- FindStat

Your data matches 124 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000422
(load all 53 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [.,.]
 => ([],1)
 => 0
[1,2] => [1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => 2
[2,1] => [1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,4,2,5,3,6] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,4,2,6,5,3] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,4,3,2,6,5] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,4,3,5,2,6] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,4,3,6,2,5] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,6,2,5,3,4] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,6,2,5,4,3] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,6,3,2,5,4] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,6,3,4,2,5] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,6,3,5,2,4] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,6,4,2,5,3] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,6,4,3,2,5] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,6,5,2,4,3] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,6,5,3,2,4] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,4,1,5,3,6] => [1,5,2,4,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,4,1,6,3,5] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,4,1,6,5,3] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,4,3,1,6,5] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,4,3,5,1,6] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,4,3,6,1,5] => [1,5,2,4,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,5,1,4,3,6] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,5,1,6,3,4] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,5,1,6,4,3] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,5,3,1,6,4] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,5,3,4,1,6] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,5,3,6,1,4] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,5,4,1,6,3] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,5,4,3,1,6] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,5,4,6,1,3] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,6,1,4,3,5] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,6,1,5,3,4] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,6,1,5,4,3] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,6,3,1,5,4] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,6,3,4,1,5] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6

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: St001721
(load all 3 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St001721: Binary words ⟶ ℤResult quality: 75% ●values known / values provided: 100%●distinct values known / distinct values provided: 75%

Values

[1] => [1] =>  =>  => ? = 0 - 1
[1,2] => [1,2] => 1 => 0 => 1 = 2 - 1
[2,1] => [1,2] => 1 => 0 => 1 = 2 - 1
[1,3,2,5,4,6] => [1,3,2,5,4,6] => 10101 => 01010 => 5 = 6 - 1
[1,3,2,6,4,5] => [1,3,2,6,4,5] => 10100 => 01011 => 5 = 6 - 1
[1,3,2,6,5,4] => [1,3,2,6,4,5] => 10100 => 01011 => 5 = 6 - 1
[1,4,2,5,3,6] => [1,4,2,5,3,6] => 10001 => 01110 => 5 = 6 - 1
[1,4,2,6,3,5] => [1,4,2,6,3,5] => 10000 => 01111 => 5 = 6 - 1
[1,4,2,6,5,3] => [1,4,2,6,3,5] => 10000 => 01111 => 5 = 6 - 1
[1,4,3,2,6,5] => [1,4,2,6,3,5] => 10000 => 01111 => 5 = 6 - 1
[1,4,3,5,2,6] => [1,4,2,6,3,5] => 10000 => 01111 => 5 = 6 - 1
[1,4,3,6,2,5] => [1,4,2,5,3,6] => 10001 => 01110 => 5 = 6 - 1
[1,5,2,4,3,6] => [1,5,2,4,3,6] => 10001 => 01110 => 5 = 6 - 1
[1,5,2,6,3,4] => [1,5,2,6,3,4] => 10000 => 01111 => 5 = 6 - 1
[1,5,2,6,4,3] => [1,5,2,6,3,4] => 10000 => 01111 => 5 = 6 - 1
[1,5,3,2,6,4] => [1,5,2,6,3,4] => 10000 => 01111 => 5 = 6 - 1
[1,5,3,4,2,6] => [1,5,2,6,3,4] => 10000 => 01111 => 5 = 6 - 1
[1,5,3,6,2,4] => [1,5,2,4,3,6] => 10001 => 01110 => 5 = 6 - 1
[1,5,4,2,6,3] => [1,5,2,6,3,4] => 10000 => 01111 => 5 = 6 - 1
[1,5,4,3,2,6] => [1,5,2,6,3,4] => 10000 => 01111 => 5 = 6 - 1
[1,6,2,4,3,5] => [1,6,2,4,3,5] => 10000 => 01111 => 5 = 6 - 1
[1,6,2,5,3,4] => [1,6,2,5,3,4] => 10000 => 01111 => 5 = 6 - 1
[1,6,2,5,4,3] => [1,6,2,5,3,4] => 10000 => 01111 => 5 = 6 - 1
[1,6,3,2,5,4] => [1,6,2,5,3,4] => 10000 => 01111 => 5 = 6 - 1
[1,6,3,4,2,5] => [1,6,2,5,3,4] => 10000 => 01111 => 5 = 6 - 1
[1,6,3,5,2,4] => [1,6,2,4,3,5] => 10000 => 01111 => 5 = 6 - 1
[1,6,4,2,5,3] => [1,6,2,5,3,4] => 10000 => 01111 => 5 = 6 - 1
[1,6,4,3,2,5] => [1,6,2,5,3,4] => 10000 => 01111 => 5 = 6 - 1
[1,6,5,2,4,3] => [1,6,2,4,3,5] => 10000 => 01111 => 5 = 6 - 1
[1,6,5,3,2,4] => [1,6,2,4,3,5] => 10000 => 01111 => 5 = 6 - 1
[2,4,1,5,3,6] => [1,5,2,4,3,6] => 10001 => 01110 => 5 = 6 - 1
[2,4,1,6,3,5] => [1,6,2,4,3,5] => 10000 => 01111 => 5 = 6 - 1
[2,4,1,6,5,3] => [1,6,2,4,3,5] => 10000 => 01111 => 5 = 6 - 1
[2,4,3,1,6,5] => [1,6,2,4,3,5] => 10000 => 01111 => 5 = 6 - 1
[2,4,3,5,1,6] => [1,6,2,4,3,5] => 10000 => 01111 => 5 = 6 - 1
[2,4,3,6,1,5] => [1,5,2,4,3,6] => 10001 => 01110 => 5 = 6 - 1
[2,5,1,4,3,6] => [1,4,2,5,3,6] => 10001 => 01110 => 5 = 6 - 1
[2,5,1,6,3,4] => [1,6,2,5,3,4] => 10000 => 01111 => 5 = 6 - 1
[2,5,1,6,4,3] => [1,6,2,5,3,4] => 10000 => 01111 => 5 = 6 - 1
[2,5,3,1,6,4] => [1,6,2,5,3,4] => 10000 => 01111 => 5 = 6 - 1
[2,5,3,4,1,6] => [1,6,2,5,3,4] => 10000 => 01111 => 5 = 6 - 1
[2,5,3,6,1,4] => [1,4,2,5,3,6] => 10001 => 01110 => 5 = 6 - 1
[2,5,4,1,6,3] => [1,6,2,5,3,4] => 10000 => 01111 => 5 = 6 - 1
[2,5,4,3,1,6] => [1,6,2,5,3,4] => 10000 => 01111 => 5 = 6 - 1
[2,5,4,6,1,3] => [1,3,2,5,4,6] => 10101 => 01010 => 5 = 6 - 1
[2,6,1,4,3,5] => [1,4,2,6,3,5] => 10000 => 01111 => 5 = 6 - 1
[2,6,1,5,3,4] => [1,5,2,6,3,4] => 10000 => 01111 => 5 = 6 - 1
[2,6,1,5,4,3] => [1,5,2,6,3,4] => 10000 => 01111 => 5 = 6 - 1
[2,6,3,1,5,4] => [1,5,2,6,3,4] => 10000 => 01111 => 5 = 6 - 1
[2,6,3,4,1,5] => [1,5,2,6,3,4] => 10000 => 01111 => 5 = 6 - 1
[2,6,3,5,1,4] => [1,4,2,6,3,5] => 10000 => 01111 => 5 = 6 - 1

Description

The degree of a binary word. A valley in a binary word is a letter $0$ which is not immediately followed by a $1$. A peak is a letter $1$ which is not immediately followed by a $0$. Let $f$ be the map that replaces every valley with a peak. The degree of a binary word $w$ is the number of times $f$ has to be applied to obtain a binary word without zeros.

Matching statistic: St000727
(load all 41 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000727: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 99%●distinct values known / distinct values provided: 50%

Values

[1] => [1] => [1] => [] => ? = 0 - 2
[1,2] => [1,2] => [1,2] => [1] => ? = 2 - 2
[2,1] => [1,2] => [1,2] => [1] => ? = 2 - 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4] => 4 = 6 - 2
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [1,3,2,5,6,4] => [1,3,2,5,4] => 4 = 6 - 2
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [1,3,2,5,6,4] => [1,3,2,5,4] => 4 = 6 - 2
[1,4,2,5,3,6] => [1,4,2,5,3,6] => [1,3,5,2,4,6] => [1,3,5,2,4] => 4 = 6 - 2
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [1,3,5,2,6,4] => [1,3,5,2,4] => 4 = 6 - 2
[1,4,2,6,5,3] => [1,4,2,6,3,5] => [1,3,5,2,6,4] => [1,3,5,2,4] => 4 = 6 - 2
[1,4,3,2,6,5] => [1,4,2,6,3,5] => [1,3,5,2,6,4] => [1,3,5,2,4] => 4 = 6 - 2
[1,4,3,5,2,6] => [1,4,2,6,3,5] => [1,3,5,2,6,4] => [1,3,5,2,4] => 4 = 6 - 2
[1,4,3,6,2,5] => [1,4,2,5,3,6] => [1,3,5,2,4,6] => [1,3,5,2,4] => 4 = 6 - 2
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [1,3,5,4,2,6] => [1,3,5,4,2] => 4 = 6 - 2
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [1,3,5,6,2,4] => [1,3,5,2,4] => 4 = 6 - 2
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [1,3,5,6,2,4] => [1,3,5,2,4] => 4 = 6 - 2
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [1,3,5,6,2,4] => [1,3,5,2,4] => 4 = 6 - 2
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [1,3,5,6,2,4] => [1,3,5,2,4] => 4 = 6 - 2
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [1,3,5,4,2,6] => [1,3,5,4,2] => 4 = 6 - 2
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [1,3,5,6,2,4] => [1,3,5,2,4] => 4 = 6 - 2
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [1,3,5,6,2,4] => [1,3,5,2,4] => 4 = 6 - 2
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [1,3,5,4,6,2] => [1,3,5,4,2] => 4 = 6 - 2
[1,6,2,5,3,4] => [1,6,2,5,3,4] => [1,3,5,6,4,2] => [1,3,5,4,2] => 4 = 6 - 2
[1,6,2,5,4,3] => [1,6,2,5,3,4] => [1,3,5,6,4,2] => [1,3,5,4,2] => 4 = 6 - 2
[1,6,3,2,5,4] => [1,6,2,5,3,4] => [1,3,5,6,4,2] => [1,3,5,4,2] => 4 = 6 - 2
[1,6,3,4,2,5] => [1,6,2,5,3,4] => [1,3,5,6,4,2] => [1,3,5,4,2] => 4 = 6 - 2
[1,6,3,5,2,4] => [1,6,2,4,3,5] => [1,3,5,4,6,2] => [1,3,5,4,2] => 4 = 6 - 2
[1,6,4,2,5,3] => [1,6,2,5,3,4] => [1,3,5,6,4,2] => [1,3,5,4,2] => 4 = 6 - 2
[1,6,4,3,2,5] => [1,6,2,5,3,4] => [1,3,5,6,4,2] => [1,3,5,4,2] => 4 = 6 - 2
[1,6,5,2,4,3] => [1,6,2,4,3,5] => [1,3,5,4,6,2] => [1,3,5,4,2] => 4 = 6 - 2
[1,6,5,3,2,4] => [1,6,2,4,3,5] => [1,3,5,4,6,2] => [1,3,5,4,2] => 4 = 6 - 2
[2,4,1,5,3,6] => [1,5,2,4,3,6] => [1,3,5,4,2,6] => [1,3,5,4,2] => 4 = 6 - 2
[2,4,1,6,3,5] => [1,6,2,4,3,5] => [1,3,5,4,6,2] => [1,3,5,4,2] => 4 = 6 - 2
[2,4,1,6,5,3] => [1,6,2,4,3,5] => [1,3,5,4,6,2] => [1,3,5,4,2] => 4 = 6 - 2
[2,4,3,1,6,5] => [1,6,2,4,3,5] => [1,3,5,4,6,2] => [1,3,5,4,2] => 4 = 6 - 2
[2,4,3,5,1,6] => [1,6,2,4,3,5] => [1,3,5,4,6,2] => [1,3,5,4,2] => 4 = 6 - 2
[2,4,3,6,1,5] => [1,5,2,4,3,6] => [1,3,5,4,2,6] => [1,3,5,4,2] => 4 = 6 - 2
[2,5,1,4,3,6] => [1,4,2,5,3,6] => [1,3,5,2,4,6] => [1,3,5,2,4] => 4 = 6 - 2
[2,5,1,6,3,4] => [1,6,2,5,3,4] => [1,3,5,6,4,2] => [1,3,5,4,2] => 4 = 6 - 2
[2,5,1,6,4,3] => [1,6,2,5,3,4] => [1,3,5,6,4,2] => [1,3,5,4,2] => 4 = 6 - 2
[2,5,3,1,6,4] => [1,6,2,5,3,4] => [1,3,5,6,4,2] => [1,3,5,4,2] => 4 = 6 - 2
[2,5,3,4,1,6] => [1,6,2,5,3,4] => [1,3,5,6,4,2] => [1,3,5,4,2] => 4 = 6 - 2
[2,5,3,6,1,4] => [1,4,2,5,3,6] => [1,3,5,2,4,6] => [1,3,5,2,4] => 4 = 6 - 2
[2,5,4,1,6,3] => [1,6,2,5,3,4] => [1,3,5,6,4,2] => [1,3,5,4,2] => 4 = 6 - 2
[2,5,4,3,1,6] => [1,6,2,5,3,4] => [1,3,5,6,4,2] => [1,3,5,4,2] => 4 = 6 - 2
[2,5,4,6,1,3] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4] => 4 = 6 - 2
[2,6,1,4,3,5] => [1,4,2,6,3,5] => [1,3,5,2,6,4] => [1,3,5,2,4] => 4 = 6 - 2
[2,6,1,5,3,4] => [1,5,2,6,3,4] => [1,3,5,6,2,4] => [1,3,5,2,4] => 4 = 6 - 2
[2,6,1,5,4,3] => [1,5,2,6,3,4] => [1,3,5,6,2,4] => [1,3,5,2,4] => 4 = 6 - 2
[2,6,3,1,5,4] => [1,5,2,6,3,4] => [1,3,5,6,2,4] => [1,3,5,2,4] => 4 = 6 - 2
[2,6,3,4,1,5] => [1,5,2,6,3,4] => [1,3,5,6,2,4] => [1,3,5,2,4] => 4 = 6 - 2
[2,6,3,5,1,4] => [1,4,2,6,3,5] => [1,3,5,2,6,4] => [1,3,5,2,4] => 4 = 6 - 2
[2,6,4,1,5,3] => [1,5,2,6,3,4] => [1,3,5,6,2,4] => [1,3,5,2,4] => 4 = 6 - 2
[2,6,4,3,1,5] => [1,5,2,6,3,4] => [1,3,5,6,2,4] => [1,3,5,2,4] => 4 = 6 - 2

Description

The largest label of a leaf in the binary search tree associated with the permutation. Alternatively, this is 1 plus the position of the last descent of the inverse of the reversal of the permutation, and 1 if there is no descent.

Matching statistic: St000641
(load all 2 compositions to match this statistic)

Mapping

Mp00252: Permutations —restriction⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000641: Posets ⟶ ℤResult quality: 50% ●values known / values provided: 99%●distinct values known / distinct values provided: 50%

Values

[1] => [] => .
 => ?
 => ? = 0 + 3
[1,2] => [1] => [.,.]
 => ([],1)
 => ? = 2 + 3
[2,1] => [1] => [.,.]
 => ([],1)
 => ? = 2 + 3
[1,3,2,5,4,6] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[1,3,2,6,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[1,3,2,6,5,4] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[1,4,2,5,3,6] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[1,4,2,6,3,5] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[1,4,2,6,5,3] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[1,4,3,2,6,5] => [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[1,4,3,5,2,6] => [1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 9 = 6 + 3
[1,4,3,6,2,5] => [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[1,5,2,4,3,6] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[1,5,2,6,3,4] => [1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[1,5,2,6,4,3] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[1,5,3,2,6,4] => [1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[1,5,3,4,2,6] => [1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 9 = 6 + 3
[1,5,3,6,2,4] => [1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[1,5,4,2,6,3] => [1,5,4,2,3] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[1,5,4,3,2,6] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 9 = 6 + 3
[1,6,2,4,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 9 = 6 + 3
[1,6,2,5,3,4] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 9 = 6 + 3
[1,6,2,5,4,3] => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 9 = 6 + 3
[1,6,3,2,5,4] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[1,6,3,4,2,5] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[1,6,3,5,2,4] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[1,6,4,2,5,3] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[1,6,4,3,2,5] => [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[1,6,5,2,4,3] => [1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[1,6,5,3,2,4] => [1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[2,4,1,5,3,6] => [2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 9 = 6 + 3
[2,4,1,6,3,5] => [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 9 = 6 + 3
[2,4,1,6,5,3] => [2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 9 = 6 + 3
[2,4,3,1,6,5] => [2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3
[2,4,3,5,1,6] => [2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 9 = 6 + 3
[2,4,3,6,1,5] => [2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3
[2,5,1,4,3,6] => [2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 9 = 6 + 3
[2,5,1,6,3,4] => [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 9 = 6 + 3
[2,5,1,6,4,3] => [2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 9 = 6 + 3
[2,5,3,1,6,4] => [2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3
[2,5,3,4,1,6] => [2,5,3,4,1] => [[.,[[.,.],[.,.]]],.]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => 9 = 6 + 3
[2,5,3,6,1,4] => [2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3
[2,5,4,1,6,3] => [2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3
[2,5,4,3,1,6] => [2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 9 = 6 + 3
[2,5,4,6,1,3] => [2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3
[2,6,1,4,3,5] => [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 9 = 6 + 3
[2,6,1,5,3,4] => [2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 9 = 6 + 3
[2,6,1,5,4,3] => [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3
[2,6,3,1,5,4] => [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 9 = 6 + 3
[2,6,3,4,1,5] => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3
[2,6,3,5,1,4] => [2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3
[2,6,4,1,5,3] => [2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 9 = 6 + 3
[2,6,4,3,1,5] => [2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3

Description

The number of non-empty boolean intervals in a poset.

Matching statistic: St000941
(load all 2 compositions to match this statistic)

Mapping

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000941: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 99%●distinct values known / distinct values provided: 50%

Values

[1] => [1] => [1]
 => []
 => ? = 0 - 5
[1,2] => [1,2] => [1,1]
 => [1]
 => ? = 2 - 5
[2,1] => [2,1] => [2]
 => []
 => ? = 2 - 5
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[1,3,2,6,4,5] => [1,3,2,6,5,4] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[1,3,2,6,5,4] => [1,3,2,6,5,4] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[1,4,2,5,3,6] => [1,5,3,4,2,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => 1 = 6 - 5
[1,4,2,6,3,5] => [1,5,3,6,2,4] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[1,4,2,6,5,3] => [1,6,3,5,4,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[1,4,3,2,6,5] => [1,4,3,2,6,5] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[1,4,3,5,2,6] => [1,5,3,4,2,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => 1 = 6 - 5
[1,4,3,6,2,5] => [1,5,3,6,2,4] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[1,5,2,4,3,6] => [1,5,3,4,2,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => 1 = 6 - 5
[1,5,2,6,3,4] => [1,6,3,5,4,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[1,5,2,6,4,3] => [1,6,3,5,4,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[1,5,3,2,6,4] => [1,6,4,3,5,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[1,5,3,4,2,6] => [1,5,4,3,2,6] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[1,5,3,6,2,4] => [1,6,5,4,3,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[1,5,4,2,6,3] => [1,6,4,3,5,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[1,5,4,3,2,6] => [1,5,4,3,2,6] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[1,6,2,4,3,5] => [1,6,3,5,4,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[1,6,2,5,3,4] => [1,6,3,5,4,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[1,6,2,5,4,3] => [1,6,3,5,4,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[1,6,3,2,5,4] => [1,6,4,3,5,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[1,6,3,4,2,5] => [1,6,5,4,3,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[1,6,3,5,2,4] => [1,6,5,4,3,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[1,6,4,2,5,3] => [1,6,5,4,3,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[1,6,4,3,2,5] => [1,6,5,4,3,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[1,6,5,2,4,3] => [1,6,5,4,3,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[1,6,5,3,2,4] => [1,6,5,4,3,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[2,4,1,5,3,6] => [3,5,1,4,2,6] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[2,4,1,6,3,5] => [3,5,1,6,2,4] => [2,2,2]
 => [2,2]
 => 1 = 6 - 5
[2,4,1,6,5,3] => [3,6,1,5,4,2] => [2,2,2]
 => [2,2]
 => 1 = 6 - 5
[2,4,3,1,6,5] => [4,3,2,1,6,5] => [2,2,2]
 => [2,2]
 => 1 = 6 - 5
[2,4,3,5,1,6] => [5,3,2,4,1,6] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[2,4,3,6,1,5] => [5,3,2,6,1,4] => [2,2,2]
 => [2,2]
 => 1 = 6 - 5
[2,5,1,4,3,6] => [3,5,1,4,2,6] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[2,5,1,6,3,4] => [3,6,1,5,4,2] => [2,2,2]
 => [2,2]
 => 1 = 6 - 5
[2,5,1,6,4,3] => [3,6,1,5,4,2] => [2,2,2]
 => [2,2]
 => 1 = 6 - 5
[2,5,3,1,6,4] => [4,6,3,1,5,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[2,5,3,4,1,6] => [5,4,3,2,1,6] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[2,5,3,6,1,4] => [5,6,3,4,1,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[2,5,4,1,6,3] => [4,6,3,1,5,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[2,5,4,3,1,6] => [5,4,3,2,1,6] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[2,5,4,6,1,3] => [5,6,3,4,1,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[2,6,1,4,3,5] => [3,6,1,5,4,2] => [2,2,2]
 => [2,2]
 => 1 = 6 - 5
[2,6,1,5,3,4] => [3,6,1,5,4,2] => [2,2,2]
 => [2,2]
 => 1 = 6 - 5
[2,6,1,5,4,3] => [3,6,1,5,4,2] => [2,2,2]
 => [2,2]
 => 1 = 6 - 5
[2,6,3,1,5,4] => [4,6,3,1,5,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[2,6,3,4,1,5] => [5,6,3,4,1,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[2,6,3,5,1,4] => [5,6,3,4,1,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[2,6,4,1,5,3] => [4,6,5,1,3,2] => [2,2,2]
 => [2,2]
 => 1 = 6 - 5
[2,6,4,3,1,5] => [5,6,4,3,1,2] => [2,2,2]
 => [2,2]
 => 1 = 6 - 5

Description

The number of characters of the symmetric group whose value on the partition is even.

Matching statistic: St000990

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000990: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 99%●distinct values known / distinct values provided: 50%

Values

[1] => [1] => [1] => [] => ? = 0 - 5
[1,2] => [1,2] => [2,1] => [1] => ? = 2 - 5
[2,1] => [1,2] => [2,1] => [1] => ? = 2 - 5
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [6,4,5,2,3,1] => [4,5,2,3,1] => 1 = 6 - 5
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [6,4,5,1,3,2] => [4,5,1,3,2] => 1 = 6 - 5
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [6,4,5,1,3,2] => [4,5,1,3,2] => 1 = 6 - 5
[1,4,2,5,3,6] => [1,4,2,5,3,6] => [6,3,5,2,4,1] => [3,5,2,4,1] => 1 = 6 - 5
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [6,3,5,1,4,2] => [3,5,1,4,2] => 1 = 6 - 5
[1,4,2,6,5,3] => [1,4,2,6,3,5] => [6,3,5,1,4,2] => [3,5,1,4,2] => 1 = 6 - 5
[1,4,3,2,6,5] => [1,4,2,6,3,5] => [6,3,5,1,4,2] => [3,5,1,4,2] => 1 = 6 - 5
[1,4,3,5,2,6] => [1,4,2,6,3,5] => [6,3,5,1,4,2] => [3,5,1,4,2] => 1 = 6 - 5
[1,4,3,6,2,5] => [1,4,2,5,3,6] => [6,3,5,2,4,1] => [3,5,2,4,1] => 1 = 6 - 5
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [6,2,5,3,4,1] => [2,5,3,4,1] => 1 = 6 - 5
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [6,2,5,1,4,3] => [2,5,1,4,3] => 1 = 6 - 5
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [6,2,5,1,4,3] => [2,5,1,4,3] => 1 = 6 - 5
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [6,2,5,1,4,3] => [2,5,1,4,3] => 1 = 6 - 5
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [6,2,5,1,4,3] => [2,5,1,4,3] => 1 = 6 - 5
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [6,2,5,3,4,1] => [2,5,3,4,1] => 1 = 6 - 5
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [6,2,5,1,4,3] => [2,5,1,4,3] => 1 = 6 - 5
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [6,2,5,1,4,3] => [2,5,1,4,3] => 1 = 6 - 5
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [6,1,5,3,4,2] => [1,5,3,4,2] => 1 = 6 - 5
[1,6,2,5,3,4] => [1,6,2,5,3,4] => [6,1,5,2,4,3] => [1,5,2,4,3] => 1 = 6 - 5
[1,6,2,5,4,3] => [1,6,2,5,3,4] => [6,1,5,2,4,3] => [1,5,2,4,3] => 1 = 6 - 5
[1,6,3,2,5,4] => [1,6,2,5,3,4] => [6,1,5,2,4,3] => [1,5,2,4,3] => 1 = 6 - 5
[1,6,3,4,2,5] => [1,6,2,5,3,4] => [6,1,5,2,4,3] => [1,5,2,4,3] => 1 = 6 - 5
[1,6,3,5,2,4] => [1,6,2,4,3,5] => [6,1,5,3,4,2] => [1,5,3,4,2] => 1 = 6 - 5
[1,6,4,2,5,3] => [1,6,2,5,3,4] => [6,1,5,2,4,3] => [1,5,2,4,3] => 1 = 6 - 5
[1,6,4,3,2,5] => [1,6,2,5,3,4] => [6,1,5,2,4,3] => [1,5,2,4,3] => 1 = 6 - 5
[1,6,5,2,4,3] => [1,6,2,4,3,5] => [6,1,5,3,4,2] => [1,5,3,4,2] => 1 = 6 - 5
[1,6,5,3,2,4] => [1,6,2,4,3,5] => [6,1,5,3,4,2] => [1,5,3,4,2] => 1 = 6 - 5
[2,4,1,5,3,6] => [1,5,2,4,3,6] => [6,2,5,3,4,1] => [2,5,3,4,1] => 1 = 6 - 5
[2,4,1,6,3,5] => [1,6,2,4,3,5] => [6,1,5,3,4,2] => [1,5,3,4,2] => 1 = 6 - 5
[2,4,1,6,5,3] => [1,6,2,4,3,5] => [6,1,5,3,4,2] => [1,5,3,4,2] => 1 = 6 - 5
[2,4,3,1,6,5] => [1,6,2,4,3,5] => [6,1,5,3,4,2] => [1,5,3,4,2] => 1 = 6 - 5
[2,4,3,5,1,6] => [1,6,2,4,3,5] => [6,1,5,3,4,2] => [1,5,3,4,2] => 1 = 6 - 5
[2,4,3,6,1,5] => [1,5,2,4,3,6] => [6,2,5,3,4,1] => [2,5,3,4,1] => 1 = 6 - 5
[2,5,1,4,3,6] => [1,4,2,5,3,6] => [6,3,5,2,4,1] => [3,5,2,4,1] => 1 = 6 - 5
[2,5,1,6,3,4] => [1,6,2,5,3,4] => [6,1,5,2,4,3] => [1,5,2,4,3] => 1 = 6 - 5
[2,5,1,6,4,3] => [1,6,2,5,3,4] => [6,1,5,2,4,3] => [1,5,2,4,3] => 1 = 6 - 5
[2,5,3,1,6,4] => [1,6,2,5,3,4] => [6,1,5,2,4,3] => [1,5,2,4,3] => 1 = 6 - 5
[2,5,3,4,1,6] => [1,6,2,5,3,4] => [6,1,5,2,4,3] => [1,5,2,4,3] => 1 = 6 - 5
[2,5,3,6,1,4] => [1,4,2,5,3,6] => [6,3,5,2,4,1] => [3,5,2,4,1] => 1 = 6 - 5
[2,5,4,1,6,3] => [1,6,2,5,3,4] => [6,1,5,2,4,3] => [1,5,2,4,3] => 1 = 6 - 5
[2,5,4,3,1,6] => [1,6,2,5,3,4] => [6,1,5,2,4,3] => [1,5,2,4,3] => 1 = 6 - 5
[2,5,4,6,1,3] => [1,3,2,5,4,6] => [6,4,5,2,3,1] => [4,5,2,3,1] => 1 = 6 - 5
[2,6,1,4,3,5] => [1,4,2,6,3,5] => [6,3,5,1,4,2] => [3,5,1,4,2] => 1 = 6 - 5
[2,6,1,5,3,4] => [1,5,2,6,3,4] => [6,2,5,1,4,3] => [2,5,1,4,3] => 1 = 6 - 5
[2,6,1,5,4,3] => [1,5,2,6,3,4] => [6,2,5,1,4,3] => [2,5,1,4,3] => 1 = 6 - 5
[2,6,3,1,5,4] => [1,5,2,6,3,4] => [6,2,5,1,4,3] => [2,5,1,4,3] => 1 = 6 - 5
[2,6,3,4,1,5] => [1,5,2,6,3,4] => [6,2,5,1,4,3] => [2,5,1,4,3] => 1 = 6 - 5
[2,6,3,5,1,4] => [1,4,2,6,3,5] => [6,3,5,1,4,2] => [3,5,1,4,2] => 1 = 6 - 5
[2,6,4,1,5,3] => [1,5,2,6,3,4] => [6,2,5,1,4,3] => [2,5,1,4,3] => 1 = 6 - 5
[2,6,4,3,1,5] => [1,5,2,6,3,4] => [6,2,5,1,4,3] => [2,5,1,4,3] => 1 = 6 - 5

Description

The first ascent of a permutation. For a permutation $\pi$, this is the smallest index such that $\pi(i) < \pi(i+1)$. For the first descent, see [[St000654]].

Matching statistic: St000455
(load all 5 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 46%●distinct values known / distinct values provided: 25%

Values

[1] => [1] => [1] => ([],1)
 => ? = 0 - 8
[1,2] => [1,2] => [2] => ([],2)
 => ? = 2 - 8
[2,1] => [1,2] => [2] => ([],2)
 => ? = 2 - 8
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,4,2,5,3,6] => [1,4,2,5,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,4,2,6,5,3] => [1,4,2,6,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,4,3,2,6,5] => [1,4,2,6,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,4,3,5,2,6] => [1,4,2,6,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,4,3,6,2,5] => [1,4,2,5,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,6,2,5,3,4] => [1,6,2,5,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,6,2,5,4,3] => [1,6,2,5,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,6,3,2,5,4] => [1,6,2,5,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,6,3,4,2,5] => [1,6,2,5,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,6,3,5,2,4] => [1,6,2,4,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,6,4,2,5,3] => [1,6,2,5,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,6,4,3,2,5] => [1,6,2,5,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,6,5,2,4,3] => [1,6,2,4,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,6,5,3,2,4] => [1,6,2,4,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[2,4,1,5,3,6] => [1,5,2,4,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[2,4,1,6,3,5] => [1,6,2,4,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[2,4,1,6,5,3] => [1,6,2,4,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[2,4,3,1,6,5] => [1,6,2,4,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[2,4,3,5,1,6] => [1,6,2,4,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[2,4,3,6,1,5] => [1,5,2,4,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[2,5,1,4,3,6] => [1,4,2,5,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[2,5,1,6,3,4] => [1,6,2,5,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[2,5,1,6,4,3] => [1,6,2,5,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[2,5,3,1,6,4] => [1,6,2,5,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[2,5,3,4,1,6] => [1,6,2,5,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[2,5,3,6,1,4] => [1,4,2,5,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[2,5,4,1,6,3] => [1,6,2,5,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[2,5,4,3,1,6] => [1,6,2,5,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[2,5,4,6,1,3] => [1,3,2,5,4,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[2,6,1,4,3,5] => [1,4,2,6,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[2,6,1,5,3,4] => [1,5,2,6,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[2,6,1,5,4,3] => [1,5,2,6,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[2,6,3,1,5,4] => [1,5,2,6,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[2,6,3,4,1,5] => [1,5,2,6,3,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 8
[1,2,4,5,3,6,7] => [1,2,4,5,3,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[1,2,4,6,3,5,7] => [1,2,4,6,3,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[1,2,4,6,5,7,3] => [1,2,4,6,3,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[1,2,4,7,3,5,6] => [1,2,4,7,3,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[1,2,4,7,5,6,3] => [1,2,4,7,3,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[1,2,4,7,6,3,5] => [1,2,4,7,3,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[1,2,4,7,6,5,3] => [1,2,4,7,3,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[1,2,5,6,3,4,7] => [1,2,5,6,3,4,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[1,2,5,6,4,7,3] => [1,2,5,6,3,4,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[1,2,5,7,3,4,6] => [1,2,5,7,3,4,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[1,2,5,7,4,6,3] => [1,2,5,7,3,4,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[1,2,5,7,6,3,4] => [1,2,5,7,3,4,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[1,2,5,7,6,4,3] => [1,2,5,7,3,4,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[1,2,6,7,3,4,5] => [1,2,6,7,3,4,5] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[1,2,6,7,4,5,3] => [1,2,6,7,3,4,5] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[1,2,6,7,5,3,4] => [1,2,6,7,3,4,5] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[1,2,6,7,5,4,3] => [1,2,6,7,3,4,5] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[2,4,5,3,6,7,1] => [1,2,4,5,3,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[2,4,6,3,5,7,1] => [1,2,4,6,3,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[2,4,6,5,7,3,1] => [1,2,4,6,3,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[2,4,7,3,5,6,1] => [1,2,4,7,3,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[2,4,7,5,6,3,1] => [1,2,4,7,3,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[2,4,7,6,3,5,1] => [1,2,4,7,3,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[2,4,7,6,5,3,1] => [1,2,4,7,3,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[2,5,6,3,4,7,1] => [1,2,5,6,3,4,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[2,5,6,4,7,3,1] => [1,2,5,6,3,4,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[2,5,7,3,4,6,1] => [1,2,5,7,3,4,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[2,5,7,4,6,3,1] => [1,2,5,7,3,4,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[2,5,7,6,3,4,1] => [1,2,5,7,3,4,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[2,5,7,6,4,3,1] => [1,2,5,7,3,4,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[2,6,7,3,4,5,1] => [1,2,6,7,3,4,5] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[2,6,7,4,5,3,1] => [1,2,6,7,3,4,5] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[2,6,7,5,3,4,1] => [1,2,6,7,3,4,5] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[2,6,7,5,4,3,1] => [1,2,6,7,3,4,5] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[3,1,2,4,6,5,7] => [1,2,4,6,3,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[3,1,2,4,7,5,6] => [1,2,4,7,3,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[3,1,2,4,7,6,5] => [1,2,4,7,3,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[3,1,2,5,6,4,7] => [1,2,5,6,3,4,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[3,1,2,5,7,4,6] => [1,2,5,7,3,4,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[3,1,2,5,7,6,4] => [1,2,5,7,3,4,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[3,1,2,6,7,4,5] => [1,2,6,7,3,4,5] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[3,1,2,6,7,5,4] => [1,2,6,7,3,4,5] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[3,2,4,6,5,7,1] => [1,2,4,6,3,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[3,2,4,7,5,6,1] => [1,2,4,7,3,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[3,2,4,7,6,5,1] => [1,2,4,7,3,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[3,2,5,6,4,7,1] => [1,2,5,6,3,4,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[3,2,5,7,4,6,1] => [1,2,5,7,3,4,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[3,2,5,7,6,4,1] => [1,2,5,7,3,4,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[3,2,6,7,4,5,1] => [1,2,6,7,3,4,5] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8
[3,2,6,7,5,4,1] => [1,2,6,7,3,4,5] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0 = 8 - 8

Description

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.

Matching statistic: St000235
(load all 11 compositions to match this statistic)

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000235: Permutations ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 75%

Values

[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 2
[2,1] => [1,2] => [1,2] => [1,2] => 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,6,5,4,3,2] => 6
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [1,3,2,6,4,5] => [1,6,5,4,3,2] => 6
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [1,3,2,6,4,5] => [1,6,5,4,3,2] => 6
[1,4,2,5,3,6] => [1,4,2,5,3,6] => [1,4,2,5,3,6] => [1,6,5,4,3,2] => 6
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [1,4,2,6,3,5] => [1,6,5,4,3,2] => 6
[1,4,2,6,5,3] => [1,4,2,6,3,5] => [1,4,2,6,3,5] => [1,6,5,4,3,2] => 6
[1,4,3,2,6,5] => [1,4,2,6,3,5] => [1,4,2,6,3,5] => [1,6,5,4,3,2] => 6
[1,4,3,5,2,6] => [1,4,2,6,3,5] => [1,4,2,6,3,5] => [1,6,5,4,3,2] => 6
[1,4,3,6,2,5] => [1,4,2,5,3,6] => [1,4,2,5,3,6] => [1,6,5,4,3,2] => 6
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [1,5,2,4,3,6] => [1,6,5,4,3,2] => 6
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [1,5,2,6,3,4] => [1,6,5,4,3,2] => 6
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [1,5,2,6,3,4] => [1,6,5,4,3,2] => 6
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [1,5,2,6,3,4] => [1,6,5,4,3,2] => 6
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [1,5,2,6,3,4] => [1,6,5,4,3,2] => 6
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [1,5,2,4,3,6] => [1,6,5,4,3,2] => 6
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [1,5,2,6,3,4] => [1,6,5,4,3,2] => 6
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [1,5,2,6,3,4] => [1,6,5,4,3,2] => 6
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [1,6,2,4,3,5] => [1,6,5,4,3,2] => 6
[1,6,2,5,3,4] => [1,6,2,5,3,4] => [1,6,2,5,3,4] => [1,6,5,4,3,2] => 6
[1,6,2,5,4,3] => [1,6,2,5,3,4] => [1,6,2,5,3,4] => [1,6,5,4,3,2] => 6
[1,6,3,2,5,4] => [1,6,2,5,3,4] => [1,6,2,5,3,4] => [1,6,5,4,3,2] => 6
[1,6,3,4,2,5] => [1,6,2,5,3,4] => [1,6,2,5,3,4] => [1,6,5,4,3,2] => 6
[1,6,3,5,2,4] => [1,6,2,4,3,5] => [1,6,2,4,3,5] => [1,6,5,4,3,2] => 6
[1,6,4,2,5,3] => [1,6,2,5,3,4] => [1,6,2,5,3,4] => [1,6,5,4,3,2] => 6
[1,6,4,3,2,5] => [1,6,2,5,3,4] => [1,6,2,5,3,4] => [1,6,5,4,3,2] => 6
[1,6,5,2,4,3] => [1,6,2,4,3,5] => [1,6,2,4,3,5] => [1,6,5,4,3,2] => 6
[1,6,5,3,2,4] => [1,6,2,4,3,5] => [1,6,2,4,3,5] => [1,6,5,4,3,2] => 6
[2,4,1,5,3,6] => [1,5,2,4,3,6] => [1,5,2,4,3,6] => [1,6,5,4,3,2] => 6
[2,4,1,6,3,5] => [1,6,2,4,3,5] => [1,6,2,4,3,5] => [1,6,5,4,3,2] => 6
[2,4,1,6,5,3] => [1,6,2,4,3,5] => [1,6,2,4,3,5] => [1,6,5,4,3,2] => 6
[2,4,3,1,6,5] => [1,6,2,4,3,5] => [1,6,2,4,3,5] => [1,6,5,4,3,2] => 6
[2,4,3,5,1,6] => [1,6,2,4,3,5] => [1,6,2,4,3,5] => [1,6,5,4,3,2] => 6
[2,4,3,6,1,5] => [1,5,2,4,3,6] => [1,5,2,4,3,6] => [1,6,5,4,3,2] => 6
[2,5,1,4,3,6] => [1,4,2,5,3,6] => [1,4,2,5,3,6] => [1,6,5,4,3,2] => 6
[2,5,1,6,3,4] => [1,6,2,5,3,4] => [1,6,2,5,3,4] => [1,6,5,4,3,2] => 6
[2,5,1,6,4,3] => [1,6,2,5,3,4] => [1,6,2,5,3,4] => [1,6,5,4,3,2] => 6
[2,5,3,1,6,4] => [1,6,2,5,3,4] => [1,6,2,5,3,4] => [1,6,5,4,3,2] => 6
[2,5,3,4,1,6] => [1,6,2,5,3,4] => [1,6,2,5,3,4] => [1,6,5,4,3,2] => 6
[2,5,3,6,1,4] => [1,4,2,5,3,6] => [1,4,2,5,3,6] => [1,6,5,4,3,2] => 6
[2,5,4,1,6,3] => [1,6,2,5,3,4] => [1,6,2,5,3,4] => [1,6,5,4,3,2] => 6
[2,5,4,3,1,6] => [1,6,2,5,3,4] => [1,6,2,5,3,4] => [1,6,5,4,3,2] => 6
[2,5,4,6,1,3] => [1,3,2,5,4,6] => [1,3,2,5,4,6] => [1,6,5,4,3,2] => 6
[2,6,1,4,3,5] => [1,4,2,6,3,5] => [1,4,2,6,3,5] => [1,6,5,4,3,2] => 6
[2,6,1,5,3,4] => [1,5,2,6,3,4] => [1,5,2,6,3,4] => [1,6,5,4,3,2] => 6
[2,6,1,5,4,3] => [1,5,2,6,3,4] => [1,5,2,6,3,4] => [1,6,5,4,3,2] => 6
[2,6,3,1,5,4] => [1,5,2,6,3,4] => [1,5,2,6,3,4] => [1,6,5,4,3,2] => 6
[2,6,3,4,1,5] => [1,5,2,6,3,4] => [1,5,2,6,3,4] => [1,6,5,4,3,2] => 6
[1,2,4,5,3,6,7] => [1,2,4,5,3,6,7] => [1,2,4,3,5,6,7] => [1,7,6,5,4,3,2] => ? = 8
[1,2,4,5,3,7,6] => [1,2,4,5,3,7,6] => [4,1,7,2,3,5,6] => [4,1,7,6,5,3,2] => ? = 8
[1,2,4,6,3,5,7] => [1,2,4,6,3,5,7] => [1,4,6,2,3,5,7] => [1,7,6,5,4,3,2] => ? = 8
[1,2,4,6,3,7,5] => [1,2,4,6,3,7,5] => [4,1,6,2,3,5,7] => [4,1,7,6,5,3,2] => ? = 8
[1,2,4,6,5,3,7] => [1,2,4,6,3,7,5] => [4,1,6,2,3,5,7] => [4,1,7,6,5,3,2] => ? = 8
[1,2,4,6,5,7,3] => [1,2,4,6,3,5,7] => [1,4,6,2,3,5,7] => [1,7,6,5,4,3,2] => ? = 8
[1,2,4,7,3,5,6] => [1,2,4,7,3,5,6] => [1,4,7,2,3,5,6] => [1,7,6,5,4,3,2] => ? = 8
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [7,4,6,1,2,3,5] => [7,4,6,1,5,3,2] => ? = 8
[1,2,4,7,5,3,6] => [1,2,4,7,3,6,5] => [7,4,6,1,2,3,5] => [7,4,6,1,5,3,2] => ? = 8
[1,2,4,7,5,6,3] => [1,2,4,7,3,5,6] => [1,4,7,2,3,5,6] => [1,7,6,5,4,3,2] => ? = 8
[1,2,4,7,6,3,5] => [1,2,4,7,3,5,6] => [1,4,7,2,3,5,6] => [1,7,6,5,4,3,2] => ? = 8
[1,2,4,7,6,5,3] => [1,2,4,7,3,5,6] => [1,4,7,2,3,5,6] => [1,7,6,5,4,3,2] => ? = 8
[1,2,5,6,3,4,7] => [1,2,5,6,3,4,7] => [1,5,6,2,3,4,7] => [1,7,6,5,4,3,2] => ? = 8
[1,2,5,6,3,7,4] => [1,2,5,6,3,7,4] => [5,1,6,2,3,4,7] => [5,1,7,6,4,3,2] => ? = 8
[1,2,5,6,4,3,7] => [1,2,5,6,3,7,4] => [5,1,6,2,3,4,7] => [5,1,7,6,4,3,2] => ? = 8
[1,2,5,6,4,7,3] => [1,2,5,6,3,4,7] => [1,5,6,2,3,4,7] => [1,7,6,5,4,3,2] => ? = 8
[1,2,5,7,3,4,6] => [1,2,5,7,3,4,6] => [1,5,7,2,3,4,6] => [1,7,6,5,4,3,2] => ? = 8
[1,2,5,7,3,6,4] => [1,2,5,7,3,6,4] => [7,5,6,1,2,3,4] => [7,5,6,1,4,3,2] => ? = 8
[1,2,5,7,4,3,6] => [1,2,5,7,3,6,4] => [7,5,6,1,2,3,4] => [7,5,6,1,4,3,2] => ? = 8
[1,2,5,7,4,6,3] => [1,2,5,7,3,4,6] => [1,5,7,2,3,4,6] => [1,7,6,5,4,3,2] => ? = 8
[1,2,5,7,6,3,4] => [1,2,5,7,3,4,6] => [1,5,7,2,3,4,6] => [1,7,6,5,4,3,2] => ? = 8
[1,2,5,7,6,4,3] => [1,2,5,7,3,4,6] => [1,5,7,2,3,4,6] => [1,7,6,5,4,3,2] => ? = 8
[1,2,6,7,3,4,5] => [1,2,6,7,3,4,5] => [1,6,7,2,3,4,5] => [1,7,6,5,4,3,2] => ? = 8
[1,2,6,7,3,5,4] => [1,2,6,7,3,5,4] => [6,5,7,1,2,3,4] => [6,5,7,1,4,3,2] => ? = 8
[1,2,6,7,4,3,5] => [1,2,6,7,3,5,4] => [6,5,7,1,2,3,4] => [6,5,7,1,4,3,2] => ? = 8
[1,2,6,7,4,5,3] => [1,2,6,7,3,4,5] => [1,6,7,2,3,4,5] => [1,7,6,5,4,3,2] => ? = 8
[1,2,6,7,5,3,4] => [1,2,6,7,3,4,5] => [1,6,7,2,3,4,5] => [1,7,6,5,4,3,2] => ? = 8
[1,2,6,7,5,4,3] => [1,2,6,7,3,4,5] => [1,6,7,2,3,4,5] => [1,7,6,5,4,3,2] => ? = 8
[2,4,5,3,6,7,1] => [1,2,4,5,3,6,7] => [1,2,4,3,5,6,7] => [1,7,6,5,4,3,2] => ? = 8
[2,4,5,3,7,6,1] => [1,2,4,5,3,7,6] => [4,1,7,2,3,5,6] => [4,1,7,6,5,3,2] => ? = 8
[2,4,6,3,5,7,1] => [1,2,4,6,3,5,7] => [1,4,6,2,3,5,7] => [1,7,6,5,4,3,2] => ? = 8
[2,4,6,3,7,5,1] => [1,2,4,6,3,7,5] => [4,1,6,2,3,5,7] => [4,1,7,6,5,3,2] => ? = 8
[2,4,6,5,3,7,1] => [1,2,4,6,3,7,5] => [4,1,6,2,3,5,7] => [4,1,7,6,5,3,2] => ? = 8
[2,4,6,5,7,3,1] => [1,2,4,6,3,5,7] => [1,4,6,2,3,5,7] => [1,7,6,5,4,3,2] => ? = 8
[2,4,7,3,5,6,1] => [1,2,4,7,3,5,6] => [1,4,7,2,3,5,6] => [1,7,6,5,4,3,2] => ? = 8
[2,4,7,3,6,5,1] => [1,2,4,7,3,6,5] => [7,4,6,1,2,3,5] => [7,4,6,1,5,3,2] => ? = 8
[2,4,7,5,3,6,1] => [1,2,4,7,3,6,5] => [7,4,6,1,2,3,5] => [7,4,6,1,5,3,2] => ? = 8
[2,4,7,5,6,3,1] => [1,2,4,7,3,5,6] => [1,4,7,2,3,5,6] => [1,7,6,5,4,3,2] => ? = 8
[2,4,7,6,3,5,1] => [1,2,4,7,3,5,6] => [1,4,7,2,3,5,6] => [1,7,6,5,4,3,2] => ? = 8
[2,4,7,6,5,3,1] => [1,2,4,7,3,5,6] => [1,4,7,2,3,5,6] => [1,7,6,5,4,3,2] => ? = 8
[2,5,6,3,4,7,1] => [1,2,5,6,3,4,7] => [1,5,6,2,3,4,7] => [1,7,6,5,4,3,2] => ? = 8
[2,5,6,3,7,4,1] => [1,2,5,6,3,7,4] => [5,1,6,2,3,4,7] => [5,1,7,6,4,3,2] => ? = 8
[2,5,6,4,3,7,1] => [1,2,5,6,3,7,4] => [5,1,6,2,3,4,7] => [5,1,7,6,4,3,2] => ? = 8
[2,5,6,4,7,3,1] => [1,2,5,6,3,4,7] => [1,5,6,2,3,4,7] => [1,7,6,5,4,3,2] => ? = 8
[2,5,7,3,4,6,1] => [1,2,5,7,3,4,6] => [1,5,7,2,3,4,6] => [1,7,6,5,4,3,2] => ? = 8
[2,5,7,3,6,4,1] => [1,2,5,7,3,6,4] => [7,5,6,1,2,3,4] => [7,5,6,1,4,3,2] => ? = 8
[2,5,7,4,3,6,1] => [1,2,5,7,3,6,4] => [7,5,6,1,2,3,4] => [7,5,6,1,4,3,2] => ? = 8
[2,5,7,4,6,3,1] => [1,2,5,7,3,4,6] => [1,5,7,2,3,4,6] => [1,7,6,5,4,3,2] => ? = 8
[2,5,7,6,3,4,1] => [1,2,5,7,3,4,6] => [1,5,7,2,3,4,6] => [1,7,6,5,4,3,2] => ? = 8
[2,5,7,6,4,3,1] => [1,2,5,7,3,4,6] => [1,5,7,2,3,4,6] => [1,7,6,5,4,3,2] => ? = 8

Description

The number of indices that are not cyclical small weak excedances. A cyclical small weak excedance is an index $i < n$ such that $\pi_i = i+1$, or the index $i = n$ if $\pi_n = 1$.

Matching statistic: St000311

Mapping

Mp00223: Permutations —runsort⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000311: Graphs ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 75%

Values

[1] => [1] => [.,.]
 => ([],1)
 => 0
[1,2] => [1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => 2
[2,1] => [1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,4,2,5,3,6] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,4,2,6,3,5] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,4,2,6,5,3] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,4,3,2,6,5] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,4,3,5,2,6] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,4,3,6,2,5] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,5,2,4,3,6] => [1,5,2,4,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,5,2,6,3,4] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,5,2,6,4,3] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,5,3,2,6,4] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,5,3,4,2,6] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,5,3,6,2,4] => [1,5,2,4,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,5,4,2,6,3] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,5,4,3,2,6] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,6,2,4,3,5] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,6,2,5,3,4] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,6,2,5,4,3] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,6,3,2,5,4] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,6,3,4,2,5] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,6,3,5,2,4] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,6,4,2,5,3] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,6,4,3,2,5] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,6,5,2,4,3] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,6,5,3,2,4] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,4,1,5,3,6] => [1,5,2,4,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,4,1,6,3,5] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,4,1,6,5,3] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,4,3,1,6,5] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,4,3,5,1,6] => [1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,4,3,6,1,5] => [1,5,2,4,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,5,1,4,3,6] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,5,1,6,3,4] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,5,1,6,4,3] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,5,3,1,6,4] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,5,3,4,1,6] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,5,3,6,1,4] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,5,4,1,6,3] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,5,4,3,1,6] => [1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,5,4,6,1,3] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,6,1,4,3,5] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,6,1,5,3,4] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,6,1,5,4,3] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,6,3,1,5,4] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[2,6,3,4,1,5] => [1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,2,4,5,3,6,7] => [1,2,4,5,3,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,4,5,3,7,6] => [1,2,4,5,3,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,4,6,3,5,7] => [1,2,4,6,3,5,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,4,6,3,7,5] => [1,2,4,6,3,7,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,4,6,5,3,7] => [1,2,4,6,3,7,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,4,6,5,7,3] => [1,2,4,6,3,5,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,4,7,3,5,6] => [1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,4,7,3,6,5] => [1,2,4,7,3,6,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,4,7,5,3,6] => [1,2,4,7,3,6,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,4,7,5,6,3] => [1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,4,7,6,3,5] => [1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,4,7,6,5,3] => [1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,5,6,3,4,7] => [1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,5,6,3,7,4] => [1,2,5,6,3,7,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,5,6,4,3,7] => [1,2,5,6,3,7,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,5,6,4,7,3] => [1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,5,7,3,4,6] => [1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,5,7,3,6,4] => [1,2,5,7,3,6,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,5,7,4,3,6] => [1,2,5,7,3,6,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,5,7,4,6,3] => [1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,5,7,6,3,4] => [1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,5,7,6,4,3] => [1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,6,7,3,4,5] => [1,2,6,7,3,4,5] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,6,7,3,5,4] => [1,2,6,7,3,5,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,6,7,4,3,5] => [1,2,6,7,3,5,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,6,7,4,5,3] => [1,2,6,7,3,4,5] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,6,7,5,3,4] => [1,2,6,7,3,4,5] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,2,6,7,5,4,3] => [1,2,6,7,3,4,5] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[2,4,5,3,6,7,1] => [1,2,4,5,3,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[2,4,5,3,7,6,1] => [1,2,4,5,3,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[2,4,6,3,5,7,1] => [1,2,4,6,3,5,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[2,4,6,3,7,5,1] => [1,2,4,6,3,7,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[2,4,6,5,3,7,1] => [1,2,4,6,3,7,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[2,4,6,5,7,3,1] => [1,2,4,6,3,5,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[2,4,7,3,5,6,1] => [1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[2,4,7,3,6,5,1] => [1,2,4,7,3,6,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[2,4,7,5,3,6,1] => [1,2,4,7,3,6,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[2,4,7,5,6,3,1] => [1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[2,4,7,6,3,5,1] => [1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[2,4,7,6,5,3,1] => [1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[2,5,6,3,4,7,1] => [1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[2,5,6,3,7,4,1] => [1,2,5,6,3,7,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[2,5,6,4,3,7,1] => [1,2,5,6,3,7,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[2,5,6,4,7,3,1] => [1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[2,5,7,3,4,6,1] => [1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[2,5,7,3,6,4,1] => [1,2,5,7,3,6,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[2,5,7,4,3,6,1] => [1,2,5,7,3,6,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[2,5,7,4,6,3,1] => [1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[2,5,7,6,3,4,1] => [1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[2,5,7,6,4,3,1] => [1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8

Description

The number of vertices of odd degree in a graph.

Matching statistic: St000456
(load all 3 compositions to match this statistic)

Mapping

Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 75%

Values

[1] => [.,.]
 => ([],1)
 => ([(0,1)],2)
 => 1 = 0 + 1
[1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[2,1] => [[.,.],.]
 => ([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,3,2,6,5,4] => [.,[[.,.],[[[.,.],.],.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,4,2,6,5,3] => [.,[[.,.],[[[.,.],.],.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,4,3,2,6,5] => [.,[[[.,.],.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,4,3,5,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,4,3,6,2,5] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,5,2,4,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,5,2,6,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,5,2,6,4,3] => [.,[[.,.],[[[.,.],.],.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,5,3,2,6,4] => [.,[[[.,.],.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,5,3,4,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,5,4,2,6,3] => [.,[[[.,.],.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,5,4,3,2,6] => [.,[[[[.,.],.],.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,6,2,4,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,6,2,5,3,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,6,2,5,4,3] => [.,[[.,.],[[[.,.],.],.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,6,3,2,5,4] => [.,[[[.,.],.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,6,3,4,2,5] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,6,3,5,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,6,4,2,5,3] => [.,[[[.,.],.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,6,4,3,2,5] => [.,[[[[.,.],.],.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,6,5,2,4,3] => [.,[[[.,.],.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,6,5,3,2,4] => [.,[[[[.,.],.],.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[2,4,1,5,3,6] => [[.,[.,.]],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[2,4,1,6,3,5] => [[.,[.,.]],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[2,4,1,6,5,3] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => 7 = 6 + 1
[2,4,3,1,6,5] => [[.,[[.,.],.]],[[.,.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => 7 = 6 + 1
[2,4,3,5,1,6] => [[.,[[.,.],[.,.]]],[.,.]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[2,4,3,6,1,5] => [[.,[[.,.],[.,.]]],[.,.]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[2,5,1,4,3,6] => [[.,[.,.]],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[2,5,1,6,3,4] => [[.,[.,.]],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[2,5,1,6,4,3] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => 7 = 6 + 1
[2,5,3,1,6,4] => [[.,[[.,.],.]],[[.,.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => 7 = 6 + 1
[2,5,3,4,1,6] => [[.,[[.,.],[.,.]]],[.,.]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[2,5,3,6,1,4] => [[.,[[.,.],[.,.]]],[.,.]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[2,5,4,1,6,3] => [[.,[[.,.],.]],[[.,.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => 7 = 6 + 1
[2,5,4,3,1,6] => [[.,[[[.,.],.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => 7 = 6 + 1
[2,5,4,6,1,3] => [[.,[[.,.],[.,.]]],[.,.]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[2,6,1,4,3,5] => [[.,[.,.]],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[2,6,1,5,3,4] => [[.,[.,.]],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[2,6,1,5,4,3] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => 7 = 6 + 1
[2,6,3,1,5,4] => [[.,[[.,.],.]],[[.,.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
 => 7 = 6 + 1
[2,6,3,4,1,5] => [[.,[[.,.],[.,.]]],[.,.]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,2,4,5,3,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,4,5,3,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,4,6,3,5,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,4,6,3,7,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,4,6,5,3,7] => [.,[.,[[.,[[.,.],.]],[.,.]]]]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,4,6,5,7,3] => [.,[.,[[.,[[.,.],[.,.]]],.]]]
 => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,4,7,3,6,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,4,7,5,3,6] => [.,[.,[[.,[[.,.],.]],[.,.]]]]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,4,7,5,6,3] => [.,[.,[[.,[[.,.],[.,.]]],.]]]
 => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,4,7,6,3,5] => [.,[.,[[.,[[.,.],.]],[.,.]]]]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,4,7,6,5,3] => [.,[.,[[.,[[[.,.],.],.]],.]]]
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,5,6,3,7,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,5,6,4,3,7] => [.,[.,[[[.,[.,.]],.],[.,.]]]]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,5,6,4,7,3] => [.,[.,[[[.,[.,.]],[.,.]],.]]]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,5,7,3,6,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,5,7,4,3,6] => [.,[.,[[[.,[.,.]],.],[.,.]]]]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,5,7,4,6,3] => [.,[.,[[[.,[.,.]],[.,.]],.]]]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,5,7,6,3,4] => [.,[.,[[.,[[.,.],.]],[.,.]]]]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,5,7,6,4,3] => [.,[.,[[[.,[[.,.],.]],.],.]]]
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,6,7,3,4,5] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,6,7,3,5,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,6,7,4,3,5] => [.,[.,[[[.,[.,.]],.],[.,.]]]]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,6,7,4,5,3] => [.,[.,[[[.,[.,.]],[.,.]],.]]]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,6,7,5,3,4] => [.,[.,[[[.,[.,.]],.],[.,.]]]]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,2,6,7,5,4,3] => [.,[.,[[[[.,[.,.]],.],.],.]]]
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 8 + 1
[2,4,5,3,6,7,1] => [[.,[[.,[.,.]],[.,[.,.]]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[2,4,5,3,7,6,1] => [[.,[[.,[.,.]],[[.,.],.]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[2,4,6,3,5,7,1] => [[.,[[.,[.,.]],[.,[.,.]]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[2,4,6,3,7,5,1] => [[.,[[.,[.,.]],[[.,.],.]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[2,4,6,5,3,7,1] => [[.,[[.,[[.,.],.]],[.,.]]],.]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[2,4,6,5,7,3,1] => [[.,[[.,[[.,.],[.,.]]],.]],.]
 => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 8 + 1
[2,4,7,3,5,6,1] => [[.,[[.,[.,.]],[.,[.,.]]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[2,4,7,3,6,5,1] => [[.,[[.,[.,.]],[[.,.],.]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[2,4,7,5,3,6,1] => [[.,[[.,[[.,.],.]],[.,.]]],.]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[2,4,7,5,6,3,1] => [[.,[[.,[[.,.],[.,.]]],.]],.]
 => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 8 + 1
[2,4,7,6,3,5,1] => [[.,[[.,[[.,.],.]],[.,.]]],.]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[2,4,7,6,5,3,1] => [[.,[[.,[[[.,.],.],.]],.]],.]
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 8 + 1
[2,5,6,3,4,7,1] => [[.,[[.,[.,.]],[.,[.,.]]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[2,5,6,3,7,4,1] => [[.,[[.,[.,.]],[[.,.],.]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[2,5,6,4,3,7,1] => [[.,[[[.,[.,.]],.],[.,.]]],.]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[2,5,6,4,7,3,1] => [[.,[[[.,[.,.]],[.,.]],.]],.]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[2,5,7,3,4,6,1] => [[.,[[.,[.,.]],[.,[.,.]]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[2,5,7,3,6,4,1] => [[.,[[.,[.,.]],[[.,.],.]]],.]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[2,5,7,4,3,6,1] => [[.,[[[.,[.,.]],.],[.,.]]],.]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[2,5,7,4,6,3,1] => [[.,[[[.,[.,.]],[.,.]],.]],.]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[2,5,7,6,3,4,1] => [[.,[[.,[[.,.],.]],[.,.]]],.]
 => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[2,5,7,6,4,3,1] => [[.,[[[.,[[.,.],.]],.],.]],.]
 => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 8 + 1

Description

The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.

The following 114 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001468The smallest fixpoint of a permutation. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000926The clique-coclique number of a graph. St000844The size of the largest block in the direct sum decomposition of a permutation. St000060The greater neighbor of the maximum. St000197The number of entries equal to positive one in the alternating sign matrix. St000653The last descent of a permutation. St000956The maximal displacement of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St000216The absolute length of a permutation. St000064The number of one-box pattern of a permutation. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000471The sum of the ascent tops of a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000673The number of non-fixed points of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000890The number of nonzero entries in an alternating sign matrix. St000495The number of inversions of distance at most 2 of a permutation. St000530The number of permutations with the same descent word as the given permutation. St000619The number of cyclic descents of a permutation. St000652The maximal difference between successive positions of a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001480The number of simple summands of the module J^2/J^3. St000226The convexity of a permutation. St000354The number of recoils of a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000538The number of even inversions of a permutation. St000795The mad of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000836The number of descents of distance 2 of a permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001061The number of indices that are both descents and recoils of a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001388The number of non-attacking neighbors of a permutation. St001430The number of positive entries in a signed permutation. St001703The villainy of a graph. St001925The minimal number of zeros in a row of an alternating sign matrix. St000045The number of linear extensions of a binary tree. St001060The distinguishing index of a graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001569The maximal modular displacement of a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000327The number of cover relations in a poset. St000264The girth of a graph, which is not a tree. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001557The number of inversions of the second entry of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001890The maximum magnitude of the Möbius function of a poset. St001948The number of augmented double ascents of a permutation. St001520The number of strict 3-descents. St001811The Castelnuovo-Mumford regularity of a permutation. St001560The product of the cardinalities of the lower order ideal and upper order ideal generated by a permutation in weak order. St000189The number of elements in the poset. St000656The number of cuts of a poset. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001717The largest size of an interval in a poset. St000080The rank of the poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001664The number of non-isomorphic subposets of a poset. St001782The order of rowmotion on the set of order ideals of a poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001568The smallest positive integer that does not appear twice in the partition. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000464The Schultz index of a connected graph. St000528The height of a poset. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St000070The number of antichains in a poset. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001817The number of flag weak exceedances of a signed permutation. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001892The flag excedance statistic of a signed permutation. St001894The depth of a signed permutation. St000699The toughness times the least common multiple of 1,. St001332The number of steps on the non-negative side of the walk associated with the permutation. St000401The size of the symmetry class of a permutation. St000625The sum of the minimal distances to a greater element.