Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001060
Mapping
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,4,3] => 110 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[1,3,2,4] => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[2,1,4,3] => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[2,3,1,4] => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[3,1,2,4] => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[3,2,1,4] => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 2
[1,2,3,5,4] => 1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,2,4,3,5] => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,2,4,5,3] => 1100 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[1,2,5,3,4] => 1100 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[1,2,5,4,3] => 1100 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[1,3,2,4,5] => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[1,3,2,5,4] => 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[1,3,4,2,5] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,4,2,3,5] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,4,3,2,5] => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,1,3,5,4] => 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,1,4,3,5] => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
[2,1,4,5,3] => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,1,5,3,4] => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,1,5,4,3] => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,3,1,4,5] => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[2,3,1,5,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,3,4,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[2,4,1,3,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[2,4,3,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[3,1,2,4,5] => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[3,1,2,5,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,1,4,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[3,2,1,4,5] => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => 2
[3,2,1,5,4] => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,2,4,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[3,4,1,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[3,4,2,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,1,2,3,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,1,3,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,2,1,3,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,2,3,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,3,1,2,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[4,3,2,1,5] => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,2,3,4,6,5] => 11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,2,3,5,4,6] => 11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,2,3,5,6,4] => 11100 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[1,2,3,6,4,5] => 11100 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[1,2,3,6,5,4] => 11100 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[1,2,4,3,5,6] => 11011 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,2,4,3,6,5] => 11010 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,2,4,5,3,6] => 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,2,5,3,4,6] => 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,2,5,4,3,6] => 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
Description
The distinguishing index of a graph. This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism. If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St001645
(load all 41 compositions to match this statistic)
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 51% ●values known / values provided: 51%●distinct values known / distinct values provided: 100%
Values
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? = 2 + 2
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 3 + 2
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => ? = 3 + 2
[2,3,1,4] => [3,2,1,4] => [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] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[3,2,1,4] => [3,2,1,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 2 + 2
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? = 3 + 2
[1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[1,2,4,5,3] => [1,2,5,4,3] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? = 2 + 2
[1,2,5,3,4] => [1,2,5,4,3] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? = 2 + 2
[1,2,5,4,3] => [1,2,5,4,3] => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? = 2 + 2
[1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 2
[1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 2
[1,3,4,2,5] => [1,4,3,2,5] => [2,5,4,3,1] => ([(0,4),(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] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[1,4,3,2,5] => [1,4,3,2,5] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[2,1,3,5,4] => [2,1,3,5,4] => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[2,1,4,3,5] => [2,1,4,3,5] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => ? = 3 + 2
[2,1,4,5,3] => [2,1,5,4,3] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 2
[2,1,5,3,4] => [2,1,5,4,3] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 2
[2,1,5,4,3] => [2,1,5,4,3] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 2
[2,3,1,4,5] => [3,2,1,4,5] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[2,3,1,5,4] => [3,2,1,5,4] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[2,3,4,1,5] => [4,2,3,1,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[2,4,1,3,5] => [3,4,1,2,5] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 5 = 3 + 2
[2,4,3,1,5] => [4,3,2,1,5] => [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] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[3,1,2,5,4] => [3,2,1,5,4] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[3,1,4,2,5] => [4,2,3,1,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[3,2,1,4,5] => [3,2,1,4,5] => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[3,2,1,5,4] => [3,2,1,5,4] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 2
[3,2,4,1,5] => [4,2,3,1,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[3,4,1,2,5] => [4,3,2,1,5] => [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
[3,4,2,1,5] => [4,3,2,1,5] => [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
[4,1,2,3,5] => [4,2,3,1,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 3 + 2
[4,1,3,2,5] => [4,2,3,1,5] => [5,3,4,2,1] => ([(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] => [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
[4,2,3,1,5] => [4,3,2,1,5] => [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
[4,3,1,2,5] => [4,3,2,1,5] => [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
[4,3,2,1,5] => [4,3,2,1,5] => [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
[1,2,3,4,6,5] => [1,2,3,4,6,5] => [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 4 + 2
[1,2,3,5,4,6] => [1,2,3,5,4,6] => [2,3,4,6,5,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 2
[1,2,3,5,6,4] => [1,2,3,6,5,4] => [2,3,4,1,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ? = 3 + 2
[1,2,3,6,4,5] => [1,2,3,6,5,4] => [2,3,4,1,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ? = 3 + 2
[1,2,3,6,5,4] => [1,2,3,6,5,4] => [2,3,4,1,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
 => ? = 3 + 2
[1,2,4,3,5,6] => [1,2,4,3,5,6] => [2,3,5,4,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [2,3,5,4,1,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 2
[1,2,4,5,3,6] => [1,2,5,4,3,6] => [2,3,6,5,4,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 2
[1,2,5,3,4,6] => [1,2,5,4,3,6] => [2,3,6,5,4,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 2
[1,2,5,4,3,6] => [1,2,5,4,3,6] => [2,3,6,5,4,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 2
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [2,4,3,5,1,6] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [2,4,3,6,5,1] => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ? = 3 + 2
[1,3,2,5,6,4] => [1,3,2,6,5,4] => [2,4,3,1,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 2
[1,3,2,6,4,5] => [1,3,2,6,5,4] => [2,4,3,1,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 2
[1,3,2,6,5,4] => [1,3,2,6,5,4] => [2,4,3,1,6,5] => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 2
[1,3,4,2,5,6] => [1,4,3,2,5,6] => [2,5,4,3,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 2
[1,3,4,2,6,5] => [1,4,3,2,6,5] => [2,5,4,3,1,6] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 2
[1,3,4,5,2,6] => [1,5,3,4,2,6] => [2,6,4,5,3,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 2
[1,3,5,2,4,6] => [1,4,5,2,3,6] => [2,5,6,3,4,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 2 + 2
[1,3,5,4,2,6] => [1,5,4,3,2,6] => [2,6,5,4,3,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 2
[1,4,2,3,5,6] => [1,4,3,2,5,6] => [2,5,4,3,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 2
[1,4,2,3,6,5] => [1,4,3,2,6,5] => [2,5,4,3,1,6] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 2
[1,4,2,5,3,6] => [1,5,3,4,2,6] => [2,6,4,5,3,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 2
[1,4,3,2,5,6] => [1,4,3,2,5,6] => [2,5,4,3,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 2
[1,4,3,2,6,5] => [1,4,3,2,6,5] => [2,5,4,3,1,6] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 2
[1,4,3,5,2,6] => [1,5,3,4,2,6] => [2,6,4,5,3,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 2
[2,3,4,5,1,6] => [5,2,3,4,1,6] => [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[2,3,5,1,4,6] => [4,2,5,1,3,6] => [5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[2,3,5,4,1,6] => [5,2,4,3,1,6] => [6,3,5,4,2,1] => ([(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
[2,4,1,5,3,6] => [3,5,1,4,2,6] => [4,6,2,5,3,1] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[2,4,3,5,1,6] => [5,3,2,4,1,6] => [6,4,3,5,2,1] => ([(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
[2,4,5,1,3,6] => [4,5,3,1,2,6] => [5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[2,4,5,3,1,6] => [5,4,3,2,1,6] => [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
[2,5,1,3,4,6] => [3,5,1,4,2,6] => [4,6,2,5,3,1] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[2,5,1,4,3,6] => [3,5,1,4,2,6] => [4,6,2,5,3,1] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[2,5,3,1,4,6] => [4,5,3,1,2,6] => [5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[2,5,3,4,1,6] => [5,4,3,2,1,6] => [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
[2,5,4,1,3,6] => [4,5,3,1,2,6] => [5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[2,5,4,3,1,6] => [5,4,3,2,1,6] => [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
[3,1,4,5,2,6] => [5,2,3,4,1,6] => [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[3,1,5,2,4,6] => [4,2,5,1,3,6] => [5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[3,1,5,4,2,6] => [5,2,4,3,1,6] => [6,3,5,4,2,1] => ([(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
[3,2,4,5,1,6] => [5,2,3,4,1,6] => [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[3,2,5,1,4,6] => [4,2,5,1,3,6] => [5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[3,2,5,4,1,6] => [5,2,4,3,1,6] => [6,3,5,4,2,1] => ([(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
[3,4,1,5,2,6] => [5,3,2,4,1,6] => [6,4,3,5,2,1] => ([(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
[3,4,2,5,1,6] => [5,3,2,4,1,6] => [6,4,3,5,2,1] => ([(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
[3,4,5,1,2,6] => [5,4,3,2,1,6] => [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
[3,4,5,2,1,6] => [5,4,3,2,1,6] => [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
[3,5,1,2,4,6] => [4,5,3,1,2,6] => [5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[3,5,1,4,2,6] => [5,4,3,2,1,6] => [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
[3,5,2,1,4,6] => [4,5,3,1,2,6] => [5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[3,5,2,4,1,6] => [5,4,3,2,1,6] => [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
[3,5,4,1,2,6] => [5,4,3,2,1,6] => [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
[3,5,4,2,1,6] => [5,4,3,2,1,6] => [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
[4,1,2,5,3,6] => [5,2,3,4,1,6] => [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[4,1,3,5,2,6] => [5,2,3,4,1,6] => [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 4 + 2
[4,1,5,2,3,6] => [5,2,4,3,1,6] => [6,3,5,4,2,1] => ([(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
[4,1,5,3,2,6] => [5,2,4,3,1,6] => [6,3,5,4,2,1] => ([(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
[4,2,1,5,3,6] => [5,3,2,4,1,6] => [6,4,3,5,2,1] => ([(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: St001232
(load all 4 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: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 100%
Values
[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]
 => ? = 3 + 1
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => ? = 3 + 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]
 => ? = 3 + 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]
 => ? = 3 + 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
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => ? = 2 + 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => ? = 2 + 1
[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]
 => ? = 3 + 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]
 => ? = 3 + 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]
 => ? = 3 + 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]
 => ? = 3 + 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
[1,2,3,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 4 + 1
[1,2,3,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 + 1
[1,2,3,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 3 + 1
[1,2,3,6,4,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[1,2,3,6,5,4] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[1,2,4,3,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 + 1
[1,2,4,3,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 + 1
[1,2,4,5,3,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 + 1
[1,2,5,3,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2 + 1
[1,2,5,4,3,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2 + 1
[1,3,2,4,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 + 1
[1,3,2,5,4,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 3 + 1
[1,3,2,5,6,4] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 3 + 1
[1,3,2,6,4,5] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[1,3,2,6,5,4] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => [1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 3 + 1
[1,3,4,2,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 + 1
[1,3,4,2,6,5] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 + 1
[1,3,4,5,2,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 2 + 1
[1,3,5,2,4,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2 + 1
[1,3,5,4,2,6] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 2 + 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: St001879
(load all 2 compositions to match this statistic)
Mapping
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00009: Binary trees —left rotate⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 100%
Values
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 3 + 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 2 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 2 + 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 2 + 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 1
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2 + 1
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 3 + 1
[2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 3 + 1
[2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 3 + 1
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 3 + 1
[2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2 + 1
[2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2 + 1
[2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3 + 1
[2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 3 + 1
[2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 3 + 1
[3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 1
[3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 2 + 1
[3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3 + 1
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 1
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 2 + 1
[3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3 + 1
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3 + 1
[3,4,2,1,5] => [[[.,.],.],[.,[.,.]]]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3 + 1
[4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[4,1,3,2,5] => [[.,[[.,.],.]],[.,.]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[4,2,1,3,5] => [[[.,.],[.,.]],[.,.]]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 3 + 1
[4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 3 + 1
[4,3,1,2,5] => [[[.,[.,.]],.],[.,.]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 3 + 1
[1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
 => [[.,.],[.,[.,[[.,.],.]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 4 + 1
[1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]]
 => [[.,.],[.,[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ? = 2 + 1
[1,2,3,5,6,4] => [.,[.,[.,[[.,.],[.,.]]]]]
 => [[.,.],[.,[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ? = 3 + 1
[1,2,3,6,4,5] => [.,[.,[.,[[.,[.,.]],.]]]]
 => [[.,.],[.,[[.,[.,.]],.]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 3 + 1
[1,2,3,6,5,4] => [.,[.,[.,[[[.,.],.],.]]]]
 => [[.,.],[.,[[[.,.],.],.]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 3 + 1
[1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]]
 => [[.,.],[[.,.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 2 + 1
[1,2,4,3,6,5] => [.,[.,[[.,.],[[.,.],.]]]]
 => [[.,.],[[.,.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 2 + 1
[1,2,4,5,3,6] => [.,[.,[[.,.],[.,[.,.]]]]]
 => [[.,.],[[.,.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 2 + 1
[1,2,5,3,4,6] => [.,[.,[[.,[.,.]],[.,.]]]]
 => [[.,.],[[.,[.,.]],[.,.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 2 + 1
[1,2,5,4,3,6] => [.,[.,[[[.,.],.],[.,.]]]]
 => [[.,.],[[[.,.],.],[.,.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 2 + 1
[1,3,2,4,6,5] => [.,[[.,.],[.,[[.,.],.]]]]
 => [[.,[.,.]],[.,[[.,.],.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 2 + 1
[1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => [[.,[.,.]],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,3,2,5,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => [[.,[.,.]],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,3,2,6,4,5] => [.,[[.,.],[[.,[.,.]],.]]]
 => [[.,[.,.]],[[.,[.,.]],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 3 + 1
[1,3,2,6,5,4] => [.,[[.,.],[[[.,.],.],.]]]
 => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 3 + 1
[1,3,4,2,5,6] => [.,[[.,.],[.,[.,[.,.]]]]]
 => [[.,[.,.]],[.,[.,[.,.]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 2 + 1
[5,1,2,3,4,6] => [[.,[.,[.,[.,.]]]],[.,.]]
 => [[[.,[.,[.,[.,.]]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
[5,1,2,4,3,6] => [[.,[.,[[.,.],.]]],[.,.]]
 => [[[.,[.,[[.,.],.]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
[5,1,4,2,3,6] => [[.,[[.,[.,.]],.]],[.,.]]
 => [[[.,[[.,[.,.]],.]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
[5,1,4,3,2,6] => [[.,[[[.,.],.],.]],[.,.]]
 => [[[.,[[[.,.],.],.]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
[5,4,1,2,3,6] => [[[.,[.,[.,.]]],.],[.,.]]
 => [[[[.,[.,[.,.]]],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
[5,4,1,3,2,6] => [[[.,[[.,.],.]],.],[.,.]]
 => [[[[.,[[.,.],.]],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
[5,4,3,1,2,6] => [[[[.,[.,.]],.],.],[.,.]]
 => [[[[[.,[.,.]],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
[5,4,3,2,1,6] => [[[[[.,.],.],.],.],[.,.]]
 => [[[[[[.,.],.],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 4 + 1
[6,1,2,3,4,5,7] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => [[[.,[.,[.,[.,[.,.]]]]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[6,1,2,3,5,4,7] => [[.,[.,[.,[[.,.],.]]]],[.,.]]
 => [[[.,[.,[.,[[.,.],.]]]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[6,1,2,5,3,4,7] => [[.,[.,[[.,[.,.]],.]]],[.,.]]
 => [[[.,[.,[[.,[.,.]],.]]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[6,1,2,5,4,3,7] => [[.,[.,[[[.,.],.],.]]],[.,.]]
 => [[[.,[.,[[[.,.],.],.]]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[6,1,5,2,3,4,7] => [[.,[[.,[.,[.,.]]],.]],[.,.]]
 => [[[.,[[.,[.,[.,.]]],.]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[6,1,5,2,4,3,7] => [[.,[[.,[[.,.],.]],.]],[.,.]]
 => [[[.,[[.,[[.,.],.]],.]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[6,1,5,4,2,3,7] => [[.,[[[.,[.,.]],.],.]],[.,.]]
 => [[[.,[[[.,[.,.]],.],.]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[6,1,5,4,3,2,7] => [[.,[[[[.,.],.],.],.]],[.,.]]
 => [[[.,[[[[.,.],.],.],.]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[6,5,1,2,3,4,7] => [[[.,[.,[.,[.,.]]]],.],[.,.]]
 => [[[[.,[.,[.,[.,.]]]],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[6,5,1,2,4,3,7] => [[[.,[.,[[.,.],.]]],.],[.,.]]
 => [[[[.,[.,[[.,.],.]]],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[6,5,1,4,2,3,7] => [[[.,[[.,[.,.]],.]],.],[.,.]]
 => [[[[.,[[.,[.,.]],.]],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[6,5,1,4,3,2,7] => [[[.,[[[.,.],.],.]],.],[.,.]]
 => [[[[.,[[[.,.],.],.]],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[6,5,4,1,2,3,7] => [[[[.,[.,[.,.]]],.],.],[.,.]]
 => [[[[[.,[.,[.,.]]],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[6,5,4,1,3,2,7] => [[[[.,[[.,.],.]],.],.],[.,.]]
 => [[[[[.,[[.,.],.]],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[6,5,4,3,1,2,7] => [[[[[.,[.,.]],.],.],.],[.,.]]
 => [[[[[[.,[.,.]],.],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 1
[6,5,4,3,2,1,7] => [[[[[[.,.],.],.],.],.],[.,.]]
 => [[[[[[[.,.],.],.],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 5 + 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
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00009: Binary trees —left rotate⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 100%
Values
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 2
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => [[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 + 2
[2,1,4,3] => [[.,.],[[.,.],.]]
 => [[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 3 + 2
[2,3,1,4] => [[.,.],[.,[.,.]]]
 => [[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 2 + 2
[3,1,2,4] => [[.,[.,.]],[.,.]]
 => [[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[3,2,1,4] => [[[.,.],.],[.,.]]
 => [[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 2 + 2
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3 + 2
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 2 + 2
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 2 + 2
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => [[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3 + 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3 + 2
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2 + 2
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2 + 2
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 3 + 2
[2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 3 + 2
[2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
 => [[[.,.],[.,[.,.]]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 3 + 2
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 3 + 2
[2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2 + 2
[2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
 => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 2 + 2
[2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ? = 3 + 2
[2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 3 + 2
[2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
 => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ? = 3 + 2
[3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
 => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 2 + 2
[3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3 + 2
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 2 + 2
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
 => [[[[.,.],.],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 2 + 2
[3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3 + 2
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
 => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3 + 2
[3,4,2,1,5] => [[[.,.],.],[.,[.,.]]]
 => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ? = 3 + 2
[4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
 => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[4,1,3,2,5] => [[.,[[.,.],.]],[.,.]]
 => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[4,2,1,3,5] => [[[.,.],[.,.]],[.,.]]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 3 + 2
[4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
 => [[[[.,.],[.,.]],.],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 3 + 2
[4,3,1,2,5] => [[[.,[.,.]],.],[.,.]]
 => [[[[.,[.,.]],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => [[[[[.,.],.],.],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 3 + 2
[1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
 => [[.,.],[.,[.,[[.,.],.]]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 4 + 2
[1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]]
 => [[.,.],[.,[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ? = 2 + 2
[1,2,3,5,6,4] => [.,[.,[.,[[.,.],[.,.]]]]]
 => [[.,.],[.,[[.,.],[.,.]]]]
 => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ? = 3 + 2
[1,2,3,6,4,5] => [.,[.,[.,[[.,[.,.]],.]]]]
 => [[.,.],[.,[[.,[.,.]],.]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 3 + 2
[1,2,3,6,5,4] => [.,[.,[.,[[[.,.],.],.]]]]
 => [[.,.],[.,[[[.,.],.],.]]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 3 + 2
[1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]]
 => [[.,.],[[.,.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 2 + 2
[1,2,4,3,6,5] => [.,[.,[[.,.],[[.,.],.]]]]
 => [[.,.],[[.,.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 2 + 2
[1,2,4,5,3,6] => [.,[.,[[.,.],[.,[.,.]]]]]
 => [[.,.],[[.,.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 2 + 2
[1,2,5,3,4,6] => [.,[.,[[.,[.,.]],[.,.]]]]
 => [[.,.],[[.,[.,.]],[.,.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 2 + 2
[1,2,5,4,3,6] => [.,[.,[[[.,.],.],[.,.]]]]
 => [[.,.],[[[.,.],.],[.,.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 2 + 2
[1,3,2,4,6,5] => [.,[[.,.],[.,[[.,.],.]]]]
 => [[.,[.,.]],[.,[[.,.],.]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 2 + 2
[1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => [[.,[.,.]],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 3 + 2
[1,3,2,5,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => [[.,[.,.]],[[.,.],[.,.]]]
 => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 3 + 2
[1,3,2,6,4,5] => [.,[[.,.],[[.,[.,.]],.]]]
 => [[.,[.,.]],[[.,[.,.]],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 3 + 2
[1,3,2,6,5,4] => [.,[[.,.],[[[.,.],.],.]]]
 => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 3 + 2
[1,3,4,2,5,6] => [.,[[.,.],[.,[.,[.,.]]]]]
 => [[.,[.,.]],[.,[.,[.,.]]]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 2 + 2
[5,1,2,3,4,6] => [[.,[.,[.,[.,.]]]],[.,.]]
 => [[[.,[.,[.,[.,.]]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[5,1,2,4,3,6] => [[.,[.,[[.,.],.]]],[.,.]]
 => [[[.,[.,[[.,.],.]]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[5,1,4,2,3,6] => [[.,[[.,[.,.]],.]],[.,.]]
 => [[[.,[[.,[.,.]],.]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[5,1,4,3,2,6] => [[.,[[[.,.],.],.]],[.,.]]
 => [[[.,[[[.,.],.],.]],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[5,4,1,2,3,6] => [[[.,[.,[.,.]]],.],[.,.]]
 => [[[[.,[.,[.,.]]],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[5,4,1,3,2,6] => [[[.,[[.,.],.]],.],[.,.]]
 => [[[[.,[[.,.],.]],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[5,4,3,1,2,6] => [[[[.,[.,.]],.],.],[.,.]]
 => [[[[[.,[.,.]],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[5,4,3,2,1,6] => [[[[[.,.],.],.],.],[.,.]]
 => [[[[[[.,.],.],.],.],.],.]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 6 = 4 + 2
[6,1,2,3,4,5,7] => [[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => [[[.,[.,[.,[.,[.,.]]]]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[6,1,2,3,5,4,7] => [[.,[.,[.,[[.,.],.]]]],[.,.]]
 => [[[.,[.,[.,[[.,.],.]]]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[6,1,2,5,3,4,7] => [[.,[.,[[.,[.,.]],.]]],[.,.]]
 => [[[.,[.,[[.,[.,.]],.]]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[6,1,2,5,4,3,7] => [[.,[.,[[[.,.],.],.]]],[.,.]]
 => [[[.,[.,[[[.,.],.],.]]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[6,1,5,2,3,4,7] => [[.,[[.,[.,[.,.]]],.]],[.,.]]
 => [[[.,[[.,[.,[.,.]]],.]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[6,1,5,2,4,3,7] => [[.,[[.,[[.,.],.]],.]],[.,.]]
 => [[[.,[[.,[[.,.],.]],.]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[6,1,5,4,2,3,7] => [[.,[[[.,[.,.]],.],.]],[.,.]]
 => [[[.,[[[.,[.,.]],.],.]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[6,1,5,4,3,2,7] => [[.,[[[[.,.],.],.],.]],[.,.]]
 => [[[.,[[[[.,.],.],.],.]],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[6,5,1,2,3,4,7] => [[[.,[.,[.,[.,.]]]],.],[.,.]]
 => [[[[.,[.,[.,[.,.]]]],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[6,5,1,2,4,3,7] => [[[.,[.,[[.,.],.]]],.],[.,.]]
 => [[[[.,[.,[[.,.],.]]],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[6,5,1,4,2,3,7] => [[[.,[[.,[.,.]],.]],.],[.,.]]
 => [[[[.,[[.,[.,.]],.]],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[6,5,1,4,3,2,7] => [[[.,[[[.,.],.],.]],.],[.,.]]
 => [[[[.,[[[.,.],.],.]],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[6,5,4,1,2,3,7] => [[[[.,[.,[.,.]]],.],.],[.,.]]
 => [[[[[.,[.,[.,.]]],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[6,5,4,1,3,2,7] => [[[[.,[[.,.],.]],.],.],[.,.]]
 => [[[[[.,[[.,.],.]],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[6,5,4,3,1,2,7] => [[[[[.,[.,.]],.],.],.],[.,.]]
 => [[[[[[.,[.,.]],.],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
[6,5,4,3,2,1,7] => [[[[[[.,.],.],.],.],.],[.,.]]
 => [[[[[[[.,.],.],.],.],.],.],.]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 5 + 2
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
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: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Values
[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)
 => ? = 3 + 3
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 + 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)
 => ? = 3 + 3
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 + 3
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 + 3
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 + 3
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 + 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)
 => ? = 3 + 3
[2,1,4,5,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 + 3
[2,1,5,3,4] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 + 3
[2,1,5,4,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 + 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
[3,2,1,5,4] => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 + 3
[3,2,4,1,5] => [3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 3 + 3
[3,4,1,2,5] => [3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 + 3
[3,4,2,1,5] => [3,5,2,1,4] => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 + 3
[4,1,2,3,5] => [4,1,5,3,2] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 + 3
[4,1,3,2,5] => [4,1,5,3,2] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 + 3
[4,2,1,3,5] => [4,2,1,5,3] => [[[.,.],.],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 + 3
[4,2,3,1,5] => [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 3 + 3
[4,3,1,2,5] => [4,3,1,5,2] => [[[.,.],.],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 + 3
[4,3,2,1,5] => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 + 3
[1,2,3,4,6,5] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 4 + 3
[1,2,3,5,4,6] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 2 + 3
[1,2,3,5,6,4] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 3 + 3
[1,2,3,6,4,5] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 3 + 3
[1,2,3,6,5,4] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 3 + 3
[1,2,4,3,5,6] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 2 + 3
[1,2,4,3,6,5] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 2 + 3
[1,2,4,5,3,6] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 2 + 3
[1,2,5,3,4,6] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 2 + 3
[1,2,5,4,3,6] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 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: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Values
[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)
 => ? = 3 - 3
[2,1,4,3] => [2,1,4,3] => [[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 3 - 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)
 => ? = 3 - 3
[1,3,2,5,4] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 - 3
[1,3,4,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 - 3
[1,4,2,3,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 - 3
[1,4,3,2,5] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 - 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)
 => ? = 3 - 3
[2,1,4,5,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 - 3
[2,1,5,3,4] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 - 3
[2,1,5,4,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 - 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
[3,2,1,5,4] => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 2 - 3
[3,2,4,1,5] => [3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 3 - 3
[3,4,1,2,5] => [3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 - 3
[3,4,2,1,5] => [3,5,2,1,4] => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 - 3
[4,1,2,3,5] => [4,1,5,3,2] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 - 3
[4,1,3,2,5] => [4,1,5,3,2] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 - 3
[4,2,1,3,5] => [4,2,1,5,3] => [[[.,.],.],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 - 3
[4,2,3,1,5] => [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 3 - 3
[4,3,1,2,5] => [4,3,1,5,2] => [[[.,.],.],[[.,.],.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 - 3
[4,3,2,1,5] => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 3 - 3
[1,2,3,4,6,5] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 4 - 3
[1,2,3,5,4,6] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 2 - 3
[1,2,3,5,6,4] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 3 - 3
[1,2,3,6,4,5] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 3 - 3
[1,2,3,6,5,4] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 3 - 3
[1,2,4,3,5,6] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 2 - 3
[1,2,4,3,6,5] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 2 - 3
[1,2,4,5,3,6] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 2 - 3
[1,2,5,3,4,6] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 2 - 3
[1,2,5,4,3,6] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ? = 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.