - 1
[5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,3,4,6,1,5] => 0 = 1 - 1
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [2,1,3,5,4,6] => 0 = 1 - 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => 0 = 1 - 1
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => 0 = 1 - 1
[3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,3,2,4,6,5] => 0 = 1 - 1
[2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [3,1,4,5,6,2] => 0 = 1 - 1
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,4,2,6,3,5] => 0 = 1 - 1
[4,4,2]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [2,1,5,3,6,4] => 0 = 1 - 1
[4,4,1,1]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,2,4,3,6,5] => 0 = 1 - 1
[4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4,6] => 0 = 1 - 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]
 => [3,4,5,1,2] => 1 = 2 - 1
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [2,1,4,3,5,6] => 0 = 1 - 1
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4,6] => 0 = 1 - 1
[3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,4,2,5,6,3] => 0 = 1 - 1
[3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,6,5] => 0 = 1 - 1
[5,4,1,1]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,2,4,6,3,5] => 0 = 1 - 1
[5,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,3,2,6,4,5] => 0 = 1 - 1
[5,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4,6] => 0 = 1 - 1
[4,4,3]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [2,3,5,1,6,4] => 0 = 1 - 1
[4,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [3,1,4,2,5,6] => 0 = 1 - 1
[3,3,3,2]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [2,4,1,5,6,3] => 0 = 1 - 1
[5,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,3,4,6,2,5] => 0 = 1 - 1
[5,3,3,1]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,4,2,5,3,6] => 0 = 1 - 1
[5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,6,3,5] => 0 = 1 - 1
[5,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [3,1,4,5,2,6] => 0 = 1 - 1
[4,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [3,1,5,2,6,4] => 0 = 1 - 1
[4,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4,6] => 0 = 1 - 1
[5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,6,3,5] => 0 = 1 - 1
[5,3,3,2]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [2,4,1,5,3,6] => 0 = 1 - 1
Description
The number of global descents of a permutation. The global descents are the integers in the set $$C(\pi)=\{i\in [n-1] : \forall 1 \leq j \leq i < k \leq n :\quad \pi(j) > \pi(k)\}.$$ In particular, if $i\in C(\pi)$ then $i$ is a descent. For the number of global ascents, see [[St000234]].
Matching statistic: St001198
(load all 27 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 91%●distinct values known / distinct values provided: 50%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => ? = 1 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? = 2 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 1 + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
[3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[4,4,2]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[4,4,1,1]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 2 = 1 + 1
[5,4,1,1]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[5,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[5,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[4,4,3]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[4,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[3,3,3,2]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[5,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[5,3,3,1]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
[5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[5,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[4,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[4,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[5,3,3,2]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
[4,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[4,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[5,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001206
(load all 27 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 91%●distinct values known / distinct values provided: 50%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => ? = 1 + 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => ? = 2 + 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2 = 1 + 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
[3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[4,4,2]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[4,4,1,1]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => 2 = 1 + 1
[5,4,1,1]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[5,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[5,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[4,4,3]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[4,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[3,3,3,2]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[5,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[5,3,3,1]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
[5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => 2 = 1 + 1
[5,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[4,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[4,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[5,3,3,2]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
[4,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[4,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[5,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
Description
The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$.
Matching statistic: St001498
(load all 14 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 91%●distinct values known / distinct values provided: 50%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => ? = 1 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 2 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 0 = 1 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 0 = 1 - 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 0 = 1 - 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => 0 = 1 - 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 - 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => 0 = 1 - 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => 0 = 1 - 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 0 = 1 - 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 0 = 1 - 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
 => 0 = 1 - 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => 0 = 1 - 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
 => 0 = 1 - 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 0 = 1 - 1
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
 => 0 = 1 - 1
[5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
 => 0 = 1 - 1
[2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => 0 = 1 - 1
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
 => 0 = 1 - 1
[4,4,2]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[4,4,1,1]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => 0 = 1 - 1
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 - 1
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
 => 0 = 1 - 1
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
 => 0 = 1 - 1
[3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
 => 0 = 1 - 1
[3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
 => 0 = 1 - 1
[5,4,1,1]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[5,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
 => 0 = 1 - 1
[5,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
 => 0 = 1 - 1
[4,4,3]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => 0 = 1 - 1
[4,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
 => 0 = 1 - 1
[3,3,3,2]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => 0 = 1 - 1
[5,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[5,3,3,1]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => 0 = 1 - 1
[5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => 0 = 1 - 1
[5,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[4,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => 0 = 1 - 1
[4,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
 => 0 = 1 - 1
[5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[5,3,3,2]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[4,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => 0 = 1 - 1
[4,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => 0 = 1 - 1
[5,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [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]
 => ? = 2 - 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: St000781
(load all 2 compositions to match this statistic)
Mapping
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 87%●distinct values known / distinct values provided: 50%
Values
[1]
 => [1]
 => [1]
 => []
 => ? = 1
[2,1]
 => [1,1,1]
 => [3]
 => []
 => ? = 2
[3,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? = 1
[2,1,1]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
[3,2]
 => [1,1,1,1,1]
 => [5]
 => []
 => ? = 1
[2,2,1]
 => [2,2,1]
 => [3,2]
 => [2]
 => 1
[4,2]
 => [2,1,1,1,1]
 => [5,1]
 => [1]
 => 1
[4,1,1]
 => [2,2,1,1]
 => [4,2]
 => [2]
 => 1
[3,2,1]
 => [1,1,1,1,1,1]
 => [6]
 => []
 => ? = 2
[3,1,1,1]
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 1
[2,2,1,1]
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 1
[4,3]
 => [2,2,1,1,1]
 => [5,2]
 => [2]
 => 1
[3,3,1]
 => [2,1,1,1,1,1]
 => [6,1]
 => [1]
 => 1
[3,2,2]
 => [3,1,1,1,1]
 => [5,1,1]
 => [1,1]
 => 1
[2,2,2,1]
 => [3,2,2]
 => [3,3,1]
 => [3,1]
 => 1
[5,3]
 => [2,2,2,1,1]
 => [5,3]
 => [3]
 => 1
[5,1,1,1]
 => [2,2,2,2]
 => [4,4]
 => [4]
 => 1
[4,2,1,1]
 => [3,1,1,1,1,1]
 => [6,1,1]
 => [1,1]
 => 1
[4,1,1,1,1]
 => [5,2,1]
 => [3,2,1,1,1]
 => [2,1,1,1]
 => 1
[3,3,2]
 => [2,2,1,1,1,1]
 => [6,2]
 => [2]
 => 1
[2,2,2,1,1]
 => [5,3]
 => [2,2,2,1,1]
 => [2,2,1,1]
 => 1
[5,4]
 => [2,2,2,2,1]
 => [5,4]
 => [4]
 => 1
[5,2,2]
 => [3,2,1,1,1,1]
 => [6,2,1]
 => [2,1]
 => 1
[4,3,1,1]
 => [2,1,1,1,1,1,1,1]
 => [8,1]
 => [1]
 => 1
[4,2,2,1]
 => [2,2,1,1,1,1,1]
 => [7,2]
 => [2]
 => 1
[3,3,1,1,1]
 => [5,2,1,1]
 => [4,2,1,1,1]
 => [2,1,1,1]
 => 1
[2,2,2,2,1]
 => [3,3,3]
 => [3,3,3]
 => [3,3]
 => 1
[5,3,1,1]
 => [2,2,1,1,1,1,1,1]
 => [8,2]
 => [2]
 => 1
[4,4,2]
 => [3,1,1,1,1,1,1,1]
 => [8,1,1]
 => [1,1]
 => 1
[4,4,1,1]
 => [3,2,1,1,1,1,1]
 => [7,2,1]
 => [2,1]
 => 1
[4,3,3]
 => [4,1,1,1,1,1,1]
 => [7,1,1,1]
 => [1,1,1]
 => 1
[4,3,2,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [10]
 => []
 => ? = 2
[4,2,2,2]
 => [3,2,2,1,1,1]
 => [6,3,1]
 => [3,1]
 => 1
[4,2,2,1,1]
 => [5,2,1,1,1]
 => [5,2,1,1,1]
 => [2,1,1,1]
 => 1
[3,3,3,1]
 => [2,2,2,2,1,1]
 => [6,4]
 => [4]
 => 1
[3,3,2,2]
 => [4,2,2,1,1]
 => [5,3,1,1]
 => [3,1,1]
 => 1
[5,4,1,1]
 => [2,2,2,1,1,1,1,1]
 => [8,3]
 => [3]
 => 1
[5,3,1,1,1]
 => [3,2,2,1,1,1,1]
 => [7,3,1]
 => [3,1]
 => 1
[5,2,2,1,1]
 => [4,2,2,1,1,1]
 => [6,3,1,1]
 => [3,1,1]
 => 1
[4,4,3]
 => [3,3,1,1,1,1,1]
 => [7,2,2]
 => [2,2]
 => 1
[4,2,2,2,1]
 => [3,3,3,1,1]
 => [5,3,3]
 => [3,3]
 => 1
[3,3,3,2]
 => [2,2,2,2,2,1]
 => [6,5]
 => [5]
 => 1
[5,4,1,1,1]
 => [2,2,2,2,1,1,1,1]
 => [8,4]
 => [4]
 => 1
[5,3,3,1]
 => [2,1,1,1,1,1,1,1,1,1,1]
 => [11,1]
 => [1]
 => 1
[5,3,2,2]
 => [3,1,1,1,1,1,1,1,1,1]
 => [10,1,1]
 => [1,1]
 => 1
[5,2,2,2,1]
 => [2,2,2,2,2,1,1]
 => [7,5]
 => [5]
 => 1
[4,4,2,1,1]
 => [4,2,1,1,1,1,1,1]
 => [8,2,1,1]
 => [2,1,1]
 => 1
[4,3,3,1,1]
 => [4,2,2,1,1,1,1]
 => [7,3,1,1]
 => [3,1,1]
 => 1
[5,4,2,2]
 => [2,1,1,1,1,1,1,1,1,1,1,1]
 => [12,1]
 => [1]
 => 1
[5,3,3,2]
 => [2,2,1,1,1,1,1,1,1,1,1]
 => [11,2]
 => [2]
 => 1
[4,4,3,1,1]
 => [3,2,2,1,1,1,1,1,1]
 => [9,3,1]
 => [3,1]
 => 1
[4,4,2,2,1]
 => [3,3,2,1,1,1,1,1]
 => [8,3,2]
 => [3,2]
 => 1
[5,4,2,2,1]
 => [2,2,1,1,1,1,1,1,1,1,1,1]
 => [12,2]
 => [2]
 => 1
[5,4,3,2,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [15]
 => []
 => ? = 2
Description
The number of proper colouring schemes of a Ferrers diagram. A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic is the number of distinct such integer partitions that occur.
Matching statistic: St001901
(load all 2 compositions to match this statistic)
Mapping
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 87%●distinct values known / distinct values provided: 50%
Values
[1]
 => [1]
 => [1]
 => []
 => ? = 1
[2,1]
 => [1,1,1]
 => [3]
 => []
 => ? = 2
[3,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? = 1
[2,1,1]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 1
[3,2]
 => [1,1,1,1,1]
 => [5]
 => []
 => ? = 1
[2,2,1]
 => [2,2,1]
 => [3,2]
 => [2]
 => 1
[4,2]
 => [2,1,1,1,1]
 => [5,1]
 => [1]
 => 1
[4,1,1]
 => [2,2,1,1]
 => [4,2]
 => [2]
 => 1
[3,2,1]
 => [1,1,1,1,1,1]
 => [6]
 => []
 => ? = 2
[3,1,1,1]
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 1
[2,2,1,1]
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 1
[4,3]
 => [2,2,1,1,1]
 => [5,2]
 => [2]
 => 1
[3,3,1]
 => [2,1,1,1,1,1]
 => [6,1]
 => [1]
 => 1
[3,2,2]
 => [3,1,1,1,1]
 => [5,1,1]
 => [1,1]
 => 1
[2,2,2,1]
 => [3,2,2]
 => [3,3,1]
 => [3,1]
 => 1
[5,3]
 => [2,2,2,1,1]
 => [5,3]
 => [3]
 => 1
[5,1,1,1]
 => [2,2,2,2]
 => [4,4]
 => [4]
 => 1
[4,2,1,1]
 => [3,1,1,1,1,1]
 => [6,1,1]
 => [1,1]
 => 1
[4,1,1,1,1]
 => [5,2,1]
 => [3,2,1,1,1]
 => [2,1,1,1]
 => 1
[3,3,2]
 => [2,2,1,1,1,1]
 => [6,2]
 => [2]
 => 1
[2,2,2,1,1]
 => [5,3]
 => [2,2,2,1,1]
 => [2,2,1,1]
 => 1
[5,4]
 => [2,2,2,2,1]
 => [5,4]
 => [4]
 => 1
[5,2,2]
 => [3,2,1,1,1,1]
 => [6,2,1]
 => [2,1]
 => 1
[4,3,1,1]
 => [2,1,1,1,1,1,1,1]
 => [8,1]
 => [1]
 => 1
[4,2,2,1]
 => [2,2,1,1,1,1,1]
 => [7,2]
 => [2]
 => 1
[3,3,1,1,1]
 => [5,2,1,1]
 => [4,2,1,1,1]
 => [2,1,1,1]
 => 1
[2,2,2,2,1]
 => [3,3,3]
 => [3,3,3]
 => [3,3]
 => 1
[5,3,1,1]
 => [2,2,1,1,1,1,1,1]
 => [8,2]
 => [2]
 => 1
[4,4,2]
 => [3,1,1,1,1,1,1,1]
 => [8,1,1]
 => [1,1]
 => 1
[4,4,1,1]
 => [3,2,1,1,1,1,1]
 => [7,2,1]
 => [2,1]
 => 1
[4,3,3]
 => [4,1,1,1,1,1,1]
 => [7,1,1,1]
 => [1,1,1]
 => 1
[4,3,2,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [10]
 => []
 => ? = 2
[4,2,2,2]
 => [3,2,2,1,1,1]
 => [6,3,1]
 => [3,1]
 => 1
[4,2,2,1,1]
 => [5,2,1,1,1]
 => [5,2,1,1,1]
 => [2,1,1,1]
 => 1
[3,3,3,1]
 => [2,2,2,2,1,1]
 => [6,4]
 => [4]
 => 1
[3,3,2,2]
 => [4,2,2,1,1]
 => [5,3,1,1]
 => [3,1,1]
 => 1
[5,4,1,1]
 => [2,2,2,1,1,1,1,1]
 => [8,3]
 => [3]
 => 1
[5,3,1,1,1]
 => [3,2,2,1,1,1,1]
 => [7,3,1]
 => [3,1]
 => 1
[5,2,2,1,1]
 => [4,2,2,1,1,1]
 => [6,3,1,1]
 => [3,1,1]
 => 1
[4,4,3]
 => [3,3,1,1,1,1,1]
 => [7,2,2]
 => [2,2]
 => 1
[4,2,2,2,1]
 => [3,3,3,1,1]
 => [5,3,3]
 => [3,3]
 => 1
[3,3,3,2]
 => [2,2,2,2,2,1]
 => [6,5]
 => [5]
 => 1
[5,4,1,1,1]
 => [2,2,2,2,1,1,1,1]
 => [8,4]
 => [4]
 => 1
[5,3,3,1]
 => [2,1,1,1,1,1,1,1,1,1,1]
 => [11,1]
 => [1]
 => 1
[5,3,2,2]
 => [3,1,1,1,1,1,1,1,1,1]
 => [10,1,1]
 => [1,1]
 => 1
[5,2,2,2,1]
 => [2,2,2,2,2,1,1]
 => [7,5]
 => [5]
 => 1
[4,4,2,1,1]
 => [4,2,1,1,1,1,1,1]
 => [8,2,1,1]
 => [2,1,1]
 => 1
[4,3,3,1,1]
 => [4,2,2,1,1,1,1]
 => [7,3,1,1]
 => [3,1,1]
 => 1
[5,4,2,2]
 => [2,1,1,1,1,1,1,1,1,1,1,1]
 => [12,1]
 => [1]
 => 1
[5,3,3,2]
 => [2,2,1,1,1,1,1,1,1,1,1]
 => [11,2]
 => [2]
 => 1
[4,4,3,1,1]
 => [3,2,2,1,1,1,1,1,1]
 => [9,3,1]
 => [3,1]
 => 1
[4,4,2,2,1]
 => [3,3,2,1,1,1,1,1]
 => [8,3,2]
 => [3,2]
 => 1
[5,4,2,2,1]
 => [2,2,1,1,1,1,1,1,1,1,1,1]
 => [12,2]
 => [2]
 => 1
[5,4,3,2,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [15]
 => []
 => ? = 2
Description
The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition.
Matching statistic: St000264
(load all 6 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 57%●distinct values known / distinct values provided: 50%
Values
[1]
 => [1,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ? = 1 + 2
[2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ? = 2 + 2
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 1 + 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 2 + 2
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 1 + 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => 3 = 1 + 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => 3 = 1 + 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 1 + 2
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [2,3,7,5,1,4,6] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [4,3,1,5,7,2,6] => ([(0,6),(1,2),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 1 + 2
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [3,1,4,7,6,2,5] => ([(0,6),(1,2),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 1 + 2
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [5,1,4,2,6,7,3] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
[5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,3,4,7,1,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ? = 1 + 2
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [2,5,4,1,7,3,6] => ([(0,6),(1,2),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => 3 = 1 + 2
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 1 + 2
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 1 + 2
[3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [3,1,6,5,2,7,4] => ([(0,6),(1,2),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => 3 = 1 + 2
[2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ? = 1 + 2
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [7,3,1,5,2,4,6] => ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => 3 = 1 + 2
[4,4,2]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [2,6,4,1,3,7,5] => ([(0,4),(1,3),(2,5),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
[4,4,1,1]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [4,3,1,6,2,7,5] => ([(0,3),(1,3),(1,4),(2,5),(2,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
[4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [2,3,7,1,6,4,5] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => 3 = 1 + 2
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2 + 2
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [2,4,1,7,6,3,5] => ([(0,3),(1,3),(1,4),(2,5),(2,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [7,1,4,2,6,3,5] => ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => 3 = 1 + 2
[3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [5,3,1,2,6,7,4] => ([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => 3 = 1 + 2
[3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,6,1,5,3,7,4] => ([(0,4),(1,3),(2,5),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
[5,4,1,1]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [4,3,1,7,2,5,6] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5)],7)
 => 3 = 1 + 2
[5,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [3,1,7,5,2,4,6] => ([(0,6),(1,2),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
[5,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [5,1,4,2,7,3,6] => ([(0,6),(1,2),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 3 = 1 + 2
[4,4,3]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ? = 1 + 2
[4,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [4,1,2,7,6,3,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5)],7)
 => 3 = 1 + 2
[3,3,3,2]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,5,1,3,6,7,4] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ? = 1 + 2
[5,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [3,1,4,7,2,5,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ? = 1 + 2
[5,3,3,1]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [5,3,1,2,7,4,6] => ([(0,4),(1,4),(1,6),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => 3 = 1 + 2
[5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,7,1,5,3,4,6] => ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => 3 = 1 + 2
[5,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [4,1,2,5,7,3,6] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ? = 1 + 2
[4,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [6,1,4,2,3,7,5] => ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => 3 = 1 + 2
[4,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [3,1,7,2,6,4,5] => ([(0,4),(1,4),(1,6),(2,5),(2,6),(3,5),(3,6),(5,6)],7)
 => 3 = 1 + 2
[5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,4,1,7,3,5,6] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => ? = 1 + 2
[5,3,3,2]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,5,1,3,7,4,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ? = 1 + 2
[4,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ? = 1 + 2
[4,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => ? = 1 + 2
[5,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ? = 1 + 2
[5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 2 + 2
Description
The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.
Matching statistic: St000772
(load all 3 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 50%
Values
[1]
 => [1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [2,1,3] => ([(1,2)],3)
 => ? = 2
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ? = 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? = 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ? = 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? = 2
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? = 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 1
[2,2,2,1]
 => [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
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [5,1,2,6,3,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [3,1,2,4,5,6] => ([(3,5),(4,5)],6)
 => ? = 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? = 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
 => ? = 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 1
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 1
[5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [5,1,6,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => 1
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ? = 1
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? = 1
[3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
[2,2,2,2,1]
 => [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)
 => 1
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [3,1,2,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
 => ? = 1
[4,4,2]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [4,5,1,2,3,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 1
[4,4,1,1]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 1
[4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [4,5,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => 1
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ? = 2
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => 1
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [2,3,1,5,6,4] => ([(0,5),(1,5),(2,4),(3,4)],6)
 => ? = 1
[3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => 1
[3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [3,4,5,1,2,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 1
[5,4,1,1]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[5,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6)
 => ? = 1
[5,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [2,1,3,5,6,4] => ([(1,2),(3,5),(4,5)],6)
 => ? = 1
[4,4,3]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [4,5,1,2,6,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 1
[4,2,2,2,1]
 => [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
[3,3,3,2]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => 1
[5,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [2,1,6,3,4,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ? = 1
[5,3,3,1]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [3,1,5,6,2,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => 1
[5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [3,1,4,5,2,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 1
[5,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [2,1,4,5,6,3] => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ? = 1
[4,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ? = 1
[4,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => 1
[5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [3,1,4,2,5,6] => ([(2,5),(3,4),(4,5)],6)
 => ? = 1
[5,3,3,2]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[4,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[4,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
 => ? = 1
[5,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
 => ? = 1
[5,4,3,2,1]
 => [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)
 => ? = 2
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: St001199
(load all 5 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 50%
Values
[1]
 => [1,0,1,0]
 => [2,1] => [1,1,0,0]
 => ? = 1
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => [1,1,1,0,0,0]
 => ? = 2
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => ? = 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => ? = 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 2
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [6,2,3,4,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [5,2,3,4,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => 1
[2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [3,4,5,2,6,1] => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 1
[5,4]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[5,2,2]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1
[3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,0,0,0]
 => [4,5,2,3,6,1] => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 1
[2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [3,4,5,6,2,1] => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 1
[5,3,1,1]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => [6,4,2,3,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[4,4,2]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [5,6,3,1,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1
[4,4,1,1]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => [5,6,2,3,1,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1
[4,3,3]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [5,4,6,1,2,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 1
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[4,2,2,2]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,1,2] => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 1
[4,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [5,3,4,2,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 1
[3,3,3,1]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => [4,5,6,2,1,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 1
[3,3,2,2]
 => [1,1,0,0,1,1,0,1,1,0,0,0]
 => [4,5,3,6,1,2] => [1,1,1,1,0,1,0,0,1,0,0,0]
 => 1
[5,4,1,1]
 => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [6,5,2,3,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[5,3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [6,4,2,3,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[5,2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [6,3,4,2,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[4,4,3]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [5,6,4,1,2,3] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1
[4,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [5,3,4,6,2,1] => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 1
[3,3,3,2]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,1,2] => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 1
[5,4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [6,5,2,3,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[5,3,3,1]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [6,4,5,2,1,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[5,3,2,2]
 => [1,1,0,0,1,1,0,1,0,0,1,0]
 => [6,4,3,5,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[5,2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [6,3,4,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[4,4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,1,0,0]
 => [5,6,3,2,4,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1
[4,3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [5,4,6,2,3,1] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 1
[5,4,2,2]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[5,3,3,2]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[4,4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [5,6,4,2,3,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1
[4,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,2,1] => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1
[5,4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[5,4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
The following 21 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000664The number of right ropes of a permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000842The breadth of a permutation. St001820The size of the image of the pop stack sorting operator. St001162The minimum jump of a permutation. St001399The distinguishing number of a poset. St000633The size of the automorphism group of a poset. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000542The number of left-to-right-minima of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000338The number of pixed points of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000990The first ascent of a permutation. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St000902 The minimal number of repetitions of an integer composition. St001423The number of distinct cubes in a binary word. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.