Your data matches 97 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000422
(load all 47 compositions to match this statistic)
Mapping
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ 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
[3,1,5,2,4,6] => [1,3,5,4,2,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,1,5,2,6,4] => [1,3,5,6,4,2] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,1,5,4,2,6] => [1,3,5,2,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,1,5,4,6,2] => [1,3,5,6,2,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,1,5,6,2,4] => [1,3,5,2,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,1,5,6,4,2] => [1,3,5,4,6,2] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,2,5,1,4,6] => [1,3,5,4,2,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,2,5,1,6,4] => [1,3,5,6,4,2] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,2,5,4,1,6] => [1,3,5,2,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,2,5,4,6,1] => [1,3,5,6,2,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,2,5,6,1,4] => [1,3,5,2,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,2,5,6,4,1] => [1,3,5,4,6,2] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,4,5,1,2,6] => [1,3,5,2,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,4,5,1,6,2] => [1,3,5,6,2,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,4,5,2,1,6] => [1,3,5,2,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,4,5,2,6,1] => [1,3,5,6,2,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,4,5,6,1,2] => [1,3,5,2,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,4,5,6,2,1] => [1,3,5,2,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,5,1,2,4,6] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,5,1,2,6,4] => [1,3,2,5,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,5,1,4,2,6] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,5,1,4,6,2] => [1,3,2,5,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,5,1,6,2,4] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,5,1,6,4,2] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,5,2,1,4,6] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,5,2,1,6,4] => [1,3,2,5,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,5,2,4,1,6] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,5,2,4,6,1] => [1,3,2,5,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,5,2,6,1,4] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,5,2,6,4,1] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,6,5,1,2,4] => [1,3,5,2,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,6,5,1,4,2] => [1,3,5,4,2,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,6,5,2,1,4] => [1,3,5,2,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,6,5,2,4,1] => [1,3,5,4,2,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,6,5,4,1,2] => [1,3,5,2,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,6,5,4,2,1] => [1,3,5,2,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[5,1,2,4,3,6] => [1,5,3,2,4,6] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[5,1,2,4,6,3] => [1,5,6,3,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[5,1,2,6,3,4] => [1,5,3,2,4,6] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[5,1,4,2,3,6] => [1,5,3,4,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[5,1,4,2,6,3] => [1,5,6,3,4,2] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[5,1,4,6,3,2] => [1,5,3,4,6,2] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[5,1,6,2,3,4] => [1,5,3,6,4,2] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[5,1,6,4,3,2] => [1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[5,2,1,4,3,6] => [1,5,3,2,4,6] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[5,2,1,4,6,3] => [1,5,6,3,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[5,2,1,6,3,4] => [1,5,3,2,4,6] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(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: 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
[3,1,5,2,4,6] => [3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 9 = 6 + 3
[3,1,5,2,6,4] => [3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 9 = 6 + 3
[3,1,5,4,2,6] => [3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3
[3,1,5,4,6,2] => [3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3
[3,1,5,6,2,4] => [3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 9 = 6 + 3
[3,1,5,6,4,2] => [3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3
[3,2,5,1,4,6] => [3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 9 = 6 + 3
[3,2,5,1,6,4] => [3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 9 = 6 + 3
[3,2,5,4,1,6] => [3,2,5,4,1] => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[3,2,5,4,6,1] => [3,2,5,4,1] => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[3,2,5,6,1,4] => [3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 9 = 6 + 3
[3,2,5,6,4,1] => [3,2,5,4,1] => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[3,4,5,1,2,6] => [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3
[3,4,5,1,6,2] => [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3
[3,4,5,2,1,6] => [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 9 = 6 + 3
[3,4,5,2,6,1] => [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 9 = 6 + 3
[3,4,5,6,1,2] => [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3
[3,4,5,6,2,1] => [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 9 = 6 + 3
[3,5,1,2,4,6] => [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 9 = 6 + 3
[3,5,1,2,6,4] => [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 9 = 6 + 3
[3,5,1,4,2,6] => [3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 9 = 6 + 3
[3,5,1,4,6,2] => [3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 9 = 6 + 3
[3,5,1,6,2,4] => [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 9 = 6 + 3
[3,5,1,6,4,2] => [3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 9 = 6 + 3
[3,5,2,1,4,6] => [3,5,2,1,4] => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3
[3,5,2,1,6,4] => [3,5,2,1,4] => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3
[3,5,2,4,1,6] => [3,5,2,4,1] => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[3,5,2,4,6,1] => [3,5,2,4,1] => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[3,5,2,6,1,4] => [3,5,2,1,4] => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3
[3,5,2,6,4,1] => [3,5,2,4,1] => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[3,6,5,1,2,4] => [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 9 = 6 + 3
[3,6,5,1,4,2] => [3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 9 = 6 + 3
[3,6,5,2,1,4] => [3,5,2,1,4] => [[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3
[3,6,5,2,4,1] => [3,5,2,4,1] => [[[.,[.,.]],[.,.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => 9 = 6 + 3
[3,6,5,4,1,2] => [3,5,4,1,2] => [[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3
[3,6,5,4,2,1] => [3,5,4,2,1] => [[[.,[[.,.],.]],.],.]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 9 = 6 + 3
[5,1,2,4,3,6] => [5,1,2,4,3] => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3
[5,1,2,4,6,3] => [5,1,2,4,3] => [[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3
[5,1,2,6,3,4] => [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3
[5,1,4,2,3,6] => [5,1,4,2,3] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 9 = 6 + 3
[5,1,4,2,6,3] => [5,1,4,2,3] => [[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 9 = 6 + 3
[5,1,4,6,3,2] => [5,1,4,3,2] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3
[5,1,6,2,3,4] => [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3
[5,1,6,4,3,2] => [5,1,4,3,2] => [[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 9 = 6 + 3
[5,2,1,4,3,6] => [5,2,1,4,3] => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 9 = 6 + 3
[5,2,1,4,6,3] => [5,2,1,4,3] => [[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 9 = 6 + 3
[5,2,1,6,3,4] => [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 9 = 6 + 3
[5,2,4,1,3,6] => [5,2,4,1,3] => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 9 = 6 + 3
[5,2,4,1,6,3] => [5,2,4,1,3] => [[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => 9 = 6 + 3
[5,2,4,6,3,1] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
 => ([(0,4),(1,2),(2,4),(4,3)],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
[3,1,5,2,4,6] => [4,2,5,1,3,6] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,1,5,2,6,4] => [4,2,6,1,5,3] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,1,5,4,2,6] => [5,2,4,3,1,6] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,1,5,4,6,2] => [6,2,4,3,5,1] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,1,5,6,2,4] => [5,2,6,4,1,3] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,1,5,6,4,2] => [6,2,5,4,3,1] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,2,5,1,4,6] => [4,2,5,1,3,6] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,2,5,1,6,4] => [4,2,6,1,5,3] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,2,5,4,1,6] => [5,2,4,3,1,6] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,2,5,4,6,1] => [6,2,4,3,5,1] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,2,5,6,1,4] => [5,2,6,4,1,3] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,2,5,6,4,1] => [6,2,5,4,3,1] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,4,5,1,2,6] => [5,4,3,2,1,6] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,4,5,1,6,2] => [6,4,3,2,5,1] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,4,5,2,1,6] => [5,4,3,2,1,6] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,4,5,2,6,1] => [6,4,3,2,5,1] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,4,5,6,1,2] => [6,5,3,4,2,1] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,4,5,6,2,1] => [6,5,3,4,2,1] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,5,1,2,4,6] => [4,5,3,1,2,6] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,5,1,2,6,4] => [4,6,3,1,5,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,5,1,4,2,6] => [5,4,3,2,1,6] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,5,1,4,6,2] => [6,4,3,2,5,1] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,5,1,6,2,4] => [5,6,3,4,1,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,5,1,6,4,2] => [6,5,3,4,2,1] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,5,2,1,4,6] => [4,5,3,1,2,6] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,5,2,1,6,4] => [4,6,3,1,5,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,5,2,4,1,6] => [5,4,3,2,1,6] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,5,2,4,6,1] => [6,4,3,2,5,1] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,5,2,6,1,4] => [5,6,3,4,1,2] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,5,2,6,4,1] => [6,5,3,4,2,1] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[3,6,5,1,2,4] => [5,6,4,3,1,2] => [2,2,2]
 => [2,2]
 => 1 = 6 - 5
[3,6,5,1,4,2] => [6,5,4,3,2,1] => [2,2,2]
 => [2,2]
 => 1 = 6 - 5
[3,6,5,2,1,4] => [5,6,4,3,1,2] => [2,2,2]
 => [2,2]
 => 1 = 6 - 5
[3,6,5,2,4,1] => [6,5,4,3,2,1] => [2,2,2]
 => [2,2]
 => 1 = 6 - 5
[3,6,5,4,1,2] => [6,5,4,3,2,1] => [2,2,2]
 => [2,2]
 => 1 = 6 - 5
[3,6,5,4,2,1] => [6,5,4,3,2,1] => [2,2,2]
 => [2,2]
 => 1 = 6 - 5
[5,1,2,4,3,6] => [5,2,3,4,1,6] => [2,1,1,1,1]
 => [1,1,1,1]
 => 1 = 6 - 5
[5,1,2,4,6,3] => [6,2,3,4,5,1] => [2,1,1,1,1]
 => [1,1,1,1]
 => 1 = 6 - 5
[5,1,2,6,3,4] => [6,2,3,5,4,1] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[5,1,4,2,3,6] => [5,2,4,3,1,6] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[5,1,4,2,6,3] => [6,2,4,3,5,1] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[5,1,4,6,3,2] => [6,2,5,4,3,1] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[5,1,6,2,3,4] => [6,2,5,4,3,1] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[5,1,6,4,3,2] => [6,2,5,4,3,1] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[5,2,1,4,3,6] => [5,3,2,4,1,6] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[5,2,1,4,6,3] => [6,3,2,4,5,1] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[5,2,1,6,3,4] => [6,3,2,5,4,1] => [2,2,2]
 => [2,2]
 => 1 = 6 - 5
[5,2,4,1,3,6] => [5,4,3,2,1,6] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[5,2,4,1,6,3] => [6,4,3,2,5,1] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
[5,2,4,6,3,1] => [6,5,3,4,2,1] => [2,2,1,1]
 => [2,1,1]
 => 1 = 6 - 5
Description
The number of characters of the symmetric group whose value on the partition is even.
Matching statistic: St001880
(load all 9 compositions to match this statistic)
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 25% ●values known / values provided: 35%●distinct values known / distinct values provided: 25%
Values
[1] => [1] => [.,.]
 => ([],1)
 => ? = 0 - 1
[1,2] => [1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => ? = 2 - 1
[2,1] => [2,1] => [[.,.],.]
 => ([(0,1)],2)
 => ? = 2 - 1
[3,1,5,2,4,6] => [3,1,6,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,1,5,2,6,4] => [3,1,6,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,1,5,4,2,6] => [3,1,6,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,1,5,4,6,2] => [3,1,6,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,1,5,6,2,4] => [3,1,6,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,1,5,6,4,2] => [3,1,6,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,2,5,1,4,6] => [3,2,6,1,5,4] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,2,5,1,6,4] => [3,2,6,1,5,4] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,2,5,4,1,6] => [3,2,6,5,1,4] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,2,5,4,6,1] => [3,2,6,5,4,1] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,2,5,6,1,4] => [3,2,6,5,1,4] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,2,5,6,4,1] => [3,2,6,5,4,1] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,4,5,1,2,6] => [3,6,5,1,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,4,5,1,6,2] => [3,6,5,1,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,4,5,2,1,6] => [3,6,5,2,1,4] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,4,5,2,6,1] => [3,6,5,2,4,1] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,4,5,6,1,2] => [3,6,5,4,1,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,4,5,6,2,1] => [3,6,5,4,2,1] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,5,1,2,4,6] => [3,6,1,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,5,1,2,6,4] => [3,6,1,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,5,1,4,2,6] => [3,6,1,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,5,1,4,6,2] => [3,6,1,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,5,1,6,2,4] => [3,6,1,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,5,1,6,4,2] => [3,6,1,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,5,2,1,4,6] => [3,6,2,1,5,4] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,5,2,1,6,4] => [3,6,2,1,5,4] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,5,2,4,1,6] => [3,6,2,5,1,4] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,5,2,4,6,1] => [3,6,2,5,4,1] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,5,2,6,1,4] => [3,6,2,5,1,4] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,5,2,6,4,1] => [3,6,2,5,4,1] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,6,5,1,2,4] => [3,6,5,1,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,6,5,1,4,2] => [3,6,5,1,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,6,5,2,1,4] => [3,6,5,2,1,4] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,6,5,2,4,1] => [3,6,5,2,4,1] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,6,5,4,1,2] => [3,6,5,4,1,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[3,6,5,4,2,1] => [3,6,5,4,2,1] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 1
[5,1,2,4,3,6] => [5,1,6,4,3,2] => [[.,[[[.,.],.],.]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 6 - 1
[5,1,2,4,6,3] => [5,1,6,4,3,2] => [[.,[[[.,.],.],.]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 6 - 1
[5,1,2,6,3,4] => [5,1,6,4,3,2] => [[.,[[[.,.],.],.]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 6 - 1
[5,1,4,2,3,6] => [5,1,6,4,3,2] => [[.,[[[.,.],.],.]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 6 - 1
[5,1,4,2,6,3] => [5,1,6,4,3,2] => [[.,[[[.,.],.],.]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 6 - 1
[5,1,4,6,3,2] => [5,1,6,4,3,2] => [[.,[[[.,.],.],.]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 6 - 1
[5,1,6,2,3,4] => [5,1,6,4,3,2] => [[.,[[[.,.],.],.]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 6 - 1
[5,1,6,4,3,2] => [5,1,6,4,3,2] => [[.,[[[.,.],.],.]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 6 - 1
[5,2,1,4,3,6] => [5,2,1,6,4,3] => [[[.,.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 6 - 1
[5,2,1,4,6,3] => [5,2,1,6,4,3] => [[[.,.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 6 - 1
[5,2,1,6,3,4] => [5,2,1,6,4,3] => [[[.,.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 6 - 1
[1,5,2,3,4,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,2,3,4,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,2,3,6,4,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,2,3,6,7,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,2,3,7,4,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,2,3,7,6,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,2,4,3,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,2,4,3,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,2,4,6,3,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,2,4,6,7,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,2,4,7,3,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,2,4,7,6,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,2,6,3,4,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,2,6,3,7,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,2,6,4,3,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,2,6,4,7,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,2,6,7,3,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,2,6,7,4,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,2,7,3,4,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,2,7,3,6,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,2,7,4,3,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,2,7,4,6,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,2,7,6,3,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,2,7,6,4,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,3,2,4,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,3,2,4,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,3,2,6,4,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,3,2,6,7,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,3,2,7,4,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,3,2,7,6,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,3,4,2,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,3,4,2,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,3,4,6,2,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,3,4,6,7,2] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,3,4,7,2,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,3,4,7,6,2] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,3,6,2,4,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,3,6,2,7,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,3,6,4,2,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,3,6,4,7,2] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,3,6,7,2,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,3,6,7,4,2] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,3,7,2,4,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,3,7,2,6,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,3,7,4,2,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,3,7,4,6,2] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,3,7,6,2,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,3,7,6,4,2] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,4,2,3,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
[1,5,4,2,3,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 7 = 8 - 1
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001879
(load all 9 compositions to match this statistic)
Mapping
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 25% ●values known / values provided: 35%●distinct values known / distinct values provided: 25%
Values
[1] => [1] => [.,.]
 => ([],1)
 => ? = 0 - 2
[1,2] => [1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => ? = 2 - 2
[2,1] => [2,1] => [[.,.],.]
 => ([(0,1)],2)
 => ? = 2 - 2
[3,1,5,2,4,6] => [3,1,6,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,1,5,2,6,4] => [3,1,6,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,1,5,4,2,6] => [3,1,6,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,1,5,4,6,2] => [3,1,6,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,1,5,6,2,4] => [3,1,6,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,1,5,6,4,2] => [3,1,6,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,2,5,1,4,6] => [3,2,6,1,5,4] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,2,5,1,6,4] => [3,2,6,1,5,4] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,2,5,4,1,6] => [3,2,6,5,1,4] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,2,5,4,6,1] => [3,2,6,5,4,1] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,2,5,6,1,4] => [3,2,6,5,1,4] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,2,5,6,4,1] => [3,2,6,5,4,1] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,4,5,1,2,6] => [3,6,5,1,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,4,5,1,6,2] => [3,6,5,1,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,4,5,2,1,6] => [3,6,5,2,1,4] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,4,5,2,6,1] => [3,6,5,2,4,1] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,4,5,6,1,2] => [3,6,5,4,1,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,4,5,6,2,1] => [3,6,5,4,2,1] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,5,1,2,4,6] => [3,6,1,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,5,1,2,6,4] => [3,6,1,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,5,1,4,2,6] => [3,6,1,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,5,1,4,6,2] => [3,6,1,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,5,1,6,2,4] => [3,6,1,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,5,1,6,4,2] => [3,6,1,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,5,2,1,4,6] => [3,6,2,1,5,4] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,5,2,1,6,4] => [3,6,2,1,5,4] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,5,2,4,1,6] => [3,6,2,5,1,4] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,5,2,4,6,1] => [3,6,2,5,4,1] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,5,2,6,1,4] => [3,6,2,5,1,4] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,5,2,6,4,1] => [3,6,2,5,4,1] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,6,5,1,2,4] => [3,6,5,1,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,6,5,1,4,2] => [3,6,5,1,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,6,5,2,1,4] => [3,6,5,2,1,4] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,6,5,2,4,1] => [3,6,5,2,4,1] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,6,5,4,1,2] => [3,6,5,4,1,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[3,6,5,4,2,1] => [3,6,5,4,2,1] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ? = 6 - 2
[5,1,2,4,3,6] => [5,1,6,4,3,2] => [[.,[[[.,.],.],.]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 6 - 2
[5,1,2,4,6,3] => [5,1,6,4,3,2] => [[.,[[[.,.],.],.]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 6 - 2
[5,1,2,6,3,4] => [5,1,6,4,3,2] => [[.,[[[.,.],.],.]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 6 - 2
[5,1,4,2,3,6] => [5,1,6,4,3,2] => [[.,[[[.,.],.],.]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 6 - 2
[5,1,4,2,6,3] => [5,1,6,4,3,2] => [[.,[[[.,.],.],.]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 6 - 2
[5,1,4,6,3,2] => [5,1,6,4,3,2] => [[.,[[[.,.],.],.]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 6 - 2
[5,1,6,2,3,4] => [5,1,6,4,3,2] => [[.,[[[.,.],.],.]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 6 - 2
[5,1,6,4,3,2] => [5,1,6,4,3,2] => [[.,[[[.,.],.],.]],[.,.]]
 => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ? = 6 - 2
[5,2,1,4,3,6] => [5,2,1,6,4,3] => [[[.,.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 6 - 2
[5,2,1,4,6,3] => [5,2,1,6,4,3] => [[[.,.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 6 - 2
[5,2,1,6,3,4] => [5,2,1,6,4,3] => [[[.,.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ? = 6 - 2
[1,5,2,3,4,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,2,3,4,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,2,3,6,4,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,2,3,6,7,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,2,3,7,4,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,2,3,7,6,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,2,4,3,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,2,4,3,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,2,4,6,3,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,2,4,6,7,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,2,4,7,3,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,2,4,7,6,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,2,6,3,4,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,2,6,3,7,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,2,6,4,3,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,2,6,4,7,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,2,6,7,3,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,2,6,7,4,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,2,7,3,4,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,2,7,3,6,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,2,7,4,3,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,2,7,4,6,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,2,7,6,3,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,2,7,6,4,3] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,3,2,4,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,3,2,4,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,3,2,6,4,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,3,2,6,7,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,3,2,7,4,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,3,2,7,6,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,3,4,2,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,3,4,2,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,3,4,6,2,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,3,4,6,7,2] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,3,4,7,2,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,3,4,7,6,2] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,3,6,2,4,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,3,6,2,7,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,3,6,4,2,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,3,6,4,7,2] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,3,6,7,2,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,3,6,7,4,2] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,3,7,2,4,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,3,7,2,6,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,3,7,4,2,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,3,7,4,6,2] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,3,7,6,2,4] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,3,7,6,4,2] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,4,2,3,6,7] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
[1,5,4,2,3,7,6] => [1,7,6,5,4,3,2] => [.,[[[[[[.,.],.],.],.],.],.]]
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 6 = 8 - 2
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: St000455
(load all 4 compositions to match this statistic)
Mapping
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 29%●distinct values known / distinct values provided: 25%
Values
[1] => [.,.]
 => ([],1)
 => ([],1)
 => ? = 0 - 7
[1,2] => [.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 7
[2,1] => [[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 7
[3,1,5,2,4,6] => [[.,[.,.]],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,1,5,2,6,4] => [[.,[.,.]],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,1,5,4,2,6] => [[.,[.,.]],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,1,5,4,6,2] => [[.,[.,.]],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,1,5,6,2,4] => [[.,[.,.]],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,1,5,6,4,2] => [[.,[.,.]],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,2,5,1,4,6] => [[[.,.],.],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,2,5,1,6,4] => [[[.,.],.],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,2,5,4,1,6] => [[[.,.],.],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,2,5,4,6,1] => [[[.,.],.],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,2,5,6,1,4] => [[[.,.],.],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,2,5,6,4,1] => [[[.,.],.],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,4,5,1,2,6] => [[.,[.,.]],[.,[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,4,5,1,6,2] => [[.,[.,.]],[.,[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,4,5,2,1,6] => [[[.,.],.],[.,[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,4,5,2,6,1] => [[[.,.],.],[.,[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,4,5,6,1,2] => [[.,[.,.]],[.,[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,4,5,6,2,1] => [[[.,.],.],[.,[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,5,1,2,4,6] => [[.,[.,.]],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,5,1,2,6,4] => [[.,[.,.]],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,5,1,4,2,6] => [[.,[.,.]],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,5,1,4,6,2] => [[.,[.,.]],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,5,1,6,2,4] => [[.,[.,.]],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,5,1,6,4,2] => [[.,[.,.]],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,5,2,1,4,6] => [[[.,.],.],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,5,2,1,6,4] => [[[.,.],.],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,5,2,4,1,6] => [[[.,.],.],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,5,2,4,6,1] => [[[.,.],.],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,5,2,6,1,4] => [[[.,.],.],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,5,2,6,4,1] => [[[.,.],.],[[.,.],[.,.]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,6,5,1,2,4] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,6,5,1,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,6,5,2,1,4] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,6,5,2,4,1] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,6,5,4,1,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[3,6,5,4,2,1] => [[[.,.],.],[[[.,.],.],.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[5,1,2,4,3,6] => [[.,[.,[[.,.],.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[5,1,2,4,6,3] => [[.,[.,[[.,.],.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[5,1,2,6,3,4] => [[.,[.,[.,[.,.]]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[5,1,4,2,3,6] => [[.,[[.,[.,.]],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[5,1,4,2,6,3] => [[.,[[.,[.,.]],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[5,1,4,6,3,2] => [[.,[[[.,.],.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[5,1,6,2,3,4] => [[.,[.,[.,[.,.]]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[5,1,6,4,3,2] => [[.,[[[.,.],.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[5,2,1,4,3,6] => [[[.,.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[5,2,1,4,6,3] => [[[.,.],[[.,.],.]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[5,2,1,6,3,4] => [[[.,.],[.,[.,.]]],[.,.]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 7
[2,5,1,3,4,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,1,3,4,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,1,3,6,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,1,3,6,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,1,3,7,4,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,1,3,7,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,1,4,3,6,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,1,4,3,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,1,4,6,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,1,4,6,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,1,4,7,3,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,1,4,7,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,1,6,3,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,1,6,3,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,1,6,4,3,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,1,6,4,7,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,1,6,7,3,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,1,6,7,4,3] => [[.,.],[[[.,.],.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,1,7,3,4,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,1,7,3,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,1,7,4,3,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,1,7,4,6,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,1,7,6,3,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,1,7,6,4,3] => [[.,.],[[[.,.],.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,3,1,4,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,3,1,4,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,3,1,6,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,3,1,6,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,3,1,7,4,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,3,1,7,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,3,4,1,6,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,3,4,1,7,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,3,4,6,1,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,3,4,6,7,1] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,3,4,7,1,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,3,4,7,6,1] => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,3,6,1,4,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,3,6,1,7,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,3,6,4,1,7] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,3,6,4,7,1] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,3,6,7,1,4] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,3,6,7,4,1] => [[.,.],[[.,[.,.]],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,3,7,1,4,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,3,7,1,6,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,3,7,4,1,6] => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,3,7,4,6,1] => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,3,7,6,1,4] => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,3,7,6,4,1] => [[.,.],[[.,[.,.]],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,4,1,3,6,7] => [[.,.],[[[.,.],.],[.,[.,.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
[2,5,4,1,3,7,6] => [[.,.],[[[.,.],.],[[.,.],.]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 1 = 8 - 7
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 7 compositions to match this statistic)
Mapping
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000235: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 75%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [1,2] => 2
[2,1] => [2,1] => [2,1] => [1,2] => 2
[3,1,5,2,4,6] => [4,2,5,1,3,6] => [5,4,3,2,1,6] => [1,6,2,3,4,5] => 6
[3,1,5,2,6,4] => [4,2,6,1,5,3] => [6,4,5,2,3,1] => [1,2,3,4,5,6] => 6
[3,1,5,4,2,6] => [5,2,4,3,1,6] => [5,4,3,2,1,6] => [1,6,2,3,4,5] => 6
[3,1,5,4,6,2] => [6,2,4,3,5,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[3,1,5,6,2,4] => [5,2,6,4,1,3] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[3,1,5,6,4,2] => [6,2,5,4,3,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[3,2,5,1,4,6] => [4,2,5,1,3,6] => [5,4,3,2,1,6] => [1,6,2,3,4,5] => 6
[3,2,5,1,6,4] => [4,2,6,1,5,3] => [6,4,5,2,3,1] => [1,2,3,4,5,6] => 6
[3,2,5,4,1,6] => [5,2,4,3,1,6] => [5,4,3,2,1,6] => [1,6,2,3,4,5] => 6
[3,2,5,4,6,1] => [6,2,4,3,5,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[3,2,5,6,1,4] => [5,2,6,4,1,3] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[3,2,5,6,4,1] => [6,2,5,4,3,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[3,4,5,1,2,6] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => [1,6,2,3,4,5] => 6
[3,4,5,1,6,2] => [6,4,3,2,5,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[3,4,5,2,1,6] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => [1,6,2,3,4,5] => 6
[3,4,5,2,6,1] => [6,4,3,2,5,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[3,4,5,6,1,2] => [6,5,3,4,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[3,4,5,6,2,1] => [6,5,3,4,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[3,5,1,2,4,6] => [4,5,3,1,2,6] => [5,4,3,2,1,6] => [1,6,2,3,4,5] => 6
[3,5,1,2,6,4] => [4,6,3,1,5,2] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[3,5,1,4,2,6] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => [1,6,2,3,4,5] => 6
[3,5,1,4,6,2] => [6,4,3,2,5,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[3,5,1,6,2,4] => [5,6,3,4,1,2] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[3,5,1,6,4,2] => [6,5,3,4,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[3,5,2,1,4,6] => [4,5,3,1,2,6] => [5,4,3,2,1,6] => [1,6,2,3,4,5] => 6
[3,5,2,1,6,4] => [4,6,3,1,5,2] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[3,5,2,4,1,6] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => [1,6,2,3,4,5] => 6
[3,5,2,4,6,1] => [6,4,3,2,5,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[3,5,2,6,1,4] => [5,6,3,4,1,2] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[3,5,2,6,4,1] => [6,5,3,4,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[3,6,5,1,2,4] => [5,6,4,3,1,2] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[3,6,5,1,4,2] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[3,6,5,2,1,4] => [5,6,4,3,1,2] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[3,6,5,2,4,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[3,6,5,4,1,2] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[3,6,5,4,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[5,1,2,4,3,6] => [5,2,3,4,1,6] => [5,4,3,2,1,6] => [1,6,2,3,4,5] => 6
[5,1,2,4,6,3] => [6,2,3,4,5,1] => [6,5,3,4,2,1] => [1,2,3,4,5,6] => 6
[5,1,2,6,3,4] => [6,2,3,5,4,1] => [6,5,3,4,2,1] => [1,2,3,4,5,6] => 6
[5,1,4,2,3,6] => [5,2,4,3,1,6] => [5,4,3,2,1,6] => [1,6,2,3,4,5] => 6
[5,1,4,2,6,3] => [6,2,4,3,5,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[5,1,4,6,3,2] => [6,2,5,4,3,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[5,1,6,2,3,4] => [6,2,5,4,3,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[5,1,6,4,3,2] => [6,2,5,4,3,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[5,2,1,4,3,6] => [5,3,2,4,1,6] => [5,4,3,2,1,6] => [1,6,2,3,4,5] => 6
[5,2,1,4,6,3] => [6,3,2,4,5,1] => [6,5,3,4,2,1] => [1,2,3,4,5,6] => 6
[5,2,1,6,3,4] => [6,3,2,5,4,1] => [6,5,3,4,2,1] => [1,2,3,4,5,6] => 6
[1,5,2,3,4,6,7] => [1,5,3,4,2,6,7] => [1,5,4,3,2,6,7] => [1,5,2,6,7,3,4] => ? = 8
[1,5,2,3,4,7,6] => [1,5,3,4,2,7,6] => [1,5,4,3,2,7,6] => [1,5,2,7,3,4,6] => ? = 8
[1,5,2,3,6,4,7] => [1,6,3,4,5,2,7] => [1,6,5,4,3,2,7] => [1,6,2,7,3,4,5] => ? = 8
[1,5,2,3,6,7,4] => [1,7,3,4,5,6,2] => [1,7,6,4,5,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,2,3,7,4,6] => [1,6,3,4,7,2,5] => [1,7,6,4,5,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,2,3,7,6,4] => [1,7,3,4,6,5,2] => [1,7,6,4,5,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,2,4,3,6,7] => [1,5,3,4,2,6,7] => [1,5,4,3,2,6,7] => [1,5,2,6,7,3,4] => ? = 8
[1,5,2,4,3,7,6] => [1,5,3,4,2,7,6] => [1,5,4,3,2,7,6] => [1,5,2,7,3,4,6] => ? = 8
[1,5,2,4,6,3,7] => [1,6,3,4,5,2,7] => [1,6,5,4,3,2,7] => [1,6,2,7,3,4,5] => ? = 8
[1,5,2,4,6,7,3] => [1,7,3,4,5,6,2] => [1,7,6,4,5,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,2,4,7,3,6] => [1,6,3,4,7,2,5] => [1,7,6,4,5,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,2,4,7,6,3] => [1,7,3,4,6,5,2] => [1,7,6,4,5,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,2,6,3,4,7] => [1,6,3,5,4,2,7] => [1,6,5,4,3,2,7] => [1,6,2,7,3,4,5] => ? = 8
[1,5,2,6,3,7,4] => [1,7,3,5,4,6,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,2,6,4,3,7] => [1,6,3,5,4,2,7] => [1,6,5,4,3,2,7] => [1,6,2,7,3,4,5] => ? = 8
[1,5,2,6,4,7,3] => [1,7,3,5,4,6,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,2,6,7,3,4] => [1,7,3,6,5,4,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,2,6,7,4,3] => [1,7,3,6,5,4,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,2,7,3,4,6] => [1,6,3,7,5,2,4] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,2,7,3,6,4] => [1,7,3,6,5,4,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,2,7,4,3,6] => [1,6,3,7,5,2,4] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,2,7,4,6,3] => [1,7,3,6,5,4,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,2,7,6,3,4] => [1,7,3,6,5,4,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,2,7,6,4,3] => [1,7,3,6,5,4,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,3,2,4,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => [1,5,2,6,7,3,4] => ? = 8
[1,5,3,2,4,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => [1,5,2,7,3,4,6] => ? = 8
[1,5,3,2,6,4,7] => [1,6,4,3,5,2,7] => [1,6,5,4,3,2,7] => [1,6,2,7,3,4,5] => ? = 8
[1,5,3,2,6,7,4] => [1,7,4,3,5,6,2] => [1,7,6,4,5,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,3,2,7,4,6] => [1,6,4,3,7,2,5] => [1,7,6,4,5,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,3,2,7,6,4] => [1,7,4,3,6,5,2] => [1,7,6,4,5,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,3,4,2,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => [1,5,2,6,7,3,4] => ? = 8
[1,5,3,4,2,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => [1,5,2,7,3,4,6] => ? = 8
[1,5,3,4,6,2,7] => [1,6,4,3,5,2,7] => [1,6,5,4,3,2,7] => [1,6,2,7,3,4,5] => ? = 8
[1,5,3,4,6,7,2] => [1,7,4,3,5,6,2] => [1,7,6,4,5,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,3,4,7,2,6] => [1,6,4,3,7,2,5] => [1,7,6,4,5,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,3,4,7,6,2] => [1,7,4,3,6,5,2] => [1,7,6,4,5,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,3,6,2,4,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,2,7,3,4,5] => ? = 8
[1,5,3,6,2,7,4] => [1,7,5,4,3,6,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,3,6,4,2,7] => [1,6,5,4,3,2,7] => [1,6,5,4,3,2,7] => [1,6,2,7,3,4,5] => ? = 8
[1,5,3,6,4,7,2] => [1,7,5,4,3,6,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,3,6,7,2,4] => [1,7,6,4,5,3,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,3,6,7,4,2] => [1,7,6,4,5,3,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,3,7,2,4,6] => [1,6,5,7,3,2,4] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,3,7,2,6,4] => [1,7,5,6,3,4,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,3,7,4,2,6] => [1,6,5,7,3,2,4] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,3,7,4,6,2] => [1,7,5,6,3,4,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,3,7,6,2,4] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,3,7,6,4,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,4,2,3,6,7] => [1,5,4,3,2,6,7] => [1,5,4,3,2,6,7] => [1,5,2,6,7,3,4] => ? = 8
[1,5,4,2,3,7,6] => [1,5,4,3,2,7,6] => [1,5,4,3,2,7,6] => [1,5,2,7,3,4,6] => ? = 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
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000311: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●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
[3,1,5,2,4,6] => [1,3,5,4,2,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,1,5,2,6,4] => [1,3,5,6,4,2] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,1,5,4,2,6] => [1,3,5,2,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,1,5,4,6,2] => [1,3,5,6,2,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,1,5,6,2,4] => [1,3,5,2,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,1,5,6,4,2] => [1,3,5,4,6,2] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,2,5,1,4,6] => [1,3,5,4,2,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,2,5,1,6,4] => [1,3,5,6,4,2] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,2,5,4,1,6] => [1,3,5,2,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,2,5,4,6,1] => [1,3,5,6,2,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,2,5,6,1,4] => [1,3,5,2,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,2,5,6,4,1] => [1,3,5,4,6,2] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,4,5,1,2,6] => [1,3,5,2,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,4,5,1,6,2] => [1,3,5,6,2,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,4,5,2,1,6] => [1,3,5,2,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,4,5,2,6,1] => [1,3,5,6,2,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,4,5,6,1,2] => [1,3,5,2,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,4,5,6,2,1] => [1,3,5,2,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,5,1,2,4,6] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,5,1,2,6,4] => [1,3,2,5,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,5,1,4,2,6] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,5,1,4,6,2] => [1,3,2,5,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,5,1,6,2,4] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,5,1,6,4,2] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,5,2,1,4,6] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,5,2,1,6,4] => [1,3,2,5,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,5,2,4,1,6] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,5,2,4,6,1] => [1,3,2,5,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,5,2,6,1,4] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,5,2,6,4,1] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,6,5,1,2,4] => [1,3,5,2,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,6,5,1,4,2] => [1,3,5,4,2,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,6,5,2,1,4] => [1,3,5,2,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,6,5,2,4,1] => [1,3,5,4,2,6] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,6,5,4,1,2] => [1,3,5,2,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[3,6,5,4,2,1] => [1,3,5,2,6,4] => [.,[[.,.],[[.,.],[.,.]]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[5,1,2,4,3,6] => [1,5,3,2,4,6] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[5,1,2,4,6,3] => [1,5,6,3,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[5,1,2,6,3,4] => [1,5,3,2,4,6] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[5,1,4,2,3,6] => [1,5,3,4,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[5,1,4,2,6,3] => [1,5,6,3,4,2] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[5,1,4,6,3,2] => [1,5,3,4,6,2] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[5,1,6,2,3,4] => [1,5,3,6,4,2] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[5,1,6,4,3,2] => [1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[5,2,1,4,3,6] => [1,5,3,2,4,6] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[5,2,1,4,6,3] => [1,5,6,3,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[5,2,1,6,3,4] => [1,5,3,2,4,6] => [.,[[[.,.],[.,.]],[.,.]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 6
[1,5,2,3,4,6,7] => [1,2,5,4,3,6,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,2,3,4,7,6] => [1,2,5,4,3,6,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,2,3,6,4,7] => [1,2,5,6,4,3,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,2,3,6,7,4] => [1,2,5,6,7,4,3] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,2,3,7,4,6] => [1,2,5,7,6,4,3] => [.,[.,[[[.,.],.],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,2,3,7,6,4] => [1,2,5,7,4,3,6] => [.,[.,[[[.,.],.],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,2,4,3,6,7] => [1,2,5,3,4,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,2,4,3,7,6] => [1,2,5,3,4,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,2,4,6,3,7] => [1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,2,4,6,7,3] => [1,2,5,6,7,3,4] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,2,4,7,3,6] => [1,2,5,7,6,3,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,2,4,7,6,3] => [1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,2,6,3,4,7] => [1,2,5,3,4,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,2,6,3,7,4] => [1,2,5,3,4,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,2,6,4,3,7] => [1,2,5,4,6,3,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,2,6,4,7,3] => [1,2,5,4,6,7,3] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,2,6,7,3,4] => [1,2,5,7,4,6,3] => [.,[.,[[[.,.],.],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,2,6,7,4,3] => [1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,2,7,3,4,6] => [1,2,5,3,4,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,2,7,3,6,4] => [1,2,5,3,4,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,2,7,4,3,6] => [1,2,5,4,7,6,3] => [.,[.,[[[.,.],.],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,2,7,4,6,3] => [1,2,5,4,7,3,6] => [.,[.,[[[.,.],.],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,2,7,6,3,4] => [1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,2,7,6,4,3] => [1,2,5,6,4,7,3] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,3,2,4,6,7] => [1,2,5,4,3,6,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,3,2,4,7,6] => [1,2,5,4,3,6,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,3,2,6,4,7] => [1,2,5,6,4,3,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,3,2,6,7,4] => [1,2,5,6,7,4,3] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,3,2,7,4,6] => [1,2,5,7,6,4,3] => [.,[.,[[[.,.],.],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,3,2,7,6,4] => [1,2,5,7,4,3,6] => [.,[.,[[[.,.],.],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,3,4,2,6,7] => [1,2,5,3,4,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,3,4,2,7,6] => [1,2,5,3,4,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,3,4,6,2,7] => [1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,3,4,6,7,2] => [1,2,5,6,7,3,4] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,3,4,7,2,6] => [1,2,5,7,6,3,4] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,3,4,7,6,2] => [1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,3,6,2,4,7] => [1,2,5,3,4,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,3,6,2,7,4] => [1,2,5,3,4,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,3,6,4,2,7] => [1,2,5,4,6,3,7] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,3,6,4,7,2] => [1,2,5,4,6,7,3] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,3,6,7,2,4] => [1,2,5,7,4,6,3] => [.,[.,[[[.,.],.],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,3,6,7,4,2] => [1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,3,7,2,4,6] => [1,2,5,3,4,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,3,7,2,6,4] => [1,2,5,3,4,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,3,7,4,2,6] => [1,2,5,4,7,6,3] => [.,[.,[[[.,.],.],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,3,7,4,6,2] => [1,2,5,4,7,3,6] => [.,[.,[[[.,.],.],[[.,.],.]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,3,7,6,2,4] => [1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,3,7,6,4,2] => [1,2,5,6,4,7,3] => [.,[.,[[[.,.],.],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,4,2,3,6,7] => [1,2,5,3,4,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
 => ? = 8
[1,5,4,2,3,7,6] => [1,2,5,3,4,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
 => ([(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 2 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: 16% ●values known / values provided: 16%●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
[3,1,5,2,4,6] => [[.,.],[[.,.],[.,[.,.]]]]
 => ([(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
[3,1,5,2,6,4] => [[.,.],[[.,.],[[.,.],.]]]
 => ([(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
[3,1,5,4,2,6] => [[.,.],[[[.,.],.],[.,.]]]
 => ([(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
[3,1,5,4,6,2] => [[.,.],[[[.,.],[.,.]],.]]
 => ([(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
[3,1,5,6,2,4] => [[.,.],[[.,[.,.]],[.,.]]]
 => ([(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
[3,1,5,6,4,2] => [[.,.],[[[.,[.,.]],.],.]]
 => ([(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
[3,2,5,1,4,6] => [[[.,.],[.,.]],[.,[.,.]]]
 => ([(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
[3,2,5,1,6,4] => [[[.,.],[.,.]],[[.,.],.]]
 => ([(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
[3,2,5,4,1,6] => [[[.,.],[[.,.],.]],[.,.]]
 => ([(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
[3,2,5,4,6,1] => [[[.,.],[[.,.],[.,.]]],.]
 => ([(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
[3,2,5,6,1,4] => [[[.,.],[.,[.,.]]],[.,.]]
 => ([(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
[3,2,5,6,4,1] => [[[.,.],[[.,[.,.]],.]],.]
 => ([(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
[3,4,5,1,2,6] => [[.,[.,[.,.]]],[.,[.,.]]]
 => ([(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
[3,4,5,1,6,2] => [[.,[.,[.,.]]],[[.,.],.]]
 => ([(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
[3,4,5,2,1,6] => [[[.,[.,[.,.]]],.],[.,.]]
 => ([(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
[3,4,5,2,6,1] => [[[.,[.,[.,.]]],[.,.]],.]
 => ([(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
[3,4,5,6,1,2] => [[.,[.,[.,[.,.]]]],[.,.]]
 => ([(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
[3,4,5,6,2,1] => [[[.,[.,[.,[.,.]]]],.],.]
 => ([(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
[3,5,1,2,4,6] => [[.,[.,.]],[.,[.,[.,.]]]]
 => ([(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
[3,5,1,2,6,4] => [[.,[.,.]],[.,[[.,.],.]]]
 => ([(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
[3,5,1,4,2,6] => [[.,[.,.]],[[.,.],[.,.]]]
 => ([(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
[3,5,1,4,6,2] => [[.,[.,.]],[[.,[.,.]],.]]
 => ([(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
[3,5,1,6,2,4] => [[.,[.,.]],[[.,.],[.,.]]]
 => ([(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
[3,5,1,6,4,2] => [[.,[.,.]],[[[.,.],.],.]]
 => ([(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
[3,5,2,1,4,6] => [[[.,[.,.]],.],[.,[.,.]]]
 => ([(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
[3,5,2,1,6,4] => [[[.,[.,.]],.],[[.,.],.]]
 => ([(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
[3,5,2,4,1,6] => [[[.,[.,.]],[.,.]],[.,.]]
 => ([(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
[3,5,2,4,6,1] => [[[.,[.,.]],[.,[.,.]]],.]
 => ([(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
[3,5,2,6,1,4] => [[[.,[.,.]],[.,.]],[.,.]]
 => ([(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
[3,5,2,6,4,1] => [[[.,[.,.]],[[.,.],.]],.]
 => ([(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
[3,6,5,1,2,4] => [[.,[[.,.],.]],[.,[.,.]]]
 => ([(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
[3,6,5,1,4,2] => [[.,[[.,.],.]],[[.,.],.]]
 => ([(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
[3,6,5,2,1,4] => [[[.,[[.,.],.]],.],[.,.]]
 => ([(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
[3,6,5,2,4,1] => [[[.,[[.,.],.]],[.,.]],.]
 => ([(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
[3,6,5,4,1,2] => [[.,[[[.,.],.],.]],[.,.]]
 => ([(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
[3,6,5,4,2,1] => [[[.,[[[.,.],.],.]],.],.]
 => ([(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
[5,1,2,4,3,6] => [[.,.],[.,[[.,.],[.,.]]]]
 => ([(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
[5,1,2,4,6,3] => [[.,.],[.,[[.,[.,.]],.]]]
 => ([(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
[5,1,2,6,3,4] => [[.,.],[.,[[.,.],[.,.]]]]
 => ([(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
[5,1,4,2,3,6] => [[.,.],[[.,.],[.,[.,.]]]]
 => ([(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
[5,1,4,2,6,3] => [[.,.],[[.,.],[[.,.],.]]]
 => ([(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
[5,1,4,6,3,2] => [[.,.],[[[.,[.,.]],.],.]]
 => ([(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
[5,1,6,2,3,4] => [[.,.],[[.,.],[.,[.,.]]]]
 => ([(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
[5,1,6,4,3,2] => [[.,.],[[[[.,.],.],.],.]]
 => ([(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
[5,2,1,4,3,6] => [[[.,.],.],[[.,.],[.,.]]]
 => ([(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
[5,2,1,4,6,3] => [[[.,.],.],[[.,[.,.]],.]]
 => ([(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
[5,2,1,6,3,4] => [[[.,.],.],[[.,.],[.,.]]]
 => ([(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
[1,5,2,3,4,6,7] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
 => ([(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
[1,5,2,3,4,7,6] => [.,[[.,.],[.,[.,[[.,.],.]]]]]
 => ([(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
[1,5,2,3,6,4,7] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,2,3,6,7,4] => [.,[[.,.],[.,[[.,[.,.]],.]]]]
 => ([(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
[1,5,2,3,7,4,6] => [.,[[.,.],[.,[[.,.],[.,.]]]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,2,3,7,6,4] => [.,[[.,.],[.,[[[.,.],.],.]]]]
 => ([(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
[1,5,2,4,3,6,7] => [.,[[.,.],[[.,.],[.,[.,.]]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,2,4,3,7,6] => [.,[[.,.],[[.,.],[[.,.],.]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,2,4,6,3,7] => [.,[[.,.],[[.,[.,.]],[.,.]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,2,4,6,7,3] => [.,[[.,.],[[.,[.,[.,.]]],.]]]
 => ([(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
[1,5,2,4,7,3,6] => [.,[[.,.],[[.,[.,.]],[.,.]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,2,4,7,6,3] => [.,[[.,.],[[.,[[.,.],.]],.]]]
 => ([(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
[1,5,2,6,3,4,7] => [.,[[.,.],[[.,.],[.,[.,.]]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,2,6,3,7,4] => [.,[[.,.],[[.,.],[[.,.],.]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,2,6,4,3,7] => [.,[[.,.],[[[.,.],.],[.,.]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,2,6,4,7,3] => [.,[[.,.],[[[.,.],[.,.]],.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,2,6,7,3,4] => [.,[[.,.],[[.,[.,.]],[.,.]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,2,6,7,4,3] => [.,[[.,.],[[[.,[.,.]],.],.]]]
 => ([(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
[1,5,2,7,3,4,6] => [.,[[.,.],[[.,.],[.,[.,.]]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,2,7,3,6,4] => [.,[[.,.],[[.,.],[[.,.],.]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,2,7,4,3,6] => [.,[[.,.],[[[.,.],.],[.,.]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,2,7,4,6,3] => [.,[[.,.],[[[.,.],[.,.]],.]]]
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ([(0,6),(0,7),(1,6),(1,7),(2,5),(2,7),(3,5),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,2,7,6,3,4] => [.,[[.,.],[[[.,.],.],[.,.]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,2,7,6,4,3] => [.,[[.,.],[[[[.,.],.],.],.]]]
 => ([(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
[1,5,3,2,4,6,7] => [.,[[[.,.],.],[.,[.,[.,.]]]]]
 => ([(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
[1,5,3,2,4,7,6] => [.,[[[.,.],.],[.,[[.,.],.]]]]
 => ([(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
[1,5,3,2,6,4,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,3,2,6,7,4] => [.,[[[.,.],.],[[.,[.,.]],.]]]
 => ([(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
[1,5,3,2,7,4,6] => [.,[[[.,.],.],[[.,.],[.,.]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,3,2,7,6,4] => [.,[[[.,.],.],[[[.,.],.],.]]]
 => ([(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
[1,5,3,4,2,6,7] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,3,4,2,7,6] => [.,[[[.,.],[.,.]],[[.,.],.]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,3,4,6,2,7] => [.,[[[.,.],[.,[.,.]]],[.,.]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,3,4,6,7,2] => [.,[[[.,.],[.,[.,[.,.]]]],.]]
 => ([(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
[1,5,3,4,7,2,6] => [.,[[[.,.],[.,[.,.]]],[.,.]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,3,4,7,6,2] => [.,[[[.,.],[.,[[.,.],.]]],.]]
 => ([(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
[1,5,3,6,2,4,7] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,3,6,2,7,4] => [.,[[[.,.],[.,.]],[[.,.],.]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,3,6,4,2,7] => [.,[[[.,.],[[.,.],.]],[.,.]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,3,6,4,7,2] => [.,[[[.,.],[[.,.],[.,.]]],.]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,3,6,7,2,4] => [.,[[[.,.],[.,[.,.]]],[.,.]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,3,6,7,4,2] => [.,[[[.,.],[[.,[.,.]],.]],.]]
 => ([(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
[1,5,3,7,2,4,6] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,3,7,2,6,4] => [.,[[[.,.],[.,.]],[[.,.],.]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,3,7,4,2,6] => [.,[[[.,.],[[.,.],.]],[.,.]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,3,7,4,6,2] => [.,[[[.,.],[[.,.],[.,.]]],.]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,3,7,6,2,4] => [.,[[[.,.],[[.,.],.]],[.,.]]]
 => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(1,5),(1,7),(2,5),(2,7),(3,4),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8 + 1
[1,5,3,7,6,4,2] => [.,[[[.,.],[[[.,.],.],.]],.]]
 => ([(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
[1,5,4,2,3,6,7] => [.,[[[.,.],.],[.,[.,[.,.]]]]]
 => ([(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
[1,5,4,2,3,7,6] => [.,[[[.,.],.],[.,[[.,.],.]]]]
 => ([(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
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.
Matching statistic: St001182
(load all 3 compositions to match this statistic)
Mapping
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St001182: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 75%
Values
[1] => [1,0]
 => [1,0]
 => [1,0]
 => 1 = 0 + 1
[1,2] => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 3 = 2 + 1
[2,1] => [1,1,0,0]
 => [1,0,1,0]
 => [1,0,1,0]
 => 3 = 2 + 1
[3,1,5,2,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,1,5,2,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,1,5,4,2,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,1,5,4,6,2] => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,1,5,6,2,4] => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,1,5,6,4,2] => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,2,5,1,4,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,2,5,1,6,4] => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,2,5,4,1,6] => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,2,5,4,6,1] => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,2,5,6,1,4] => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,2,5,6,4,1] => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,4,5,1,2,6] => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,4,5,1,6,2] => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,4,5,2,1,6] => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,4,5,2,6,1] => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,4,5,6,1,2] => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,4,5,6,2,1] => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,5,1,2,4,6] => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,5,1,2,6,4] => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,5,1,4,2,6] => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,5,1,4,6,2] => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,5,1,6,2,4] => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,5,1,6,4,2] => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,5,2,1,4,6] => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,5,2,1,6,4] => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,5,2,4,1,6] => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,5,2,4,6,1] => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,5,2,6,1,4] => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,5,2,6,4,1] => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,6,5,1,2,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,6,5,1,4,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,6,5,2,1,4] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,6,5,2,4,1] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,6,5,4,1,2] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[3,6,5,4,2,1] => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[5,1,2,4,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[5,1,2,4,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[5,1,2,6,3,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[5,1,4,2,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[5,1,4,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[5,1,4,6,3,2] => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[5,1,6,2,3,4] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[5,1,6,4,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[5,2,1,4,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[5,2,1,4,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[5,2,1,6,3,4] => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 7 = 6 + 1
[1,5,2,3,4,6,7] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 8 + 1
[1,5,2,3,4,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,2,3,6,4,7] => [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 8 + 1
[1,5,2,3,6,7,4] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,2,3,7,4,6] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,2,3,7,6,4] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,2,4,3,6,7] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 8 + 1
[1,5,2,4,3,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,2,4,6,3,7] => [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 8 + 1
[1,5,2,4,6,7,3] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,2,4,7,3,6] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,2,4,7,6,3] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,2,6,3,4,7] => [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,2,6,3,7,4] => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,2,6,4,3,7] => [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,2,6,4,7,3] => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,2,6,7,3,4] => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,2,6,7,4,3] => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,2,7,3,4,6] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,2,7,3,6,4] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,2,7,4,3,6] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,2,7,4,6,3] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,2,7,6,3,4] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,2,7,6,4,3] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,3,2,4,6,7] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 8 + 1
[1,5,3,2,4,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,3,2,6,4,7] => [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 8 + 1
[1,5,3,2,6,7,4] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,3,2,7,4,6] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,3,2,7,6,4] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,3,4,2,6,7] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 8 + 1
[1,5,3,4,2,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,3,4,6,2,7] => [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 8 + 1
[1,5,3,4,6,7,2] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,3,4,7,2,6] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,3,4,7,6,2] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,3,6,2,4,7] => [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,3,6,2,7,4] => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,3,6,4,2,7] => [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,3,6,4,7,2] => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,3,6,7,2,4] => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,3,6,7,4,2] => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,3,7,2,4,6] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,3,7,2,6,4] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,3,7,4,2,6] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,3,7,4,6,2] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,3,7,6,2,4] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,3,7,6,4,2] => [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
[1,5,4,2,3,6,7] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 8 + 1
[1,5,4,2,3,7,6] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 8 + 1
Description
Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra.
The following 87 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. 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. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000956The maximal displacement of a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000625The sum of the minimal distances to a greater element. 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. St001074The number of inversions of the cyclic embedding of a permutation. St000216The absolute length of a permutation. St000619The number of cyclic descents of a permutation. St000652The maximal difference between successive positions of a 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. St001246The maximal difference between two consecutive entries of a permutation. St001480The number of simple summands of the module J^2/J^3. St000435The number of occurrences of the pattern 213 or of the pattern 231 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. 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. 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. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001890The maximum magnitude of the Möbius function of a poset. St000890The number of nonzero entries in an alternating sign matrix. St000219The number of occurrences of the pattern 231 in a permutation. St000327The number of cover relations in a poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. 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. St001960The number of descents of a permutation minus one if its first entry is not one. 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 $eA$. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001569The maximal modular displacement 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. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001948The number of augmented double ascents of a permutation. 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. 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. St000643The size of the largest orbit of antichains under Panyushev complementation. 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. St000189The number of elements in the poset. St000656The number of cuts of 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. 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. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000896The number of zeros on the main diagonal of an alternating sign matrix. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000993The multiplicity of the largest part of an integer partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000929The constant term of the character polynomial of an integer partition. St000401The size of the symmetry class of a permutation. St001893The flag descent of a signed permutation.