Your data matches 10 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000454
(load all 4 compositions to match this statistic)
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1] => ([],1)
 => 0
[2,1] => [2,1] => [1] => ([],1)
 => 0
[1,2,3] => [1,2,3] => [1,2] => ([],2)
 => 0
[1,3,2] => [1,3,2] => [1,2] => ([],2)
 => 0
[2,1,3] => [2,1,3] => [2,1] => ([(0,1)],2)
 => 1
[2,3,1] => [3,2,1] => [2,1] => ([(0,1)],2)
 => 1
[3,1,2] => [3,2,1] => [2,1] => ([(0,1)],2)
 => 1
[3,2,1] => [3,2,1] => [2,1] => ([(0,1)],2)
 => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3] => ([],3)
 => 0
[1,2,4,3] => [1,2,4,3] => [1,2,3] => ([],3)
 => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2] => ([(1,2)],3)
 => 1
[1,3,4,2] => [1,4,3,2] => [1,3,2] => ([(1,2)],3)
 => 1
[1,4,2,3] => [1,4,3,2] => [1,3,2] => ([(1,2)],3)
 => 1
[1,4,3,2] => [1,4,3,2] => [1,3,2] => ([(1,2)],3)
 => 1
[2,1,3,4] => [2,1,3,4] => [2,1,3] => ([(1,2)],3)
 => 1
[2,1,4,3] => [2,1,4,3] => [2,1,3] => ([(1,2)],3)
 => 1
[2,3,1,4] => [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[2,4,3,1] => [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,1,2,4] => [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,2,1,4] => [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,4,1,2] => [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[3,4,2,1] => [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[4,2,1,3] => [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[4,2,3,1] => [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[4,3,1,2] => [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[4,3,2,1] => [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4] => ([],4)
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4] => ([],4)
 => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3] => ([(2,3)],4)
 => 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,3] => ([(2,3)],4)
 => 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,4,3] => ([(2,3)],4)
 => 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,3] => ([(2,3)],4)
 => 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4] => ([(2,3)],4)
 => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,4] => ([(2,3)],4)
 => 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,3,5,4,2] => [1,5,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,4,5,2,3] => [1,5,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,4,5,3,2] => [1,5,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,5,3,2,4] => [1,5,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,5,3,4,2] => [1,5,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,5,4,2,3] => [1,5,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,5,4,3,2] => [1,5,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4] => ([(2,3)],4)
 => 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,4] => ([(2,3)],4)
 => 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
[2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
[2,1,5,3,4] => [2,1,5,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => 1
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000771
(load all 4 compositions to match this statistic)
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000771: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 86%
Values
[1,2] => [1,2] => ([],2)
 => ? = 0 + 1
[2,1] => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[1,2,3] => [1,2,3] => ([],3)
 => ? = 0 + 1
[1,3,2] => [1,3,2] => ([(1,2)],3)
 => ? = 0 + 1
[2,1,3] => [2,1,3] => ([(1,2)],3)
 => ? = 1 + 1
[2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => ([],4)
 => ? = 0 + 1
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => ? = 0 + 1
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ? = 1 + 1
[1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => ? = 1 + 1
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? = 1 + 1
[2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[2,4,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,4,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,2,1,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,3,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ? = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ? = 1 + 1
[1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => ? = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? = 1 + 1
[1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,3,5,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,4,5,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,4,5,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,5,3,2,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,5,4,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => ? = 1 + 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? = 1 + 1
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ? = 1 + 1
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[2,1,5,3,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[2,4,1,3,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 2 + 1
[2,4,3,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[2,4,5,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[2,5,3,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[2,5,4,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[3,1,2,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[3,1,2,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[3,2,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[3,4,1,2,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[3,4,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[3,4,5,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[3,4,5,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[3,5,1,4,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[3,5,2,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[3,5,4,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[3,5,4,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[4,2,1,3,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[4,2,5,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[4,2,5,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[4,3,5,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[4,3,5,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[4,5,1,2,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[4,5,1,3,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[4,5,2,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[4,5,2,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[4,5,3,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[4,5,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,2,3,1,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,2,3,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,2,4,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,2,4,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,3,1,2,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,3,1,4,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,3,2,1,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,3,2,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,3,4,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,3,4,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,4,1,2,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,4,1,3,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,4,2,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,4,2,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,4,3,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[2,3,4,5,6,1] => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[2,4,6,5,3,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
[2,5,6,3,4,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
[2,5,6,4,3,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 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: St000772
(load all 4 compositions to match this statistic)
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 86%
Values
[1,2] => [1,2] => ([],2)
 => ? = 0 + 1
[2,1] => [2,1] => ([(0,1)],2)
 => 1 = 0 + 1
[1,2,3] => [1,2,3] => ([],3)
 => ? = 0 + 1
[1,3,2] => [1,3,2] => ([(1,2)],3)
 => ? = 0 + 1
[2,1,3] => [2,1,3] => ([(1,2)],3)
 => ? = 1 + 1
[2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => ([],4)
 => ? = 0 + 1
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => ? = 0 + 1
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ? = 1 + 1
[1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => ? = 1 + 1
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? = 1 + 1
[2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[2,4,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,4,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,2,1,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,3,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ? = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ? = 1 + 1
[1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => ? = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? = 1 + 1
[1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,3,5,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,4,5,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,4,5,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,5,3,2,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,5,4,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => ? = 1 + 1
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? = 1 + 1
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ? = 1 + 1
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[2,1,5,3,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[2,4,1,3,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 2 + 1
[2,4,3,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[2,4,5,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[2,5,3,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[2,5,4,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[3,1,2,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[3,1,2,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[3,2,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[3,4,1,2,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[3,4,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[3,4,5,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[3,4,5,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[3,5,1,4,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[3,5,2,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[3,5,4,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[3,5,4,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[4,2,1,3,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[4,2,5,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[4,2,5,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[4,3,5,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[4,3,5,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[4,5,1,2,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[4,5,1,3,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[4,5,2,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[4,5,2,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[4,5,3,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[4,5,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,2,3,1,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,2,3,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,2,4,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,2,4,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,3,1,2,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,3,1,4,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,3,2,1,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,3,2,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,3,4,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,3,4,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,4,1,2,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,4,1,3,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,4,2,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,4,2,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,4,3,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 3 + 1
[2,3,4,5,6,1] => [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 3 = 2 + 1
[2,4,6,5,3,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
[2,5,6,3,4,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 1
[2,5,6,4,3,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 5 = 4 + 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: St001645
(load all 6 compositions to match this statistic)
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 86%
Values
[1,2] => [1,2] => ([],2)
 => ? = 0 + 2
[2,1] => [2,1] => ([(0,1)],2)
 => 2 = 0 + 2
[1,2,3] => [1,2,3] => ([],3)
 => ? = 0 + 2
[1,3,2] => [1,3,2] => ([(1,2)],3)
 => ? = 0 + 2
[2,1,3] => [2,1,3] => ([(1,2)],3)
 => ? = 1 + 2
[2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 1 + 2
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 1 + 2
[3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 1 + 2
[1,2,3,4] => [1,2,3,4] => ([],4)
 => ? = 0 + 2
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => ? = 0 + 2
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ? = 1 + 2
[1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 1 + 2
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => ? = 1 + 2
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? = 1 + 2
[2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 2
[2,4,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 2
[3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 2
[3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[3,4,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[4,2,1,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[4,3,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 0 + 2
[1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ? = 0 + 2
[1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ? = 1 + 2
[1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[1,2,5,3,4] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[1,2,5,4,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => ? = 1 + 2
[1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? = 1 + 2
[1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[1,3,5,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[1,4,3,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[1,4,5,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[1,4,5,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[1,5,3,2,4] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[1,5,3,4,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[1,5,4,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[1,5,4,3,2] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => ? = 1 + 2
[2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? = 1 + 2
[2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ? = 1 + 2
[2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[2,1,5,3,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[2,1,5,4,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[2,4,1,3,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 2 + 2
[2,4,3,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 2
[2,4,5,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[2,5,3,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[2,5,4,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[3,1,2,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[3,1,2,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[3,2,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[3,2,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[3,4,1,2,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 2
[3,4,2,1,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 2
[3,4,5,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[3,4,5,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[3,5,1,4,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[3,5,2,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[3,5,4,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[3,5,4,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[4,2,1,3,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 2
[4,2,5,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[4,2,5,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[4,3,5,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[4,3,5,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[4,5,1,2,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[4,5,1,3,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[4,5,2,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[4,5,2,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[4,5,3,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[4,5,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[5,2,3,1,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[5,2,3,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[5,2,4,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[5,2,4,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[5,3,1,2,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[5,3,1,4,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[5,3,2,1,4] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[5,3,2,4,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[5,3,4,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[5,3,4,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[5,4,1,2,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[5,4,1,3,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[5,4,2,1,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[5,4,2,3,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[5,4,3,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[2,4,6,5,3,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[2,5,6,3,4,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[2,5,6,4,3,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[2,6,4,3,5,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
Description
The pebbling number of a connected graph.
Matching statistic: St001879
(load all 2 compositions to match this statistic)
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 71%
Values
[1,2] => [1,2] => [2,1] => ([],2)
 => ? = 0 + 1
[2,1] => [2,1] => [1,2] => ([(0,1)],2)
 => ? = 0 + 1
[1,2,3] => [1,2,3] => [2,3,1] => ([(1,2)],3)
 => ? = 0 + 1
[1,3,2] => [1,3,2] => [2,1,3] => ([(0,2),(1,2)],3)
 => ? = 0 + 1
[2,1,3] => [2,1,3] => [3,2,1] => ([],3)
 => ? = 1 + 1
[2,3,1] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 1 + 1
[3,1,2] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 1 + 1
[3,2,1] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ? = 0 + 1
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 1
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ? = 1 + 1
[1,3,4,2] => [1,4,3,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ? = 1 + 1
[1,4,2,3] => [1,4,3,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ? = 1 + 1
[1,4,3,2] => [1,4,3,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ? = 1 + 1
[2,1,3,4] => [2,1,3,4] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ? = 1 + 1
[2,3,1,4] => [3,2,1,4] => [4,3,2,1] => ([],4)
 => ? = 2 + 1
[2,4,3,1] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[3,1,2,4] => [3,2,1,4] => [4,3,2,1] => ([],4)
 => ? = 2 + 1
[3,2,1,4] => [3,2,1,4] => [4,3,2,1] => ([],4)
 => ? = 2 + 1
[3,4,1,2] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[3,4,2,1] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[4,2,1,3] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[4,2,3,1] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[4,3,1,2] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ? = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
 => ? = 1 + 1
[1,2,4,5,3] => [1,2,5,4,3] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,3,4,2,5] => [1,4,3,2,5] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ? = 2 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 2 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ? = 2 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ? = 2 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 2 + 1
[1,4,5,3,2] => [1,5,4,3,2] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 2 + 1
[1,5,3,2,4] => [1,5,4,3,2] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 2 + 1
[1,5,3,4,2] => [1,5,4,3,2] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 2 + 1
[1,5,4,2,3] => [1,5,4,3,2] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 2 + 1
[1,5,4,3,2] => [1,5,4,3,2] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 2 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => ? = 1 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 1 + 1
[2,1,4,5,3] => [2,1,5,4,3] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ? = 1 + 1
[2,1,5,3,4] => [2,1,5,4,3] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ? = 1 + 1
[2,1,5,4,3] => [2,1,5,4,3] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ? = 1 + 1
[2,3,1,4,5] => [3,2,1,4,5] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ? = 2 + 1
[2,3,1,5,4] => [3,2,1,5,4] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 2 + 1
[2,4,1,3,5] => [3,4,1,2,5] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => ? = 2 + 1
[2,4,3,1,5] => [4,3,2,1,5] => [5,4,3,2,1] => ([],5)
 => ? = 3 + 1
[2,4,5,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[2,5,3,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[2,5,4,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[3,1,2,4,5] => [3,2,1,4,5] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ? = 2 + 1
[3,1,2,5,4] => [3,2,1,5,4] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 2 + 1
[3,2,1,4,5] => [3,2,1,4,5] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ? = 2 + 1
[3,2,1,5,4] => [3,2,1,5,4] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 2 + 1
[3,4,1,2,5] => [4,3,2,1,5] => [5,4,3,2,1] => ([],5)
 => ? = 3 + 1
[3,4,2,1,5] => [4,3,2,1,5] => [5,4,3,2,1] => ([],5)
 => ? = 3 + 1
[3,4,5,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[3,4,5,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[3,5,1,4,2] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[3,5,2,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[3,5,4,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[3,5,4,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[4,2,5,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[4,2,5,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[4,3,5,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[4,3,5,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[4,5,1,2,3] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[4,5,1,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[4,5,2,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[4,5,2,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[4,5,3,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[4,5,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[5,2,3,1,4] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[5,2,3,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[5,2,4,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[5,2,4,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[5,3,1,2,4] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[5,3,1,4,2] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[5,3,2,1,4] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[5,3,2,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[5,3,4,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[5,3,4,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[5,4,1,2,3] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[5,4,1,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[5,4,2,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[5,4,2,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[5,4,3,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[2,4,6,5,3,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
[2,5,6,3,4,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
[2,5,6,4,3,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
[2,6,4,3,5,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
[2,6,4,5,3,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001880
(load all 2 compositions to match this statistic)
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 71%
Values
[1,2] => [1,2] => [2,1] => ([],2)
 => ? = 0 + 2
[2,1] => [2,1] => [1,2] => ([(0,1)],2)
 => ? = 0 + 2
[1,2,3] => [1,2,3] => [2,3,1] => ([(1,2)],3)
 => ? = 0 + 2
[1,3,2] => [1,3,2] => [2,1,3] => ([(0,2),(1,2)],3)
 => ? = 0 + 2
[2,1,3] => [2,1,3] => [3,2,1] => ([],3)
 => ? = 1 + 2
[2,3,1] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[3,1,2] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[3,2,1] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3 = 1 + 2
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ? = 0 + 2
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ? = 0 + 2
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ? = 1 + 2
[1,3,4,2] => [1,4,3,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ? = 1 + 2
[1,4,2,3] => [1,4,3,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ? = 1 + 2
[1,4,3,2] => [1,4,3,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ? = 1 + 2
[2,1,3,4] => [2,1,3,4] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ? = 1 + 2
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ? = 1 + 2
[2,3,1,4] => [3,2,1,4] => [4,3,2,1] => ([],4)
 => ? = 2 + 2
[2,4,3,1] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[3,1,2,4] => [3,2,1,4] => [4,3,2,1] => ([],4)
 => ? = 2 + 2
[3,2,1,4] => [3,2,1,4] => [4,3,2,1] => ([],4)
 => ? = 2 + 2
[3,4,1,2] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[3,4,2,1] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[4,2,1,3] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[4,2,3,1] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[4,3,1,2] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ? = 0 + 2
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 0 + 2
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
 => ? = 1 + 2
[1,2,4,5,3] => [1,2,5,4,3] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 2
[1,2,5,3,4] => [1,2,5,4,3] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 2
[1,2,5,4,3] => [1,2,5,4,3] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 1 + 2
[1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[1,3,4,2,5] => [1,4,3,2,5] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ? = 2 + 2
[1,3,5,4,2] => [1,5,4,3,2] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 2 + 2
[1,4,2,3,5] => [1,4,3,2,5] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ? = 2 + 2
[1,4,3,2,5] => [1,4,3,2,5] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ? = 2 + 2
[1,4,5,2,3] => [1,5,4,3,2] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 2 + 2
[1,4,5,3,2] => [1,5,4,3,2] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 2 + 2
[1,5,3,2,4] => [1,5,4,3,2] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 2 + 2
[1,5,3,4,2] => [1,5,4,3,2] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 2 + 2
[1,5,4,2,3] => [1,5,4,3,2] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 2 + 2
[1,5,4,3,2] => [1,5,4,3,2] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 2 + 2
[2,1,3,4,5] => [2,1,3,4,5] => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => ? = 1 + 2
[2,1,3,5,4] => [2,1,3,5,4] => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 1 + 2
[2,1,4,3,5] => [2,1,4,3,5] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ? = 1 + 2
[2,1,4,5,3] => [2,1,5,4,3] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ? = 1 + 2
[2,1,5,3,4] => [2,1,5,4,3] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ? = 1 + 2
[2,1,5,4,3] => [2,1,5,4,3] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ? = 1 + 2
[2,3,1,4,5] => [3,2,1,4,5] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ? = 2 + 2
[2,3,1,5,4] => [3,2,1,5,4] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 2 + 2
[2,4,1,3,5] => [3,4,1,2,5] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => ? = 2 + 2
[2,4,3,1,5] => [4,3,2,1,5] => [5,4,3,2,1] => ([],5)
 => ? = 3 + 2
[2,4,5,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[2,5,3,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[2,5,4,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[3,1,2,4,5] => [3,2,1,4,5] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ? = 2 + 2
[3,1,2,5,4] => [3,2,1,5,4] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 2 + 2
[3,2,1,4,5] => [3,2,1,4,5] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ? = 2 + 2
[3,2,1,5,4] => [3,2,1,5,4] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 2 + 2
[3,4,1,2,5] => [4,3,2,1,5] => [5,4,3,2,1] => ([],5)
 => ? = 3 + 2
[3,4,2,1,5] => [4,3,2,1,5] => [5,4,3,2,1] => ([],5)
 => ? = 3 + 2
[3,4,5,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[3,4,5,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[3,5,1,4,2] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[3,5,2,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[3,5,4,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[3,5,4,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[4,2,5,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[4,2,5,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[4,3,5,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[4,3,5,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[4,5,1,2,3] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[4,5,1,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[4,5,2,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[4,5,2,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[4,5,3,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[4,5,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[5,2,3,1,4] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[5,2,3,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[5,2,4,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[5,2,4,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[5,3,1,2,4] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[5,3,1,4,2] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[5,3,2,1,4] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[5,3,2,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[5,3,4,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[5,3,4,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[5,4,1,2,3] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[5,4,1,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[5,4,2,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[5,4,2,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[5,4,3,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[2,4,6,5,3,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[2,5,6,3,4,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[2,5,6,4,3,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[2,6,4,3,5,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[2,6,4,5,3,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001207
(load all 2 compositions to match this statistic)
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 57%
Values
[1,2] => [1,2] => [1,2] => [1] => ? = 0
[2,1] => [2,1] => [2,1] => [1] => ? = 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,2] => 0
[1,3,2] => [1,3,2] => [1,3,2] => [1,2] => 0
[2,1,3] => [2,1,3] => [2,1,3] => [2,1] => 1
[2,3,1] => [3,2,1] => [2,3,1] => [2,1] => 1
[3,1,2] => [3,2,1] => [2,3,1] => [2,1] => 1
[3,2,1] => [3,2,1] => [2,3,1] => [2,1] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,3] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2] => 1
[1,3,4,2] => [1,4,3,2] => [1,3,4,2] => [1,3,2] => 1
[1,4,2,3] => [1,4,3,2] => [1,3,4,2] => [1,3,2] => 1
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => [1,3,2] => 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,3] => 1
[2,3,1,4] => [3,2,1,4] => [2,3,1,4] => [2,3,1] => 2
[2,4,3,1] => [4,3,2,1] => [3,2,4,1] => [3,2,1] => 2
[3,1,2,4] => [3,2,1,4] => [2,3,1,4] => [2,3,1] => 2
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => [2,3,1] => 2
[3,4,1,2] => [4,3,2,1] => [3,2,4,1] => [3,2,1] => 2
[3,4,2,1] => [4,3,2,1] => [3,2,4,1] => [3,2,1] => 2
[4,2,1,3] => [4,3,2,1] => [3,2,4,1] => [3,2,1] => 2
[4,2,3,1] => [4,3,2,1] => [3,2,4,1] => [3,2,1] => 2
[4,3,1,2] => [4,3,2,1] => [3,2,4,1] => [3,2,1] => 2
[4,3,2,1] => [4,3,2,1] => [3,2,4,1] => [3,2,1] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3] => 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,3] => 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,3] => 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,3] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,4] => 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2] => 2
[1,3,5,4,2] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,2] => 2
[1,4,2,3,5] => [1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2] => 2
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2] => 2
[1,4,5,2,3] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,2] => 2
[1,4,5,3,2] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,2] => 2
[1,5,3,2,4] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,2] => 2
[1,5,3,4,2] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,2] => 2
[1,5,4,2,3] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,2] => 2
[1,5,4,3,2] => [1,5,4,3,2] => [1,4,3,5,2] => [1,4,3,2] => 2
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4] => 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,4] => 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3] => 1
[2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,5,3] => [2,1,4,3] => 1
[2,1,5,3,4] => [2,1,5,4,3] => [2,1,4,5,3] => [2,1,4,3] => 1
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => [2,1,4,3] => 1
[2,3,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4] => 2
[2,3,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => [2,3,1,4] => 2
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5] => ? = 0
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,5] => ? = 0
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4] => ? = 1
[1,2,3,5,6,4] => [1,2,3,6,5,4] => [1,2,3,5,6,4] => [1,2,3,5,4] => ? = 1
[1,2,3,6,4,5] => [1,2,3,6,5,4] => [1,2,3,5,6,4] => [1,2,3,5,4] => ? = 1
[1,2,3,6,5,4] => [1,2,3,6,5,4] => [1,2,3,5,6,4] => [1,2,3,5,4] => ? = 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5] => ? = 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,2,4,3,5] => ? = 1
[1,2,4,5,3,6] => [1,2,5,4,3,6] => [1,2,4,5,3,6] => [1,2,4,5,3] => ? = 2
[1,2,4,6,5,3] => [1,2,6,5,4,3] => [1,2,5,4,6,3] => [1,2,5,4,3] => ? = 2
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [1,2,4,5,3,6] => [1,2,4,5,3] => ? = 2
[1,2,5,4,3,6] => [1,2,5,4,3,6] => [1,2,4,5,3,6] => [1,2,4,5,3] => ? = 2
[1,2,5,6,3,4] => [1,2,6,5,4,3] => [1,2,5,4,6,3] => [1,2,5,4,3] => ? = 2
[1,2,5,6,4,3] => [1,2,6,5,4,3] => [1,2,5,4,6,3] => [1,2,5,4,3] => ? = 2
[1,2,6,4,3,5] => [1,2,6,5,4,3] => [1,2,5,4,6,3] => [1,2,5,4,3] => ? = 2
[1,2,6,4,5,3] => [1,2,6,5,4,3] => [1,2,5,4,6,3] => [1,2,5,4,3] => ? = 2
[1,2,6,5,3,4] => [1,2,6,5,4,3] => [1,2,5,4,6,3] => [1,2,5,4,3] => ? = 2
[1,2,6,5,4,3] => [1,2,6,5,4,3] => [1,2,5,4,6,3] => [1,2,5,4,3] => ? = 2
[1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => [1,3,2,4,5] => ? = 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => [1,3,2,4,5] => ? = 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4] => ? = 1
[1,3,2,5,6,4] => [1,3,2,6,5,4] => [1,3,2,5,6,4] => [1,3,2,5,4] => ? = 1
[1,3,2,6,4,5] => [1,3,2,6,5,4] => [1,3,2,5,6,4] => [1,3,2,5,4] => ? = 1
[1,3,2,6,5,4] => [1,3,2,6,5,4] => [1,3,2,5,6,4] => [1,3,2,5,4] => ? = 1
[1,3,4,2,5,6] => [1,4,3,2,5,6] => [1,3,4,2,5,6] => [1,3,4,2,5] => ? = 2
[1,3,4,2,6,5] => [1,4,3,2,6,5] => [1,3,4,2,6,5] => [1,3,4,2,5] => ? = 2
[1,3,5,2,4,6] => [1,4,5,2,3,6] => [1,4,2,5,3,6] => [1,4,2,5,3] => ? = 2
[1,3,5,4,2,6] => [1,5,4,3,2,6] => [1,4,3,5,2,6] => [1,4,3,5,2] => ? = 3
[1,3,5,6,4,2] => [1,6,5,4,3,2] => [1,4,5,3,6,2] => [1,4,5,3,2] => ? = 3
[1,3,6,4,5,2] => [1,6,5,4,3,2] => [1,4,5,3,6,2] => [1,4,5,3,2] => ? = 3
[1,3,6,5,4,2] => [1,6,5,4,3,2] => [1,4,5,3,6,2] => [1,4,5,3,2] => ? = 3
[1,4,2,3,5,6] => [1,4,3,2,5,6] => [1,3,4,2,5,6] => [1,3,4,2,5] => ? = 2
[1,4,2,3,6,5] => [1,4,3,2,6,5] => [1,3,4,2,6,5] => [1,3,4,2,5] => ? = 2
[1,4,3,2,5,6] => [1,4,3,2,5,6] => [1,3,4,2,5,6] => [1,3,4,2,5] => ? = 2
[1,4,3,2,6,5] => [1,4,3,2,6,5] => [1,3,4,2,6,5] => [1,3,4,2,5] => ? = 2
[1,4,5,2,3,6] => [1,5,4,3,2,6] => [1,4,3,5,2,6] => [1,4,3,5,2] => ? = 3
[1,4,5,3,2,6] => [1,5,4,3,2,6] => [1,4,3,5,2,6] => [1,4,3,5,2] => ? = 3
[1,4,5,6,2,3] => [1,6,5,4,3,2] => [1,4,5,3,6,2] => [1,4,5,3,2] => ? = 3
[1,4,5,6,3,2] => [1,6,5,4,3,2] => [1,4,5,3,6,2] => [1,4,5,3,2] => ? = 3
[1,4,6,2,5,3] => [1,6,5,4,3,2] => [1,4,5,3,6,2] => [1,4,5,3,2] => ? = 3
[1,4,6,3,5,2] => [1,6,5,4,3,2] => [1,4,5,3,6,2] => [1,4,5,3,2] => ? = 3
[1,4,6,5,2,3] => [1,6,5,4,3,2] => [1,4,5,3,6,2] => [1,4,5,3,2] => ? = 3
[1,4,6,5,3,2] => [1,6,5,4,3,2] => [1,4,5,3,6,2] => [1,4,5,3,2] => ? = 3
[1,5,3,2,4,6] => [1,5,4,3,2,6] => [1,4,3,5,2,6] => [1,4,3,5,2] => ? = 3
[1,5,3,4,2,6] => [1,5,4,3,2,6] => [1,4,3,5,2,6] => [1,4,3,5,2] => ? = 3
[1,5,3,6,2,4] => [1,6,5,4,3,2] => [1,4,5,3,6,2] => [1,4,5,3,2] => ? = 3
[1,5,3,6,4,2] => [1,6,5,4,3,2] => [1,4,5,3,6,2] => [1,4,5,3,2] => ? = 3
[1,5,4,2,3,6] => [1,5,4,3,2,6] => [1,4,3,5,2,6] => [1,4,3,5,2] => ? = 3
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St001896
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00168: Signed permutations —Kreweras complement⟶ Signed permutations
St001896: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 57%
Values
[1,2] => [1,2] => [1,2] => [2,-1] => 1 = 0 + 1
[2,1] => [2,1] => [2,1] => [-1,2] => 1 = 0 + 1
[1,2,3] => [1,2,3] => [1,2,3] => [2,3,-1] => 1 = 0 + 1
[1,3,2] => [1,3,2] => [1,3,2] => [2,-1,3] => 1 = 0 + 1
[2,1,3] => [2,1,3] => [2,1,3] => [3,2,-1] => 2 = 1 + 1
[2,3,1] => [3,2,1] => [3,2,1] => [-1,3,2] => 2 = 1 + 1
[3,1,2] => [3,2,1] => [3,2,1] => [-1,3,2] => 2 = 1 + 1
[3,2,1] => [3,2,1] => [3,2,1] => [-1,3,2] => 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [2,3,-1,4] => 1 = 0 + 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [2,4,3,-1] => 2 = 1 + 1
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => [2,-1,4,3] => 2 = 1 + 1
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => [2,-1,4,3] => 2 = 1 + 1
[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [2,-1,4,3] => 2 = 1 + 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [3,2,4,-1] => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [3,2,-1,4] => 2 = 1 + 1
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => [4,3,2,-1] => 3 = 2 + 1
[2,4,3,1] => [4,3,2,1] => [4,3,2,1] => [-1,4,3,2] => 3 = 2 + 1
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => [4,3,2,-1] => 3 = 2 + 1
[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [4,3,2,-1] => 3 = 2 + 1
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => [-1,4,3,2] => 3 = 2 + 1
[3,4,2,1] => [4,3,2,1] => [4,3,2,1] => [-1,4,3,2] => 3 = 2 + 1
[4,2,1,3] => [4,3,2,1] => [4,3,2,1] => [-1,4,3,2] => 3 = 2 + 1
[4,2,3,1] => [4,3,2,1] => [4,3,2,1] => [-1,4,3,2] => 3 = 2 + 1
[4,3,1,2] => [4,3,2,1] => [4,3,2,1] => [-1,4,3,2] => 3 = 2 + 1
[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [-1,4,3,2] => 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => ? = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => ? = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,-1] => ? = 1 + 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => [2,3,-1,5,4] => ? = 1 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => [2,3,-1,5,4] => ? = 1 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,4,3] => [2,3,-1,5,4] => ? = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,-1] => ? = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,-1,5] => ? = 1 + 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [2,5,4,3,-1] => ? = 2 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => [2,-1,5,4,3] => ? = 2 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => [2,5,4,3,-1] => ? = 2 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => [2,5,4,3,-1] => ? = 2 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => [2,-1,5,4,3] => ? = 2 + 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [2,-1,5,4,3] => ? = 2 + 1
[1,5,3,2,4] => [1,5,4,3,2] => [1,5,4,3,2] => [2,-1,5,4,3] => ? = 2 + 1
[1,5,3,4,2] => [1,5,4,3,2] => [1,5,4,3,2] => [2,-1,5,4,3] => ? = 2 + 1
[1,5,4,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => [2,-1,5,4,3] => ? = 2 + 1
[1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [2,-1,5,4,3] => ? = 2 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [3,2,4,5,-1] => ? = 1 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [3,2,4,-1,5] => ? = 1 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [3,2,5,4,-1] => ? = 1 + 1
[2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => [3,2,-1,5,4] => ? = 1 + 1
[2,1,5,3,4] => [2,1,5,4,3] => [2,1,5,4,3] => [3,2,-1,5,4] => ? = 1 + 1
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => [3,2,-1,5,4] => ? = 1 + 1
[2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => [4,3,2,5,-1] => ? = 2 + 1
[2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [4,3,2,-1,5] => ? = 2 + 1
[2,4,1,3,5] => [3,4,1,2,5] => [3,4,1,2,5] => [4,5,2,3,-1] => ? = 2 + 1
[2,4,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => [5,4,3,2,-1] => ? = 3 + 1
[2,4,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => [-1,5,4,3,2] => 4 = 3 + 1
[2,5,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => [-1,5,4,3,2] => 4 = 3 + 1
[2,5,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => [-1,5,4,3,2] => 4 = 3 + 1
[3,1,2,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => [4,3,2,5,-1] => ? = 2 + 1
[3,1,2,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [4,3,2,-1,5] => ? = 2 + 1
[3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => [4,3,2,5,-1] => ? = 2 + 1
[3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [4,3,2,-1,5] => ? = 2 + 1
[3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => [5,4,3,2,-1] => ? = 3 + 1
[3,4,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => [5,4,3,2,-1] => ? = 3 + 1
[3,4,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => [-1,5,4,3,2] => 4 = 3 + 1
[3,4,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [-1,5,4,3,2] => 4 = 3 + 1
[3,5,1,4,2] => [5,4,3,2,1] => [5,4,3,2,1] => [-1,5,4,3,2] => 4 = 3 + 1
[3,5,2,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => [-1,5,4,3,2] => 4 = 3 + 1
[3,5,4,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => [-1,5,4,3,2] => 4 = 3 + 1
[3,5,4,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [-1,5,4,3,2] => 4 = 3 + 1
[4,2,1,3,5] => [4,3,2,1,5] => [4,3,2,1,5] => [5,4,3,2,-1] => ? = 3 + 1
[4,2,3,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => [5,4,3,2,-1] => ? = 3 + 1
[4,2,5,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => [-1,5,4,3,2] => 4 = 3 + 1
[4,2,5,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => [-1,5,4,3,2] => 4 = 3 + 1
[4,3,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => [5,4,3,2,-1] => ? = 3 + 1
[4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => [5,4,3,2,-1] => ? = 3 + 1
[4,3,5,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => [-1,5,4,3,2] => 4 = 3 + 1
[4,3,5,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [-1,5,4,3,2] => 4 = 3 + 1
[4,5,1,2,3] => [5,4,3,2,1] => [5,4,3,2,1] => [-1,5,4,3,2] => 4 = 3 + 1
[4,5,1,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => [-1,5,4,3,2] => 4 = 3 + 1
[4,5,2,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => [-1,5,4,3,2] => 4 = 3 + 1
[4,5,2,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => [-1,5,4,3,2] => 4 = 3 + 1
[4,5,3,1,2] => [5,4,3,2,1] => [5,4,3,2,1] => [-1,5,4,3,2] => 4 = 3 + 1
[4,5,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => [-1,5,4,3,2] => 4 = 3 + 1
[5,2,3,1,4] => [5,4,3,2,1] => [5,4,3,2,1] => [-1,5,4,3,2] => 4 = 3 + 1
[5,2,3,4,1] => [5,4,3,2,1] => [5,4,3,2,1] => [-1,5,4,3,2] => 4 = 3 + 1
[5,2,4,1,3] => [5,4,3,2,1] => [5,4,3,2,1] => [-1,5,4,3,2] => 4 = 3 + 1
[5,2,4,3,1] => [5,4,3,2,1] => [5,4,3,2,1] => [-1,5,4,3,2] => 4 = 3 + 1
[5,3,1,2,4] => [5,4,3,2,1] => [5,4,3,2,1] => [-1,5,4,3,2] => 4 = 3 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [2,3,4,5,6,-1] => ? = 0 + 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [2,3,4,5,-1,6] => ? = 0 + 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [2,3,4,6,5,-1] => ? = 1 + 1
[1,2,3,5,6,4] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => [2,3,4,-1,6,5] => ? = 1 + 1
[1,2,3,6,4,5] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => [2,3,4,-1,6,5] => ? = 1 + 1
[1,2,3,6,5,4] => [1,2,3,6,5,4] => [1,2,3,6,5,4] => [2,3,4,-1,6,5] => ? = 1 + 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => [2,3,5,4,6,-1] => ? = 1 + 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => [2,3,5,4,-1,6] => ? = 1 + 1
[1,2,4,5,3,6] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => [2,3,6,5,4,-1] => ? = 2 + 1
[1,2,4,6,5,3] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => [2,3,-1,6,5,4] => ? = 2 + 1
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => [2,3,6,5,4,-1] => ? = 2 + 1
[1,2,5,4,3,6] => [1,2,5,4,3,6] => [1,2,5,4,3,6] => [2,3,6,5,4,-1] => ? = 2 + 1
Description
The number of right descents of a signed permutations. An index is a right descent if it is a left descent of the inverse signed permutation.
Matching statistic: St001491
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 57%
Values
[1,2] => [1,2] => [2,1] => 0 => ? = 0 + 1
[2,1] => [2,1] => [1,2] => 1 => 1 = 0 + 1
[1,2,3] => [1,2,3] => [3,2,1] => 00 => ? = 0 + 1
[1,3,2] => [1,3,2] => [2,3,1] => 00 => ? = 0 + 1
[2,1,3] => [2,1,3] => [3,1,2] => 00 => ? = 1 + 1
[2,3,1] => [3,2,1] => [1,2,3] => 11 => 2 = 1 + 1
[3,1,2] => [3,2,1] => [1,2,3] => 11 => 2 = 1 + 1
[3,2,1] => [3,2,1] => [1,2,3] => 11 => 2 = 1 + 1
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 000 => ? = 0 + 1
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 000 => ? = 0 + 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 000 => ? = 1 + 1
[1,3,4,2] => [1,4,3,2] => [2,3,4,1] => 000 => ? = 1 + 1
[1,4,2,3] => [1,4,3,2] => [2,3,4,1] => 000 => ? = 1 + 1
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => 000 => ? = 1 + 1
[2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 000 => ? = 1 + 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 000 => ? = 1 + 1
[2,3,1,4] => [3,2,1,4] => [4,1,2,3] => 000 => ? = 2 + 1
[2,4,3,1] => [4,3,2,1] => [1,2,3,4] => 111 => 3 = 2 + 1
[3,1,2,4] => [3,2,1,4] => [4,1,2,3] => 000 => ? = 2 + 1
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => 000 => ? = 2 + 1
[3,4,1,2] => [4,3,2,1] => [1,2,3,4] => 111 => 3 = 2 + 1
[3,4,2,1] => [4,3,2,1] => [1,2,3,4] => 111 => 3 = 2 + 1
[4,2,1,3] => [4,3,2,1] => [1,2,3,4] => 111 => 3 = 2 + 1
[4,2,3,1] => [4,3,2,1] => [1,2,3,4] => 111 => 3 = 2 + 1
[4,3,1,2] => [4,3,2,1] => [1,2,3,4] => 111 => 3 = 2 + 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 111 => 3 = 2 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0000 => ? = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => 0000 => ? = 0 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => 0000 => ? = 1 + 1
[1,2,4,5,3] => [1,2,5,4,3] => [3,4,5,2,1] => 0000 => ? = 1 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [3,4,5,2,1] => 0000 => ? = 1 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [3,4,5,2,1] => 0000 => ? = 1 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => 0000 => ? = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => 0000 => ? = 1 + 1
[1,3,4,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => 0000 => ? = 2 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,3,4,5,1] => 0000 => ? = 2 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [5,2,3,4,1] => 0000 => ? = 2 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => 0000 => ? = 2 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,3,4,5,1] => 0000 => ? = 2 + 1
[1,4,5,3,2] => [1,5,4,3,2] => [2,3,4,5,1] => 0000 => ? = 2 + 1
[1,5,3,2,4] => [1,5,4,3,2] => [2,3,4,5,1] => 0000 => ? = 2 + 1
[1,5,3,4,2] => [1,5,4,3,2] => [2,3,4,5,1] => 0000 => ? = 2 + 1
[1,5,4,2,3] => [1,5,4,3,2] => [2,3,4,5,1] => 0000 => ? = 2 + 1
[1,5,4,3,2] => [1,5,4,3,2] => [2,3,4,5,1] => 0000 => ? = 2 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [5,4,3,1,2] => 0000 => ? = 1 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 0000 => ? = 1 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [5,3,4,1,2] => 0000 => ? = 1 + 1
[2,1,4,5,3] => [2,1,5,4,3] => [3,4,5,1,2] => 0000 => ? = 1 + 1
[2,1,5,3,4] => [2,1,5,4,3] => [3,4,5,1,2] => 0000 => ? = 1 + 1
[2,1,5,4,3] => [2,1,5,4,3] => [3,4,5,1,2] => 0000 => ? = 1 + 1
[2,3,1,4,5] => [3,2,1,4,5] => [5,4,1,2,3] => 0000 => ? = 2 + 1
[2,3,1,5,4] => [3,2,1,5,4] => [4,5,1,2,3] => 0000 => ? = 2 + 1
[2,4,1,3,5] => [3,4,1,2,5] => [5,2,1,4,3] => 0000 => ? = 2 + 1
[2,4,3,1,5] => [4,3,2,1,5] => [5,1,2,3,4] => 0000 => ? = 3 + 1
[2,4,5,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[2,5,3,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[2,5,4,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[3,1,2,4,5] => [3,2,1,4,5] => [5,4,1,2,3] => 0000 => ? = 2 + 1
[3,1,2,5,4] => [3,2,1,5,4] => [4,5,1,2,3] => 0000 => ? = 2 + 1
[3,2,1,4,5] => [3,2,1,4,5] => [5,4,1,2,3] => 0000 => ? = 2 + 1
[3,2,1,5,4] => [3,2,1,5,4] => [4,5,1,2,3] => 0000 => ? = 2 + 1
[3,4,1,2,5] => [4,3,2,1,5] => [5,1,2,3,4] => 0000 => ? = 3 + 1
[3,4,2,1,5] => [4,3,2,1,5] => [5,1,2,3,4] => 0000 => ? = 3 + 1
[3,4,5,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[3,4,5,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[3,5,1,4,2] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[3,5,2,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[3,5,4,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[3,5,4,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[4,2,1,3,5] => [4,3,2,1,5] => [5,1,2,3,4] => 0000 => ? = 3 + 1
[4,2,5,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[4,2,5,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[4,3,5,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[4,3,5,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[4,5,1,2,3] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[4,5,1,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[4,5,2,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[4,5,2,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[4,5,3,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[4,5,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[5,2,3,1,4] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[5,2,3,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[5,2,4,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[5,2,4,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[5,3,1,2,4] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[5,3,1,4,2] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[5,3,2,1,4] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[5,3,2,4,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[5,3,4,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[5,3,4,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[5,4,1,2,3] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[5,4,1,3,2] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[5,4,2,1,3] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[5,4,2,3,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[5,4,3,1,2] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
[5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 1111 => 4 = 3 + 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let $A_n=K[x]/(x^n)$. We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St001514
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001514: Dyck paths ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 43%
Values
[1,2] => [1,0,1,0]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
[2,1] => [1,1,0,0]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
[1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
[2,1,3] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[2,3,1] => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2 = 1 + 1
[1,2,3,4] => [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]
 => 1 = 0 + 1
[1,2,4,3] => [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]
 => 1 = 0 + 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,3,4,2] => [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]
 => 2 = 1 + 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2 = 1 + 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2 = 1 + 1
[2,1,3,4] => [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]
 => 2 = 1 + 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2 = 1 + 1
[2,3,1,4] => [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]
 => 3 = 2 + 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3 = 2 + 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,2,3,4,5] => [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]
 => ? = 0 + 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 0 + 1
[1,2,4,3,5] => [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]
 => ? = 1 + 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 1 + 1
[1,2,5,3,4] => [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]
 => ? = 1 + 1
[1,2,5,4,3] => [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]
 => ? = 1 + 1
[1,3,2,4,5] => [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]
 => ? = 1 + 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => ? = 1 + 1
[1,3,4,2,5] => [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]
 => ? = 2 + 1
[1,3,5,4,2] => [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]
 => ? = 2 + 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 2 + 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 2 + 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => ? = 2 + 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => ? = 2 + 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 2 + 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 2 + 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 2 + 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => ? = 2 + 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[2,1,3,5,4] => [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]
 => ? = 1 + 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => ? = 1 + 1
[2,1,4,5,3] => [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]
 => ? = 1 + 1
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => ? = 1 + 1
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => ? = 1 + 1
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 2 + 1
[2,3,1,5,4] => [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]
 => ? = 2 + 1
[2,4,1,3,5] => [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]
 => ? = 2 + 1
[2,4,3,1,5] => [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]
 => ? = 3 + 1
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => ? = 3 + 1
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => ? = 3 + 1
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => ? = 3 + 1
[3,1,2,4,5] => [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]
 => ? = 2 + 1
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 2 + 1
[3,2,1,4,5] => [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]
 => ? = 2 + 1
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => ? = 2 + 1
[3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => ? = 3 + 1
[3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => ? = 3 + 1
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => ? = 3 + 1
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => ? = 3 + 1
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => ? = 3 + 1
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => ? = 3 + 1
[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => ? = 3 + 1
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => ? = 3 + 1
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 3 + 1
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 3 + 1
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => ? = 3 + 1
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => ? = 3 + 1
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 3 + 1
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => ? = 3 + 1
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => ? = 3 + 1
Description
The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule.