Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001498
Mapping
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,3,2] => 10 => [1,1] => [1,0,1,0]
 => 1
[2,1,3] => 01 => [1,1] => [1,0,1,0]
 => 1
[1,2,4,3] => 110 => [2,1] => [1,1,0,0,1,0]
 => 2
[1,3,2,4] => 101 => [1,1,1] => [1,0,1,0,1,0]
 => 1
[1,3,4,2] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[1,4,2,3] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[1,4,3,2] => 100 => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,3,4] => 011 => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,4,3] => 010 => [1,1,1] => [1,0,1,0,1,0]
 => 1
[2,3,1,4] => 001 => [2,1] => [1,1,0,0,1,0]
 => 2
[3,1,2,4] => 001 => [2,1] => [1,1,0,0,1,0]
 => 2
[3,2,1,4] => 001 => [2,1] => [1,1,0,0,1,0]
 => 2
[1,2,3,5,4] => 1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[1,2,4,3,5] => 1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,2,4,5,3] => 1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,2,5,3,4] => 1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,2,5,4,3] => 1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,3,2,4,5] => 1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[1,3,2,5,4] => 1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
[1,3,4,2,5] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[1,3,4,5,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,3,5,2,4] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,3,5,4,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,2,3,5] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[1,4,2,5,3] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,3,2,5] => 1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[1,4,3,5,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,5,2,3] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,5,3,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,5,2,3,4] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,5,2,4,3] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,5,3,2,4] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,5,3,4,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,5,4,2,3] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,5,4,3,2] => 1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,1,3,4,5] => 0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,1,3,5,4] => 0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[2,1,4,3,5] => 0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
[2,1,4,5,3] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[2,1,5,3,4] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[2,1,5,4,3] => 0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[2,3,1,4,5] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[2,3,1,5,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[2,3,4,1,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[2,4,1,3,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[2,4,3,1,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[3,1,2,4,5] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[3,1,2,5,4] => 0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[3,1,4,2,5] => 0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[3,2,1,4,5] => 0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
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: St001645
(load all 9 compositions to match this statistic)
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 100%
Values
[1,3,2] => [1,3,2] => [1,2] => ([],2)
 => ? = 1 + 1
[2,1,3] => [2,1,3] => [2,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,2,4,3] => [1,2,4,3] => [1,2,3] => ([],3)
 => ? = 2 + 1
[1,3,2,4] => [1,3,2,4] => [1,3,2] => ([(1,2)],3)
 => ? = 1 + 1
[1,3,4,2] => [1,4,3,2] => [1,3,2] => ([(1,2)],3)
 => ? = 1 + 1
[1,4,2,3] => [1,4,3,2] => [1,3,2] => ([(1,2)],3)
 => ? = 1 + 1
[1,4,3,2] => [1,4,3,2] => [1,3,2] => ([(1,2)],3)
 => ? = 1 + 1
[2,1,3,4] => [2,1,3,4] => [2,1,3] => ([(1,2)],3)
 => ? = 1 + 1
[2,1,4,3] => [2,1,4,3] => [2,1,3] => ([(1,2)],3)
 => ? = 1 + 1
[2,3,1,4] => [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,1,2,4] => [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[3,2,1,4] => [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4] => ([],4)
 => ? = 3 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3] => ([(2,3)],4)
 => ? = 2 + 1
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,3] => ([(2,3)],4)
 => ? = 2 + 1
[1,2,5,3,4] => [1,2,5,4,3] => [1,2,4,3] => ([(2,3)],4)
 => ? = 2 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,3] => ([(2,3)],4)
 => ? = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4] => ([(2,3)],4)
 => ? = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,4] => ([(2,3)],4)
 => ? = 1 + 1
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[1,3,5,2,4] => [1,4,5,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[1,4,2,3,5] => [1,4,3,2,5] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[1,4,2,5,3] => [1,5,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[1,4,3,5,2] => [1,5,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[1,4,5,3,2] => [1,5,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[1,5,2,3,4] => [1,5,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[1,5,2,4,3] => [1,5,3,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? = 1 + 1
[1,5,3,2,4] => [1,5,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[1,5,3,4,2] => [1,5,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[1,5,4,2,3] => [1,5,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[1,5,4,3,2] => [1,5,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4] => ([(2,3)],4)
 => ? = 1 + 1
[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,4] => ([(2,3)],4)
 => ? = 2 + 1
[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? = 1 + 1
[2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? = 1 + 1
[2,1,5,3,4] => [2,1,5,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? = 1 + 1
[2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? = 1 + 1
[2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[2,3,4,1,5] => [4,2,3,1,5] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 + 1
[2,4,1,3,5] => [3,4,1,2,5] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4 = 3 + 1
[2,4,3,1,5] => [4,3,2,1,5] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,1,2,4,5] => [3,2,1,4,5] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[3,1,2,5,4] => [3,2,1,5,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[3,1,4,2,5] => [4,2,3,1,5] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 + 1
[3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[3,2,4,1,5] => [4,2,3,1,5] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 + 1
[3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[3,4,2,1,5] => [4,3,2,1,5] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,1,2,3,5] => [4,2,3,1,5] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 + 1
[4,1,3,2,5] => [4,2,3,1,5] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 + 1
[4,2,1,3,5] => [4,3,2,1,5] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,2,3,1,5] => [4,3,2,1,5] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,3,1,2,5] => [4,3,2,1,5] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,2,3,4,5] => ([],5)
 => ? = 4 + 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4] => ([(3,4)],5)
 => ? = 3 + 1
[2,4,5,1,3,6] => [4,5,3,1,2,6] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 5 = 4 + 1
[2,4,5,3,1,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[2,5,3,1,4,6] => [4,5,3,1,2,6] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 5 = 4 + 1
[2,5,3,4,1,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[2,5,4,1,3,6] => [4,5,3,1,2,6] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 5 = 4 + 1
[2,5,4,3,1,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[3,4,5,1,2,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[3,4,5,2,1,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[3,5,1,2,4,6] => [4,5,3,1,2,6] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 5 = 4 + 1
[3,5,1,4,2,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[3,5,2,1,4,6] => [4,5,3,1,2,6] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 5 = 4 + 1
[3,5,2,4,1,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[3,5,4,1,2,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[3,5,4,2,1,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[4,2,5,1,3,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[4,2,5,3,1,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[4,3,5,1,2,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[4,3,5,2,1,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[4,5,1,2,3,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[4,5,1,3,2,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[4,5,2,1,3,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[4,5,2,3,1,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[4,5,3,1,2,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[4,5,3,2,1,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[5,2,3,1,4,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[5,2,3,4,1,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[5,2,4,1,3,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[5,2,4,3,1,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[5,3,1,2,4,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[5,3,1,4,2,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[5,3,2,1,4,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[5,3,2,4,1,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[5,3,4,1,2,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[5,3,4,2,1,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[5,4,1,2,3,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[5,4,1,3,2,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[5,4,2,1,3,6] => [5,4,3,2,1,6] => [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 = 4 + 1
[5,4,2,3,1,6] => [5,4,3,2,1,6] => [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 = 4 + 1
Description
The pebbling number of a connected graph.
Matching statistic: St001232
(load all 3 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 67%
Values
[1,3,2] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => ? = 1 + 1
[2,1,3] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => ? = 1 + 1
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => ? = 2 + 1
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => ? = 1 + 1
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => ? = 1 + 1
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => ? = 1 + 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => ? = 1 + 1
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => ? = 2 + 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 3 + 1
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 2 + 1
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 2 + 1
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => ? = 2 + 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 1 + 1
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 1 + 1
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 2 + 1
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 1 + 1
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => ? = 2 + 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => ? = 2 + 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => ? = 1 + 1
[1,5,2,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,5,2,4,3] => [1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,5,3,2,4] => [1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,5,3,4,2] => [1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,5,4,2,3] => [1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 1
[1,5,4,3,2] => [1,2,5,3,4] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => ? = 1 + 1
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 1 + 1
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 2 + 1
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 1 + 1
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 1 + 1
[2,1,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[2,1,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => ? = 1 + 1
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 2 + 1
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 2 + 1
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => ? = 3 + 1
[2,4,1,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => ? = 3 + 1
[2,4,3,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => ? = 3 + 1
[3,1,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => ? = 2 + 1
[3,1,2,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => ? = 2 + 1
[3,1,4,2,5] => [1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[3,2,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => ? = 2 + 1
[3,2,1,5,4] => [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => ? = 2 + 1
[3,2,4,1,5] => [1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[3,4,1,2,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => ? = 3 + 1
[3,4,2,1,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => ? = 3 + 1
[4,1,2,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[4,1,3,2,5] => [1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[4,2,1,3,5] => [1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[4,2,3,1,5] => [1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[4,3,1,2,5] => [1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[4,3,2,1,5] => [1,4,2,3,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[3,1,4,5,2,6] => [1,3,4,5,2,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[3,1,5,2,4,6] => [1,3,5,4,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[3,2,4,5,1,6] => [1,3,4,5,2,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[3,2,5,1,4,6] => [1,3,5,4,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[3,5,1,2,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[3,5,1,4,2,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[3,5,2,1,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[3,5,2,4,1,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => 5 = 4 + 1
[4,1,2,5,3,6] => [1,4,5,3,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[4,1,3,5,2,6] => [1,4,5,2,3,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[4,1,5,3,2,6] => [1,4,3,5,2,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[4,2,1,5,3,6] => [1,4,5,3,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[4,2,3,5,1,6] => [1,4,5,2,3,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[4,2,5,3,1,6] => [1,4,3,5,2,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[4,3,1,5,2,6] => [1,4,5,2,3,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[4,3,2,5,1,6] => [1,4,5,2,3,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[4,5,1,2,3,6] => [1,4,2,5,3,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[4,5,2,1,3,6] => [1,4,2,5,3,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[4,5,3,1,2,6] => [1,4,2,5,3,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[4,5,3,2,1,6] => [1,4,2,5,3,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[5,1,2,3,4,6] => [1,5,4,3,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[5,1,2,4,3,6] => [1,5,3,2,4,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[5,1,3,2,4,6] => [1,5,4,2,3,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[5,1,3,4,2,6] => [1,5,2,3,4,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[5,1,4,2,3,6] => [1,5,3,4,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[5,1,4,3,2,6] => [1,5,2,3,4,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[5,2,1,3,4,6] => [1,5,4,3,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[5,2,1,4,3,6] => [1,5,3,2,4,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[5,2,3,1,4,6] => [1,5,4,2,3,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[5,2,3,4,1,6] => [1,5,2,3,4,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[5,2,4,1,3,6] => [1,5,3,4,2,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[5,2,4,3,1,6] => [1,5,2,3,4,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[5,3,1,2,4,6] => [1,5,4,2,3,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[5,3,1,4,2,6] => [1,5,2,3,4,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[5,3,2,1,4,6] => [1,5,4,2,3,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[5,3,2,4,1,6] => [1,5,2,3,4,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[5,3,4,1,2,6] => [1,5,2,3,4,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[5,3,4,2,1,6] => [1,5,2,3,4,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[5,4,1,2,3,6] => [1,5,3,2,4,6] => [1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[5,4,1,3,2,6] => [1,5,2,4,3,6] => [1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001876
Mapping
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00196: Lattices —The modular quotient of a lattice.⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 17%
Values
[1,3,2] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,1,3] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? = 1
[1,2,4,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,1)],2)
 => ? = 2
[1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,1)],2)
 => ? = 1
[1,3,4,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,4,2,3] => [3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ([(0,1)],2)
 => ? = 1
[1,4,3,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,1,3,4] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,1)],2)
 => ? = 1
[2,1,4,3] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ([(0,1)],2)
 => ? = 1
[2,3,1,4] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ([(0,1)],2)
 => ? = 2
[3,1,2,4] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 2
[3,2,1,4] => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ([],1)
 => ? = 2
[1,2,3,5,4] => [2,3,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(0,1)],2)
 => ? = 3
[1,2,4,3,5] => [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
 => ([(0,1)],2)
 => ? = 2
[1,2,4,5,3] => [2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
 => ([(0,1)],2)
 => ? = 2
[1,2,5,3,4] => [2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
 => ([(0,1)],2)
 => ? = 2
[1,2,5,4,3] => [2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ([(0,1)],2)
 => ? = 2
[1,3,2,4,5] => [3,2,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(0,1)],2)
 => ? = 1
[1,3,2,5,4] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
 => ([(0,1)],2)
 => ? = 1
[1,3,4,2,5] => [4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
 => ([(0,1)],2)
 => ? = 2
[1,3,4,5,2] => [5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,3,5,2,4] => [4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ([(0,1)],2)
 => ? = 1
[1,3,5,4,2] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,4,2,3,5] => [3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
 => ([(0,1)],2)
 => ? = 2
[1,4,2,5,3] => [3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ([(0,1)],2)
 => ? = 1
[1,4,3,2,5] => [4,3,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ([(0,1)],2)
 => ? = 2
[1,4,3,5,2] => [5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,4,5,2,3] => [4,5,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
 => ([(0,1)],2)
 => ? = 1
[1,4,5,3,2] => [5,4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,5,2,3,4] => [3,4,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
 => ([(0,1)],2)
 => ? = 1
[1,5,2,4,3] => [3,5,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(0,1)],2)
 => ? = 1
[1,5,3,2,4] => [4,3,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(0,1)],2)
 => ? = 1
[1,5,3,4,2] => [5,3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,5,4,2,3] => [4,5,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ([(0,1)],2)
 => ? = 1
[1,5,4,3,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[2,1,3,4,5] => [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ([(0,1)],2)
 => ? = 1
[2,1,3,5,4] => [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
 => ([(0,1)],2)
 => ? = 2
[2,1,4,3,5] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
 => ([(0,1)],2)
 => ? = 1
[2,1,4,5,3] => [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
 => ([(0,1)],2)
 => ? = 1
[2,1,5,3,4] => [1,4,5,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
 => ([(0,1)],2)
 => ? = 1
[2,1,5,4,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ([(0,1)],2)
 => ? = 1
[2,3,1,4,5] => [1,2,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(0,1)],2)
 => ? = 2
[2,3,1,5,4] => [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
 => ([(0,1)],2)
 => ? = 2
[2,3,4,1,5] => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ([(0,1)],2)
 => ? = 3
[2,4,1,3,5] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8)
 => ([(0,1)],2)
 => ? = 3
[2,4,3,1,5] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
 => ([(0,1)],2)
 => ? = 3
[3,1,2,4,5] => [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([],1)
 => ? = 2
[3,1,2,5,4] => [3,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([],1)
 => ? = 2
[3,1,4,2,5] => [4,1,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 3
[3,2,1,4,5] => [2,1,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ([],1)
 => ? = 2
[3,2,1,5,4] => [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
 => ([],1)
 => ? = 2
[3,2,4,1,5] => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
 => ([],1)
 => ? = 3
[3,4,1,2,5] => [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([],1)
 => ? = 3
[3,4,2,1,5] => [3,1,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ([],1)
 => ? = 3
[4,1,2,3,5] => [3,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([],1)
 => ? = 3
[4,1,3,2,5] => [4,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([],1)
 => ? = 3
[4,2,1,3,5] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 3
[4,2,3,1,5] => [2,3,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ([],1)
 => ? = 3
[4,3,1,2,5] => [4,2,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ([],1)
 => ? = 3
[1,3,4,5,6,2] => [6,2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,12),(5,12),(5,15),(6,13),(6,15),(8,14),(9,14),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,9)],16)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,3,4,6,5,2] => [6,2,3,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,11),(4,12),(5,12),(6,8),(6,11),(8,13),(9,7),(10,7),(11,13),(12,8),(13,9),(13,10)],14)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,3,5,4,6,2] => [6,2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,8),(3,11),(4,10),(5,13),(6,13),(8,12),(9,12),(10,7),(11,7),(12,10),(12,11),(13,8),(13,9)],14)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,3,5,6,4,2] => [6,2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,3,6,4,5,2] => [6,2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,3,6,5,4,2] => [6,2,5,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,4,3,5,6,2] => [6,3,2,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,11),(4,12),(5,12),(6,8),(6,11),(8,13),(9,7),(10,7),(11,13),(12,8),(13,9),(13,10)],14)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,4,3,6,5,2] => [6,3,2,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,11),(9,12),(10,12),(11,9),(11,10)],13)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,4,5,3,6,2] => [6,4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,4,5,6,3,2] => [6,5,2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,12),(3,11),(4,10),(5,12),(5,13),(6,11),(6,15),(8,7),(9,7),(10,9),(11,8),(12,14),(13,14),(14,10),(14,15),(15,8),(15,9)],16)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,4,6,3,5,2] => [6,4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,8),(6,7),(7,9),(8,9),(10,7),(10,8)],11)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,4,6,5,3,2] => [6,5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,5,3,4,6,2] => [6,3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,11),(6,10),(7,10),(8,12),(9,12),(10,11),(11,8),(11,9)],13)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,5,3,6,4,2] => [6,3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,8),(6,7),(7,9),(8,9),(10,7),(10,8)],11)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,5,4,3,6,2] => [6,4,3,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,13),(6,11),(6,12),(8,13),(9,7),(10,7),(11,8),(12,8),(13,9),(13,10)],14)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,5,4,6,3,2] => [6,5,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,5,6,3,4,2] => [6,4,5,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,10),(3,13),(4,13),(5,14),(6,14),(8,7),(9,7),(10,8),(11,9),(12,8),(12,9),(13,10),(13,12),(14,11),(14,12)],15)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,5,6,4,3,2] => [6,5,4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,17),(3,17),(4,12),(5,15),(5,16),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,9),(13,8),(14,10),(14,11),(15,9),(15,14),(16,8),(16,14),(17,12),(17,15)],18)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,6,3,4,5,2] => [6,3,4,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,12),(3,11),(4,10),(5,12),(5,13),(6,11),(6,15),(8,7),(9,7),(10,9),(11,8),(12,14),(13,14),(14,10),(14,15),(15,8),(15,9)],16)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,6,3,5,4,2] => [6,3,5,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,6,4,3,5,2] => [6,4,3,5,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,14),(3,12),(4,12),(5,10),(6,11),(6,13),(8,7),(9,7),(10,9),(11,8),(12,14),(13,8),(13,9),(14,10),(14,13)],15)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,6,4,5,3,2] => [6,5,3,4,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,17),(2,17),(3,13),(4,12),(5,12),(5,15),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,10),(14,11),(15,8),(15,14),(16,9),(16,14),(17,15),(17,16)],18)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,6,5,3,4,2] => [6,4,5,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,17),(3,17),(4,12),(5,15),(5,16),(6,13),(6,16),(8,10),(9,11),(10,7),(11,7),(12,9),(13,8),(14,10),(14,11),(15,9),(15,14),(16,8),(16,14),(17,12),(17,15)],18)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,4,3,8,7,6,5,2] => [8,3,2,7,6,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,14),(2,13),(3,16),(4,15),(5,17),(6,17),(7,15),(7,19),(8,16),(8,19),(10,12),(11,12),(12,18),(13,9),(14,9),(15,10),(16,11),(17,18),(18,13),(18,14),(19,10),(19,11)],20)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,6,5,4,3,8,7,2] => [8,5,4,3,2,7,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,14),(2,13),(3,16),(4,15),(5,17),(6,17),(7,15),(7,19),(8,16),(8,19),(10,12),(11,12),(12,18),(13,9),(14,9),(15,10),(16,11),(17,18),(18,13),(18,14),(19,10),(19,11)],20)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,4,3,6,5,8,7,2] => [8,3,2,5,4,7,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,17),(2,17),(3,14),(4,14),(5,15),(6,15),(7,13),(8,12),(10,16),(11,16),(12,9),(13,9),(14,11),(15,10),(16,12),(16,13),(17,10),(17,11)],18)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,5,6,3,4,8,7,2] => [8,4,5,2,3,7,6,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,12),(3,11),(4,11),(5,10),(6,10),(7,9),(8,9),(9,15),(10,14),(11,14),(12,16),(13,16),(14,15),(15,12),(15,13)],17)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,7,5,4,8,3,6,2] => [8,6,4,3,7,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,13),(3,13),(4,13),(5,11),(6,10),(7,9),(8,9),(9,13),(10,12),(11,12),(13,10),(13,11)],14)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,7,6,8,4,3,5,2] => [8,6,5,7,3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,14),(2,14),(3,15),(4,15),(5,13),(6,12),(7,17),(8,18),(10,9),(11,9),(12,10),(13,11),(14,17),(15,18),(16,10),(16,11),(17,12),(17,16),(18,13),(18,16)],19)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,5,7,3,8,4,6,2] => [8,4,6,2,7,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,10),(8,9),(9,11),(10,11),(12,9),(12,10)],13)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,4,3,7,8,5,6,2] => [8,3,2,6,7,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,12),(3,11),(4,11),(5,10),(6,10),(7,9),(8,9),(9,15),(10,14),(11,14),(12,16),(13,16),(14,15),(15,12),(15,13)],17)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,5,8,3,7,6,4,2] => [8,4,7,2,6,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,13),(3,13),(4,13),(5,11),(6,10),(7,9),(8,9),(9,13),(10,12),(11,12),(13,10),(13,11)],14)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,6,8,7,3,5,4,2] => [8,5,7,6,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,14),(2,14),(3,15),(4,15),(5,13),(6,12),(7,17),(8,18),(10,9),(11,9),(12,10),(13,11),(14,17),(15,18),(16,10),(16,11),(17,12),(17,16),(18,13),(18,16)],19)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,7,4,8,6,3,5,2] => [8,6,3,7,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,10),(8,9),(9,11),(10,11),(12,9),(12,10)],13)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,6,4,8,3,7,5,2] => [8,5,3,7,2,6,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,10),(8,9),(9,11),(10,11),(12,9),(12,10)],13)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,5,7,3,8,6,4,2] => [8,4,7,2,6,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,10),(8,9),(9,11),(10,11),(12,9),(12,10)],13)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,6,8,5,3,7,4,2] => [8,5,7,4,2,6,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,10),(8,9),(9,11),(10,11),(12,9),(12,10)],13)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,6,4,8,3,5,7,2] => [8,5,3,6,2,7,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,10),(8,9),(9,11),(10,11),(12,9),(12,10)],13)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,5,7,8,3,6,4,2] => [8,5,7,2,6,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,13),(2,13),(3,13),(4,13),(5,11),(6,10),(7,9),(8,9),(9,13),(10,12),(11,12),(13,10),(13,11)],14)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,7,5,3,8,6,4,2] => [8,4,7,3,6,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,10),(8,9),(9,11),(10,11),(12,9),(12,10)],13)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[1,7,5,8,3,6,4,2] => [8,5,7,3,6,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,12),(2,12),(3,12),(4,12),(5,12),(6,12),(7,10),(8,9),(9,11),(10,11),(12,9),(12,10)],13)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St001964
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 17%
Values
[1,3,2] => [1,3,2] => [3,2,1] => ([(0,2),(2,1)],3)
 => 0 = 1 - 1
[2,1,3] => [2,1,3] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0 = 1 - 1
[1,2,4,3] => [1,2,4,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 2 - 1
[1,3,2,4] => [1,3,2,4] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 1 - 1
[1,3,4,2] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[1,4,2,3] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[1,4,3,2] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
 => 0 = 1 - 1
[2,1,3,4] => [2,1,3,4] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => ? = 1 - 1
[2,1,4,3] => [2,1,4,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1 - 1
[2,3,1,4] => [3,2,1,4] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 2 - 1
[3,1,2,4] => [3,2,1,4] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 2 - 1
[3,2,1,4] => [3,2,1,4] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ? = 2 - 1
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 3 - 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ? = 2 - 1
[1,2,4,5,3] => [1,2,5,4,3] => [2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2 - 1
[1,2,5,3,4] => [1,2,5,4,3] => [2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2 - 1
[1,2,5,4,3] => [1,2,5,4,3] => [2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2 - 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 1 - 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
 => ? = 1 - 1
[1,3,4,2,5] => [1,4,3,2,5] => [4,3,2,5,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2 - 1
[1,3,4,5,2] => [1,5,3,4,2] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1 - 1
[1,3,5,2,4] => [1,4,5,2,3] => [4,5,2,3,1] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
 => ? = 1 - 1
[1,3,5,4,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,4,2,3,5] => [1,4,3,2,5] => [4,3,2,5,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2 - 1
[1,4,2,5,3] => [1,5,3,4,2] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1 - 1
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,2,5,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2 - 1
[1,4,3,5,2] => [1,5,3,4,2] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1 - 1
[1,4,5,2,3] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,4,5,3,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,5,2,3,4] => [1,5,3,4,2] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1 - 1
[1,5,2,4,3] => [1,5,3,4,2] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1 - 1
[1,5,3,2,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,5,3,4,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,5,4,2,3] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 1 - 1
[2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
 => ? = 2 - 1
[2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ? = 1 - 1
[2,1,4,5,3] => [2,1,5,4,3] => [1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[2,1,5,3,4] => [2,1,5,4,3] => [1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[2,1,5,4,3] => [2,1,5,4,3] => [1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1 - 1
[2,3,1,4,5] => [3,2,1,4,5] => [2,1,4,5,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 2 - 1
[2,3,1,5,4] => [3,2,1,5,4] => [2,1,5,4,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
 => ? = 2 - 1
[2,3,4,1,5] => [4,2,3,1,5] => [2,3,1,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 3 - 1
[2,4,1,3,5] => [3,4,1,2,5] => [4,1,2,5,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
 => ? = 3 - 1
[2,4,3,1,5] => [4,3,2,1,5] => [3,2,1,5,4] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
 => ? = 3 - 1
[3,1,2,4,5] => [3,2,1,4,5] => [2,1,4,5,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 2 - 1
[3,1,2,5,4] => [3,2,1,5,4] => [2,1,5,4,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
 => ? = 2 - 1
[3,1,4,2,5] => [4,2,3,1,5] => [2,3,1,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 3 - 1
[3,2,1,4,5] => [3,2,1,4,5] => [2,1,4,5,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 2 - 1
[3,2,1,5,4] => [3,2,1,5,4] => [2,1,5,4,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
 => ? = 2 - 1
[3,2,4,1,5] => [4,2,3,1,5] => [2,3,1,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 3 - 1
[3,4,1,2,5] => [4,3,2,1,5] => [3,2,1,5,4] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
 => ? = 3 - 1
[3,4,2,1,5] => [4,3,2,1,5] => [3,2,1,5,4] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
 => ? = 3 - 1
[4,1,2,3,5] => [4,2,3,1,5] => [2,3,1,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 3 - 1
[4,1,3,2,5] => [4,2,3,1,5] => [2,3,1,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 3 - 1
[4,2,1,3,5] => [4,3,2,1,5] => [3,2,1,5,4] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
 => ? = 3 - 1
[4,2,3,1,5] => [4,3,2,1,5] => [3,2,1,5,4] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
 => ? = 3 - 1
[4,3,1,2,5] => [4,3,2,1,5] => [3,2,1,5,4] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
 => ? = 3 - 1
[4,3,2,1,5] => [4,3,2,1,5] => [3,2,1,5,4] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
 => ? = 3 - 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [2,3,4,6,5,1] => ([(0,3),(0,4),(0,5),(1,14),(2,1),(2,6),(2,7),(3,9),(3,11),(4,9),(4,10),(5,2),(5,10),(5,11),(6,13),(6,14),(7,13),(7,14),(9,12),(10,6),(10,12),(11,7),(11,12),(12,13),(13,8),(14,8)],15)
 => ? = 4 - 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [2,3,5,4,6,1] => ([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
 => ? = 3 - 1
[1,3,5,6,4,2] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,3,6,4,5,2] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,3,6,5,4,2] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,4,5,6,2,3] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,4,5,6,3,2] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,4,6,2,5,3] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,4,6,3,5,2] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,4,6,5,2,3] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,4,6,5,3,2] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,5,3,6,2,4] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,5,3,6,4,2] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,5,4,6,2,3] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,5,4,6,3,2] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,5,6,2,3,4] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,5,6,2,4,3] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,5,6,3,2,4] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,5,6,3,4,2] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,5,6,4,2,3] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,5,6,4,3,2] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,6,3,4,2,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,6,3,4,5,2] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,6,3,5,2,4] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,6,3,5,4,2] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,6,4,2,3,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,6,4,2,5,3] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,6,4,3,2,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,6,4,3,5,2] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,6,4,5,2,3] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,6,4,5,3,2] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,6,5,2,3,4] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,6,5,2,4,3] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,6,5,3,2,4] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,6,5,3,4,2] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,6,5,4,2,3] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,6,5,4,3,2] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
Description
The interval resolution global dimension of a poset. This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Matching statistic: St000422
(load all 2 compositions to match this statistic)
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 17%
Values
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
 => ([(0,2),(1,2)],3)
 => ? = 1 + 3
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 1 + 3
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 3
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 3
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 3
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 3
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 3
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 3
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 + 3
[2,3,1,4] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 3
[3,1,2,4] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 3
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 3
[1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 + 3
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 + 3
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 + 3
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 + 3
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 + 3
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 3
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 3
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 + 3
[1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 3
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 3
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 3
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 + 3
[1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 3
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 + 3
[1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 3
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 3
[1,4,5,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 3
[1,5,2,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 3
[1,5,2,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 3
[1,5,3,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 3
[1,5,3,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 3
[1,5,4,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 3
[1,5,4,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 3
[2,1,3,4,5] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 3
[2,1,3,5,4] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 + 3
[2,1,4,3,5] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 3
[2,1,4,5,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 3
[2,1,5,3,4] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 3
[2,1,5,4,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 + 3
[2,3,1,4,5] => [2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 + 3
[2,3,1,5,4] => [2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 + 3
[2,3,4,1,5] => [2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 + 3
[2,4,1,3,5] => [2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 + 3
[2,4,3,1,5] => [2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 + 3
[3,1,2,4,5] => [3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 + 3
[3,1,2,5,4] => [3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 + 3
[3,1,4,2,5] => [3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 + 3
[3,2,1,4,5] => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 + 3
[3,4,2,5,6,1,7] => [3,7,2,6,5,1,4] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 5 + 3
[3,4,2,6,5,1,7] => [3,7,2,6,5,1,4] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 5 + 3
[3,5,2,4,6,1,7] => [3,7,2,6,5,1,4] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 5 + 3
[3,5,2,6,4,1,7] => [3,7,2,6,5,1,4] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 5 + 3
[3,6,2,4,5,1,7] => [3,7,2,6,5,1,4] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 5 + 3
[3,6,2,5,4,1,7] => [3,7,2,6,5,1,4] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 5 + 3
[4,3,2,5,6,1,7] => [4,3,2,7,6,1,5] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 5 + 3
[4,3,2,6,5,1,7] => [4,3,2,7,6,1,5] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 5 + 3
[4,5,2,3,6,1,7] => [4,7,2,6,5,1,3] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 5 + 3
[4,5,2,6,3,1,7] => [4,7,2,6,5,1,3] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 5 + 3
[4,6,2,3,5,1,7] => [4,7,2,6,5,1,3] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 5 + 3
[4,6,2,5,3,1,7] => [4,7,2,6,5,1,3] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 5 + 3
[5,3,2,4,6,1,7] => [5,3,2,7,6,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 5 + 3
[5,3,2,6,4,1,7] => [5,3,2,7,6,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 5 + 3
[5,4,2,3,6,1,7] => [5,4,2,7,6,1,3] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 5 + 3
[5,4,2,6,3,1,7] => [5,4,2,7,6,1,3] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 5 + 3
[5,6,2,3,4,1,7] => [5,7,2,6,4,1,3] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 5 + 3
[5,6,2,4,3,1,7] => [5,7,2,6,4,1,3] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 5 + 3
[6,3,2,4,5,1,7] => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 5 + 3
[6,3,2,5,4,1,7] => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 5 + 3
[6,4,2,3,5,1,7] => [6,4,2,7,5,1,3] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 5 + 3
[6,4,2,5,3,1,7] => [6,4,2,7,5,1,3] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 5 + 3
[6,5,2,3,4,1,7] => [6,5,2,7,4,1,3] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 5 + 3
[6,5,2,4,3,1,7] => [6,5,2,7,4,1,3] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 8 = 5 + 3
Description
The energy of a graph, if it is integral. The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3]. The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St000454
(load all 2 compositions to match this statistic)
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 17%
Values
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
 => ([(0,2),(1,2)],3)
 => ? = 1 - 3
[2,1,3] => [2,1,3] => [[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ? = 1 - 3
[1,2,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 - 3
[1,3,2,4] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 - 3
[1,3,4,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 - 3
[1,4,2,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 - 3
[1,4,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 - 3
[2,1,3,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 - 3
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 1 - 3
[2,3,1,4] => [2,4,1,3] => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 - 3
[3,1,2,4] => [3,1,4,2] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 - 3
[3,2,1,4] => [3,2,1,4] => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 - 3
[1,2,3,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 - 3
[1,2,4,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 - 3
[1,2,4,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 - 3
[1,2,5,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 - 3
[1,2,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 - 3
[1,3,2,4,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 - 3
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 - 3
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 - 3
[1,3,4,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 - 3
[1,3,5,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 - 3
[1,3,5,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 - 3
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 - 3
[1,4,2,5,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 - 3
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 - 3
[1,4,3,5,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 - 3
[1,4,5,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 - 3
[1,4,5,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 - 3
[1,5,2,3,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 - 3
[1,5,2,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 - 3
[1,5,3,2,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 - 3
[1,5,3,4,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 - 3
[1,5,4,2,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 - 3
[1,5,4,3,2] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 - 3
[2,1,3,4,5] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 - 3
[2,1,3,5,4] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 - 3
[2,1,4,3,5] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 - 3
[2,1,4,5,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 - 3
[2,1,5,3,4] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 - 3
[2,1,5,4,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 1 - 3
[2,3,1,4,5] => [2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 - 3
[2,3,1,5,4] => [2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 - 3
[2,3,4,1,5] => [2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 - 3
[2,4,1,3,5] => [2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 - 3
[2,4,3,1,5] => [2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 - 3
[3,1,2,4,5] => [3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 - 3
[3,1,2,5,4] => [3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 - 3
[3,1,4,2,5] => [3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 - 3
[3,2,1,4,5] => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 - 3
[3,4,2,5,6,1,7] => [3,7,2,6,5,1,4] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 5 - 3
[3,4,2,6,5,1,7] => [3,7,2,6,5,1,4] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 5 - 3
[3,5,2,4,6,1,7] => [3,7,2,6,5,1,4] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 5 - 3
[3,5,2,6,4,1,7] => [3,7,2,6,5,1,4] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 5 - 3
[3,6,2,4,5,1,7] => [3,7,2,6,5,1,4] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 5 - 3
[3,6,2,5,4,1,7] => [3,7,2,6,5,1,4] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 5 - 3
[4,3,2,5,6,1,7] => [4,3,2,7,6,1,5] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 5 - 3
[4,3,2,6,5,1,7] => [4,3,2,7,6,1,5] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 5 - 3
[4,5,2,3,6,1,7] => [4,7,2,6,5,1,3] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 5 - 3
[4,5,2,6,3,1,7] => [4,7,2,6,5,1,3] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 5 - 3
[4,6,2,3,5,1,7] => [4,7,2,6,5,1,3] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 5 - 3
[4,6,2,5,3,1,7] => [4,7,2,6,5,1,3] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 5 - 3
[5,3,2,4,6,1,7] => [5,3,2,7,6,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 5 - 3
[5,3,2,6,4,1,7] => [5,3,2,7,6,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 5 - 3
[5,4,2,3,6,1,7] => [5,4,2,7,6,1,3] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 5 - 3
[5,4,2,6,3,1,7] => [5,4,2,7,6,1,3] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 5 - 3
[5,6,2,3,4,1,7] => [5,7,2,6,4,1,3] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 5 - 3
[5,6,2,4,3,1,7] => [5,7,2,6,4,1,3] => [[[.,[.,.]],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 5 - 3
[6,3,2,4,5,1,7] => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 5 - 3
[6,3,2,5,4,1,7] => [6,3,2,7,5,1,4] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 5 - 3
[6,4,2,3,5,1,7] => [6,4,2,7,5,1,3] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 5 - 3
[6,4,2,5,3,1,7] => [6,4,2,7,5,1,3] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 5 - 3
[6,5,2,3,4,1,7] => [6,5,2,7,4,1,3] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 5 - 3
[6,5,2,4,3,1,7] => [6,5,2,7,4,1,3] => [[[[.,.],.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => 2 = 5 - 3
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.