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Your data matches 102 different statistics following compositions of up to 3 maps.
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Matching statistic: St000422
(load all 41 compositions to match this statistic)
(load all 41 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000422: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> ([],1)
=> 0
[1,2] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 2
[2,1] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 2
[3,5,1,2,4,6] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[3,5,1,4,2,6] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(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
[3,5,2,4,1,6] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(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
[3,6,1,2,5,4] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[3,6,1,4,5,2] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[3,6,1,5,2,4] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[3,6,1,5,4,2] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[3,6,2,1,5,4] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[3,6,2,4,5,1] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[3,6,2,5,1,4] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[3,6,2,5,4,1] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,1,5,3,2,6] => [1,4,3,5,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,1,6,3,5,2] => [1,4,3,6,2,5] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,2,5,3,1,6] => [1,4,3,5,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,2,6,3,5,1] => [1,4,3,6,2,5] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,5,1,2,3,6] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,5,2,1,3,6] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,5,3,1,2,6] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,5,3,2,1,6] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,5,6,1,2,3] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,5,6,1,3,2] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,5,6,2,1,3] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,5,6,2,3,1] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,5,6,3,1,2] => [1,4,3,6,2,5] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,5,6,3,2,1] => [1,4,3,6,2,5] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,6,1,2,5,3] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,6,2,1,5,3] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,6,3,1,5,2] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,6,3,2,5,1] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,6,5,1,2,3] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,6,5,1,3,2] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,6,5,2,1,3] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,6,5,2,3,1] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,6,5,3,1,2] => [1,4,3,5,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,6,5,3,2,1] => [1,4,3,5,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[5,1,3,6,4,2] => [1,5,4,6,2,3] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(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
[5,1,6,4,3,2] => [1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[5,2,3,6,4,1] => [1,5,4,6,2,3] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[5,2,4,1,3,6] => [1,5,3,4,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[5,2,6,4,3,1] => [1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[5,3,1,6,4,2] => [1,5,4,6,2,3] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(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)
(load all 2 compositions to match this statistic)
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%
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
[3,5,1,2,4,6] => [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 9 = 6 + 3
[3,5,1,4,2,6] => [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
[3,5,2,4,1,6] => [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
[3,6,1,2,5,4] => [3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 9 = 6 + 3
[3,6,1,4,5,2] => [3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 9 = 6 + 3
[3,6,1,5,2,4] => [3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 9 = 6 + 3
[3,6,1,5,4,2] => [3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 9 = 6 + 3
[3,6,2,1,5,4] => [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 9 = 6 + 3
[3,6,2,4,5,1] => [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 9 = 6 + 3
[3,6,2,5,1,4] => [3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> 9 = 6 + 3
[3,6,2,5,4,1] => [3,2,5,4,1] => [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 9 = 6 + 3
[4,1,5,3,2,6] => [4,1,5,3,2] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 9 = 6 + 3
[4,1,6,3,5,2] => [4,1,3,5,2] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 9 = 6 + 3
[4,2,5,3,1,6] => [4,2,5,3,1] => [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 9 = 6 + 3
[4,2,6,3,5,1] => [4,2,3,5,1] => [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 9 = 6 + 3
[4,5,1,2,3,6] => [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 9 = 6 + 3
[4,5,2,1,3,6] => [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 9 = 6 + 3
[4,5,3,1,2,6] => [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 9 = 6 + 3
[4,5,3,2,1,6] => [4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 9 = 6 + 3
[4,5,6,1,2,3] => [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 9 = 6 + 3
[4,5,6,1,3,2] => [4,5,1,3,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 9 = 6 + 3
[4,5,6,2,1,3] => [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 9 = 6 + 3
[4,5,6,2,3,1] => [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 9 = 6 + 3
[4,5,6,3,1,2] => [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 9 = 6 + 3
[4,5,6,3,2,1] => [4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 9 = 6 + 3
[4,6,1,2,5,3] => [4,1,2,5,3] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 9 = 6 + 3
[4,6,2,1,5,3] => [4,2,1,5,3] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 9 = 6 + 3
[4,6,3,1,5,2] => [4,3,1,5,2] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 9 = 6 + 3
[4,6,3,2,5,1] => [4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 9 = 6 + 3
[4,6,5,1,2,3] => [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 9 = 6 + 3
[4,6,5,1,3,2] => [4,5,1,3,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 9 = 6 + 3
[4,6,5,2,1,3] => [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 9 = 6 + 3
[4,6,5,2,3,1] => [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 9 = 6 + 3
[4,6,5,3,1,2] => [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 9 = 6 + 3
[4,6,5,3,2,1] => [4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 9 = 6 + 3
[5,1,3,6,4,2] => [5,1,3,4,2] => [[.,.],[[.,[.,.]],.]]
=> ([(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
[5,1,6,4,3,2] => [5,1,4,3,2] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> 9 = 6 + 3
[5,2,3,6,4,1] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],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
[5,2,6,4,3,1] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 9 = 6 + 3
[5,3,1,6,4,2] => [5,3,1,4,2] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 9 = 6 + 3
[5,3,2,6,4,1] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 9 = 6 + 3
[5,4,1,3,2,6] => [5,4,1,3,2] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> 9 = 6 + 3
[5,4,2,3,1,6] => [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
=> ([(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
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%
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
[3,5,1,2,4,6] => [4,5,3,1,2,6] => [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
[3,5,1,6,2,4] => [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
[3,5,2,4,1,6] => [5,4,3,2,1,6] => [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
[3,6,1,2,5,4] => [4,6,3,1,5,2] => [2,2,1,1]
=> [2,1,1]
=> 1 = 6 - 5
[3,6,1,4,5,2] => [6,5,3,4,2,1] => [2,2,1,1]
=> [2,1,1]
=> 1 = 6 - 5
[3,6,1,5,2,4] => [5,6,3,4,1,2] => [2,2,1,1]
=> [2,1,1]
=> 1 = 6 - 5
[3,6,1,5,4,2] => [6,5,3,4,2,1] => [2,2,1,1]
=> [2,1,1]
=> 1 = 6 - 5
[3,6,2,1,5,4] => [4,6,3,1,5,2] => [2,2,1,1]
=> [2,1,1]
=> 1 = 6 - 5
[3,6,2,4,5,1] => [6,5,3,4,2,1] => [2,2,1,1]
=> [2,1,1]
=> 1 = 6 - 5
[3,6,2,5,1,4] => [5,6,3,4,1,2] => [2,2,1,1]
=> [2,1,1]
=> 1 = 6 - 5
[3,6,2,5,4,1] => [6,5,3,4,2,1] => [2,2,1,1]
=> [2,1,1]
=> 1 = 6 - 5
[4,1,5,3,2,6] => [5,2,4,3,1,6] => [2,2,1,1]
=> [2,1,1]
=> 1 = 6 - 5
[4,1,6,3,5,2] => [6,2,5,4,3,1] => [2,2,1,1]
=> [2,1,1]
=> 1 = 6 - 5
[4,2,5,3,1,6] => [5,4,3,2,1,6] => [2,2,1,1]
=> [2,1,1]
=> 1 = 6 - 5
[4,2,6,3,5,1] => [6,4,5,2,3,1] => [2,2,2]
=> [2,2]
=> 1 = 6 - 5
[4,5,1,2,3,6] => [5,4,3,2,1,6] => [2,2,1,1]
=> [2,1,1]
=> 1 = 6 - 5
[4,5,2,1,3,6] => [5,4,3,2,1,6] => [2,2,1,1]
=> [2,1,1]
=> 1 = 6 - 5
[4,5,3,1,2,6] => [5,4,3,2,1,6] => [2,2,1,1]
=> [2,1,1]
=> 1 = 6 - 5
[4,5,3,2,1,6] => [5,4,3,2,1,6] => [2,2,1,1]
=> [2,1,1]
=> 1 = 6 - 5
[4,5,6,1,2,3] => [6,5,4,3,2,1] => [2,2,2]
=> [2,2]
=> 1 = 6 - 5
[4,5,6,1,3,2] => [6,5,4,3,2,1] => [2,2,2]
=> [2,2]
=> 1 = 6 - 5
[4,5,6,2,1,3] => [6,5,4,3,2,1] => [2,2,2]
=> [2,2]
=> 1 = 6 - 5
[4,5,6,2,3,1] => [6,5,4,3,2,1] => [2,2,2]
=> [2,2]
=> 1 = 6 - 5
[4,5,6,3,1,2] => [6,5,4,3,2,1] => [2,2,2]
=> [2,2]
=> 1 = 6 - 5
[4,5,6,3,2,1] => [6,5,4,3,2,1] => [2,2,2]
=> [2,2]
=> 1 = 6 - 5
[4,6,1,2,5,3] => [6,5,3,4,2,1] => [2,2,1,1]
=> [2,1,1]
=> 1 = 6 - 5
[4,6,2,1,5,3] => [6,5,4,3,2,1] => [2,2,2]
=> [2,2]
=> 1 = 6 - 5
[4,6,3,1,5,2] => [6,5,4,3,2,1] => [2,2,2]
=> [2,2]
=> 1 = 6 - 5
[4,6,3,2,5,1] => [6,5,4,3,2,1] => [2,2,2]
=> [2,2]
=> 1 = 6 - 5
[4,6,5,1,2,3] => [6,5,4,3,2,1] => [2,2,2]
=> [2,2]
=> 1 = 6 - 5
[4,6,5,1,3,2] => [6,5,4,3,2,1] => [2,2,2]
=> [2,2]
=> 1 = 6 - 5
[4,6,5,2,1,3] => [6,5,4,3,2,1] => [2,2,2]
=> [2,2]
=> 1 = 6 - 5
[4,6,5,2,3,1] => [6,5,4,3,2,1] => [2,2,2]
=> [2,2]
=> 1 = 6 - 5
[4,6,5,3,1,2] => [6,5,4,3,2,1] => [2,2,2]
=> [2,2]
=> 1 = 6 - 5
[4,6,5,3,2,1] => [6,5,4,3,2,1] => [2,2,2]
=> [2,2]
=> 1 = 6 - 5
[5,1,3,6,4,2] => [6,2,5,4,3,1] => [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
[5,1,6,4,3,2] => [6,2,5,4,3,1] => [2,2,1,1]
=> [2,1,1]
=> 1 = 6 - 5
[5,2,3,6,4,1] => [6,5,3,4,2,1] => [2,2,1,1]
=> [2,1,1]
=> 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
[5,2,6,4,3,1] => [6,5,4,3,2,1] => [2,2,2]
=> [2,2]
=> 1 = 6 - 5
[5,3,1,6,4,2] => [6,5,3,4,2,1] => [2,2,1,1]
=> [2,1,1]
=> 1 = 6 - 5
[5,3,2,6,4,1] => [6,5,3,4,2,1] => [2,2,1,1]
=> [2,1,1]
=> 1 = 6 - 5
[5,4,1,3,2,6] => [5,4,3,2,1,6] => [2,2,1,1]
=> [2,1,1]
=> 1 = 6 - 5
[5,4,2,3,1,6] => [5,4,3,2,1,6] => [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: St000455
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 44%●distinct values known / distinct values provided: 25%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 44%●distinct values known / distinct values provided: 25%
Values
[1] => [1] => [1] => ([],1)
=> ? = 0 - 8
[1,2] => [1,2] => [2] => ([],2)
=> ? = 2 - 8
[2,1] => [1,2] => [2] => ([],2)
=> ? = 2 - 8
[3,5,1,2,4,6] => [1,3,2,5,4,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[3,5,1,4,2,6] => [1,3,2,5,4,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[3,5,1,6,2,4] => [1,3,2,5,4,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[3,5,1,6,4,2] => [1,3,2,5,4,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[3,5,2,1,4,6] => [1,3,2,5,4,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[3,5,2,4,1,6] => [1,3,2,5,4,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[3,5,2,6,1,4] => [1,3,2,5,4,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[3,5,2,6,4,1] => [1,3,2,5,4,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[3,6,1,2,5,4] => [1,3,2,6,4,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[3,6,1,4,5,2] => [1,3,2,6,4,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[3,6,1,5,2,4] => [1,3,2,6,4,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[3,6,1,5,4,2] => [1,3,2,6,4,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[3,6,2,1,5,4] => [1,3,2,6,4,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[3,6,2,4,5,1] => [1,3,2,6,4,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[3,6,2,5,1,4] => [1,3,2,6,4,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[3,6,2,5,4,1] => [1,3,2,6,4,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[4,1,5,3,2,6] => [1,4,3,5,2,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[4,1,6,3,5,2] => [1,4,3,6,2,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[4,2,5,3,1,6] => [1,4,3,5,2,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[4,2,6,3,5,1] => [1,4,3,6,2,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[4,5,1,2,3,6] => [1,4,2,5,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[4,5,2,1,3,6] => [1,4,2,5,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[4,5,3,1,2,6] => [1,4,2,5,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[4,5,3,2,1,6] => [1,4,2,5,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[4,5,6,1,2,3] => [1,4,2,5,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[4,5,6,1,3,2] => [1,4,2,5,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[4,5,6,2,1,3] => [1,4,2,5,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[4,5,6,2,3,1] => [1,4,2,5,3,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[4,5,6,3,1,2] => [1,4,3,6,2,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[4,5,6,3,2,1] => [1,4,3,6,2,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[4,6,1,2,5,3] => [1,4,2,6,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[4,6,2,1,5,3] => [1,4,2,6,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[4,6,3,1,5,2] => [1,4,2,6,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[4,6,3,2,5,1] => [1,4,2,6,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[4,6,5,1,2,3] => [1,4,2,6,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[4,6,5,1,3,2] => [1,4,2,6,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[4,6,5,2,1,3] => [1,4,2,6,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[4,6,5,2,3,1] => [1,4,2,6,3,5] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[4,6,5,3,1,2] => [1,4,3,5,2,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[4,6,5,3,2,1] => [1,4,3,5,2,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[5,1,3,6,4,2] => [1,5,4,6,2,3] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[5,1,4,2,3,6] => [1,5,3,4,2,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[5,1,6,4,3,2] => [1,5,3,6,2,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[5,2,3,6,4,1] => [1,5,4,6,2,3] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[5,2,4,1,3,6] => [1,5,3,4,2,6] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[5,2,6,4,3,1] => [1,5,3,6,2,4] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[5,3,1,6,4,2] => [1,5,4,6,2,3] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 6 - 8
[1,4,2,5,3,6,7] => [1,2,4,5,3,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,4,2,5,3,7,6] => [1,2,4,5,3,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,4,2,6,5,3,7] => [1,2,4,6,3,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,4,2,6,7,3,5] => [1,2,4,6,3,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,4,2,7,5,6,3] => [1,2,4,7,3,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,4,2,7,6,5,3] => [1,2,4,7,3,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,4,3,5,2,6,7] => [1,2,4,5,3,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,4,3,5,2,7,6] => [1,2,4,5,3,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,4,3,6,5,2,7] => [1,2,4,6,3,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,4,3,6,7,2,5] => [1,2,4,6,3,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,4,3,7,5,6,2] => [1,2,4,7,3,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,4,3,7,6,5,2] => [1,2,4,7,3,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,4,5,6,2,3,7] => [1,2,4,6,3,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,4,5,6,3,2,7] => [1,2,4,6,3,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,4,5,6,7,2,3] => [1,2,4,6,3,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,4,5,6,7,3,2] => [1,2,4,6,3,5,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,4,5,7,2,6,3] => [1,2,4,7,3,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,4,5,7,3,6,2] => [1,2,4,7,3,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,4,5,7,6,2,3] => [1,2,4,7,3,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,4,5,7,6,3,2] => [1,2,4,7,3,5,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,4,6,5,2,3,7] => [1,2,4,5,3,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,4,6,5,2,7,3] => [1,2,4,5,3,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,4,6,5,3,2,7] => [1,2,4,5,3,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,4,6,5,3,7,2] => [1,2,4,5,3,6,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,5,2,3,4,6,7] => [1,2,5,4,3,6,7] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,5,2,3,4,7,6] => [1,2,5,4,3,6,7] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,5,2,4,6,3,7] => [1,2,5,6,3,4,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,5,2,4,7,6,3] => [1,2,5,7,3,4,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,5,2,6,7,4,3] => [1,2,5,7,3,4,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,5,2,7,6,3,4] => [1,2,5,6,3,4,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,5,3,2,4,6,7] => [1,2,5,4,3,6,7] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,5,3,2,4,7,6] => [1,2,5,4,3,6,7] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,5,3,4,6,2,7] => [1,2,5,6,3,4,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,5,3,4,7,6,2] => [1,2,5,7,3,4,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,5,3,6,7,4,2] => [1,2,5,7,3,4,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,5,3,7,6,2,4] => [1,2,5,6,3,4,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,5,4,2,6,3,7] => [1,2,5,6,3,4,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,5,4,2,7,6,3] => [1,2,5,7,3,4,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,5,4,3,6,2,7] => [1,2,5,6,3,4,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,5,4,3,7,6,2] => [1,2,5,7,3,4,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,5,4,6,7,2,3] => [1,2,5,7,3,4,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,5,4,6,7,3,2] => [1,2,5,7,3,4,6] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,5,4,7,6,2,3] => [1,2,5,6,3,4,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,5,4,7,6,3,2] => [1,2,5,6,3,4,7] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,5,6,2,4,3,7] => [1,2,5,4,3,6,7] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,5,6,2,4,7,3] => [1,2,5,4,3,6,7] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,5,6,3,4,2,7] => [1,2,5,4,3,6,7] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,5,6,3,4,7,2] => [1,2,5,4,3,6,7] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,6,2,3,5,4,7] => [1,2,6,4,3,5,7] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 8 - 8
[1,6,2,3,7,4,5] => [1,2,6,4,3,5,7] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0 = 8 - 8
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001880
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 25% ●values known / values provided: 33%●distinct values known / distinct values provided: 25%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 25% ●values known / values provided: 33%●distinct values known / distinct values provided: 25%
Values
[1] => [1] => [1] => ([],1)
=> ? = 0 - 1
[1,2] => [1,2] => [2,1] => ([(0,1)],2)
=> ? = 2 - 1
[2,1] => [2,1] => [1,2] => ([(0,1)],2)
=> ? = 2 - 1
[3,5,1,2,4,6] => [3,6,1,5,4,2] => [6,1,5,4,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ? = 6 - 1
[3,5,1,4,2,6] => [3,6,1,5,4,2] => [6,1,5,4,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ? = 6 - 1
[3,5,1,6,2,4] => [3,6,1,5,4,2] => [6,1,5,4,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ? = 6 - 1
[3,5,1,6,4,2] => [3,6,1,5,4,2] => [6,1,5,4,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ? = 6 - 1
[3,5,2,1,4,6] => [3,6,2,1,5,4] => [3,1,6,5,2,4] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(1,16),(1,18),(2,7),(2,17),(2,19),(3,11),(3,12),(3,16),(3,18),(4,10),(4,13),(4,18),(5,9),(5,10),(5,12),(5,18),(6,2),(6,9),(6,11),(6,13),(6,16),(7,20),(9,15),(9,17),(9,19),(9,22),(10,14),(10,19),(11,15),(11,17),(11,19),(11,22),(12,14),(12,15),(12,22),(13,7),(13,19),(13,22),(14,21),(15,20),(15,21),(16,17),(16,22),(17,20),(17,21),(18,14),(18,22),(19,20),(19,21),(20,8),(21,8),(22,20),(22,21)],23)
=> ? = 6 - 1
[3,5,2,4,1,6] => [3,6,2,5,1,4] => [3,1,6,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,17),(1,20),(2,13),(2,15),(2,17),(2,18),(2,20),(3,12),(3,14),(3,17),(3,18),(4,8),(4,10),(4,12),(4,20),(5,8),(5,11),(5,14),(5,15),(5,20),(6,9),(6,10),(6,11),(6,13),(6,18),(8,16),(8,22),(8,26),(9,21),(9,26),(10,21),(10,25),(10,26),(11,19),(11,21),(11,26),(12,16),(12,25),(13,21),(13,22),(13,26),(14,16),(14,19),(14,25),(15,19),(15,22),(15,25),(15,26),(16,23),(17,25),(17,26),(18,19),(18,22),(18,25),(18,26),(19,23),(19,24),(20,21),(20,22),(20,25),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24),(26,23),(26,24)],27)
=> ? = 6 - 1
[3,5,2,6,1,4] => [3,6,2,5,1,4] => [3,1,6,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,17),(1,20),(2,13),(2,15),(2,17),(2,18),(2,20),(3,12),(3,14),(3,17),(3,18),(4,8),(4,10),(4,12),(4,20),(5,8),(5,11),(5,14),(5,15),(5,20),(6,9),(6,10),(6,11),(6,13),(6,18),(8,16),(8,22),(8,26),(9,21),(9,26),(10,21),(10,25),(10,26),(11,19),(11,21),(11,26),(12,16),(12,25),(13,21),(13,22),(13,26),(14,16),(14,19),(14,25),(15,19),(15,22),(15,25),(15,26),(16,23),(17,25),(17,26),(18,19),(18,22),(18,25),(18,26),(19,23),(19,24),(20,21),(20,22),(20,25),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24),(26,23),(26,24)],27)
=> ? = 6 - 1
[3,5,2,6,4,1] => [3,6,2,5,4,1] => [3,1,5,4,2,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,18),(1,23),(2,13),(2,14),(2,19),(3,11),(3,12),(3,19),(4,9),(4,10),(4,14),(4,19),(5,1),(5,9),(5,12),(5,15),(5,19),(6,10),(6,11),(6,13),(6,15),(7,21),(7,22),(9,17),(9,18),(9,23),(10,16),(10,17),(10,20),(11,20),(11,23),(12,18),(12,23),(13,16),(13,20),(14,16),(14,23),(15,7),(15,17),(15,20),(15,23),(16,22),(17,21),(17,22),(18,21),(19,18),(19,20),(19,23),(20,21),(20,22),(21,8),(22,8),(23,21),(23,22)],24)
=> ? = 6 - 1
[3,6,1,2,5,4] => [3,6,1,5,4,2] => [6,1,5,4,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ? = 6 - 1
[3,6,1,4,5,2] => [3,6,1,5,4,2] => [6,1,5,4,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ? = 6 - 1
[3,6,1,5,2,4] => [3,6,1,5,4,2] => [6,1,5,4,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ? = 6 - 1
[3,6,1,5,4,2] => [3,6,1,5,4,2] => [6,1,5,4,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ? = 6 - 1
[3,6,2,1,5,4] => [3,6,2,1,5,4] => [3,1,6,5,2,4] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(1,16),(1,18),(2,7),(2,17),(2,19),(3,11),(3,12),(3,16),(3,18),(4,10),(4,13),(4,18),(5,9),(5,10),(5,12),(5,18),(6,2),(6,9),(6,11),(6,13),(6,16),(7,20),(9,15),(9,17),(9,19),(9,22),(10,14),(10,19),(11,15),(11,17),(11,19),(11,22),(12,14),(12,15),(12,22),(13,7),(13,19),(13,22),(14,21),(15,20),(15,21),(16,17),(16,22),(17,20),(17,21),(18,14),(18,22),(19,20),(19,21),(20,8),(21,8),(22,20),(22,21)],23)
=> ? = 6 - 1
[3,6,2,4,5,1] => [3,6,2,5,4,1] => [3,1,5,4,2,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,18),(1,23),(2,13),(2,14),(2,19),(3,11),(3,12),(3,19),(4,9),(4,10),(4,14),(4,19),(5,1),(5,9),(5,12),(5,15),(5,19),(6,10),(6,11),(6,13),(6,15),(7,21),(7,22),(9,17),(9,18),(9,23),(10,16),(10,17),(10,20),(11,20),(11,23),(12,18),(12,23),(13,16),(13,20),(14,16),(14,23),(15,7),(15,17),(15,20),(15,23),(16,22),(17,21),(17,22),(18,21),(19,18),(19,20),(19,23),(20,21),(20,22),(21,8),(22,8),(23,21),(23,22)],24)
=> ? = 6 - 1
[3,6,2,5,1,4] => [3,6,2,5,1,4] => [3,1,6,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,17),(1,20),(2,13),(2,15),(2,17),(2,18),(2,20),(3,12),(3,14),(3,17),(3,18),(4,8),(4,10),(4,12),(4,20),(5,8),(5,11),(5,14),(5,15),(5,20),(6,9),(6,10),(6,11),(6,13),(6,18),(8,16),(8,22),(8,26),(9,21),(9,26),(10,21),(10,25),(10,26),(11,19),(11,21),(11,26),(12,16),(12,25),(13,21),(13,22),(13,26),(14,16),(14,19),(14,25),(15,19),(15,22),(15,25),(15,26),(16,23),(17,25),(17,26),(18,19),(18,22),(18,25),(18,26),(19,23),(19,24),(20,21),(20,22),(20,25),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24),(26,23),(26,24)],27)
=> ? = 6 - 1
[3,6,2,5,4,1] => [3,6,2,5,4,1] => [3,1,5,4,2,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,18),(1,23),(2,13),(2,14),(2,19),(3,11),(3,12),(3,19),(4,9),(4,10),(4,14),(4,19),(5,1),(5,9),(5,12),(5,15),(5,19),(6,10),(6,11),(6,13),(6,15),(7,21),(7,22),(9,17),(9,18),(9,23),(10,16),(10,17),(10,20),(11,20),(11,23),(12,18),(12,23),(13,16),(13,20),(14,16),(14,23),(15,7),(15,17),(15,20),(15,23),(16,22),(17,21),(17,22),(18,21),(19,18),(19,20),(19,23),(20,21),(20,22),(21,8),(22,8),(23,21),(23,22)],24)
=> ? = 6 - 1
[4,1,5,3,2,6] => [4,1,6,5,3,2] => [6,5,1,4,3,2] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 6 - 1
[4,1,6,3,5,2] => [4,1,6,5,3,2] => [6,5,1,4,3,2] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 6 - 1
[4,2,5,3,1,6] => [4,2,6,5,1,3] => [2,6,1,4,3,5] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,21),(1,22),(2,8),(2,10),(2,17),(3,12),(3,13),(3,15),(3,17),(4,11),(4,14),(4,15),(4,17),(5,8),(5,9),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(8,19),(8,23),(9,1),(9,19),(9,20),(9,23),(10,18),(10,19),(11,18),(11,20),(12,18),(12,20),(12,23),(13,16),(13,23),(14,16),(14,20),(14,23),(15,16),(15,19),(15,20),(16,22),(17,18),(17,19),(17,23),(18,21),(19,21),(19,22),(20,21),(20,22),(21,7),(22,7),(23,21),(23,22)],24)
=> ? = 6 - 1
[4,2,6,3,5,1] => [4,2,6,5,3,1] => [2,5,1,4,3,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,18),(1,23),(2,13),(2,14),(2,19),(3,11),(3,12),(3,19),(4,9),(4,10),(4,14),(4,19),(5,1),(5,9),(5,12),(5,15),(5,19),(6,10),(6,11),(6,13),(6,15),(7,21),(7,22),(9,17),(9,18),(9,23),(10,16),(10,17),(10,20),(11,20),(11,23),(12,18),(12,23),(13,16),(13,20),(14,16),(14,23),(15,7),(15,17),(15,20),(15,23),(16,22),(17,21),(17,22),(18,21),(19,18),(19,20),(19,23),(20,21),(20,22),(21,8),(22,8),(23,21),(23,22)],24)
=> ? = 6 - 1
[4,5,1,2,3,6] => [4,6,1,5,3,2] => [6,5,1,4,2,3] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ? = 6 - 1
[4,5,2,1,3,6] => [4,6,2,1,5,3] => [3,6,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,9),(1,18),(1,22),(2,11),(2,14),(2,16),(2,18),(3,9),(3,14),(3,15),(3,22),(4,12),(4,13),(4,16),(4,22),(5,10),(5,13),(5,15),(5,18),(5,22),(6,8),(6,10),(6,11),(6,12),(6,22),(8,20),(8,25),(9,19),(9,25),(10,20),(10,21),(10,25),(10,26),(11,17),(11,25),(11,26),(12,17),(12,20),(12,26),(13,21),(13,26),(14,19),(14,26),(15,19),(15,21),(15,25),(16,17),(16,26),(17,24),(18,19),(18,25),(18,26),(19,23),(20,23),(20,24),(21,23),(21,24),(22,20),(22,21),(22,25),(22,26),(23,7),(24,7),(25,23),(25,24),(26,23),(26,24)],27)
=> ? = 6 - 1
[4,5,3,1,2,6] => [4,6,3,1,5,2] => [6,3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,19),(2,10),(2,13),(2,19),(2,20),(3,9),(3,13),(3,18),(3,20),(4,12),(4,14),(4,18),(4,19),(4,20),(5,11),(5,14),(5,18),(5,19),(5,20),(6,8),(6,9),(6,10),(6,11),(6,12),(8,21),(8,22),(9,15),(9,21),(9,25),(10,15),(10,22),(10,25),(11,16),(11,21),(11,22),(11,25),(12,16),(12,21),(12,22),(12,25),(13,15),(13,25),(14,16),(14,17),(14,25),(15,24),(16,23),(16,24),(17,23),(18,17),(18,21),(18,25),(19,17),(19,22),(19,25),(20,17),(20,25),(21,23),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24)],26)
=> ? = 6 - 1
[4,5,3,2,1,6] => [4,6,3,2,1,5] => [4,3,1,6,2,5] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(1,16),(1,18),(2,7),(2,17),(2,19),(3,11),(3,12),(3,16),(3,18),(4,10),(4,13),(4,18),(5,9),(5,10),(5,12),(5,18),(6,2),(6,9),(6,11),(6,13),(6,16),(7,20),(9,15),(9,17),(9,19),(9,22),(10,14),(10,19),(11,15),(11,17),(11,19),(11,22),(12,14),(12,15),(12,22),(13,7),(13,19),(13,22),(14,21),(15,20),(15,21),(16,17),(16,22),(17,20),(17,21),(18,14),(18,22),(19,20),(19,21),(20,8),(21,8),(22,20),(22,21)],23)
=> ? = 6 - 1
[4,5,6,1,2,3] => [4,6,5,1,3,2] => [6,5,1,3,2,4] => ([(0,1),(0,2),(0,3),(0,5),(1,11),(1,14),(2,10),(2,13),(2,14),(3,10),(3,12),(3,14),(4,7),(4,8),(4,9),(5,4),(5,11),(5,12),(5,13),(7,17),(8,17),(8,18),(9,17),(9,18),(10,15),(11,7),(11,16),(12,8),(12,15),(12,16),(13,9),(13,15),(13,16),(14,15),(14,16),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
=> ? = 6 - 1
[4,5,6,1,3,2] => [4,6,5,1,3,2] => [6,5,1,3,2,4] => ([(0,1),(0,2),(0,3),(0,5),(1,11),(1,14),(2,10),(2,13),(2,14),(3,10),(3,12),(3,14),(4,7),(4,8),(4,9),(5,4),(5,11),(5,12),(5,13),(7,17),(8,17),(8,18),(9,17),(9,18),(10,15),(11,7),(11,16),(12,8),(12,15),(12,16),(13,9),(13,15),(13,16),(14,15),(14,16),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
=> ? = 6 - 1
[4,5,6,2,1,3] => [4,6,5,2,1,3] => [4,6,1,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(1,8),(1,18),(2,11),(2,12),(2,14),(2,18),(3,10),(3,13),(3,14),(3,18),(4,7),(4,9),(4,12),(4,13),(5,8),(5,9),(5,10),(5,11),(7,16),(7,23),(8,16),(8,19),(9,16),(9,17),(9,20),(9,23),(10,19),(10,20),(11,19),(11,20),(11,23),(12,15),(12,23),(13,15),(13,20),(13,23),(14,15),(14,17),(14,20),(15,22),(16,21),(16,22),(17,21),(17,22),(18,17),(18,19),(18,23),(19,21),(20,21),(20,22),(21,6),(22,6),(23,21),(23,22)],24)
=> ? = 6 - 1
[4,5,6,2,3,1] => [4,6,5,2,3,1] => [4,5,1,3,2,6] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,15),(1,18),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,9),(4,10),(4,13),(5,1),(5,8),(5,11),(5,13),(6,19),(6,20),(8,14),(8,18),(9,14),(9,16),(10,16),(10,17),(11,15),(11,17),(11,18),(12,15),(12,16),(12,18),(13,6),(13,14),(13,17),(14,19),(15,20),(16,19),(16,20),(17,19),(17,20),(18,19),(18,20),(19,7),(20,7)],21)
=> ? = 6 - 1
[4,5,6,3,1,2] => [4,6,5,3,1,2] => [6,4,1,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
=> ? = 6 - 1
[4,5,6,3,2,1] => [4,6,5,3,2,1] => [5,4,1,3,2,6] => ([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
=> ? = 6 - 1
[4,6,1,2,5,3] => [4,6,1,5,3,2] => [6,5,1,4,2,3] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ? = 6 - 1
[4,6,2,1,5,3] => [4,6,2,1,5,3] => [3,6,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,9),(1,18),(1,22),(2,11),(2,14),(2,16),(2,18),(3,9),(3,14),(3,15),(3,22),(4,12),(4,13),(4,16),(4,22),(5,10),(5,13),(5,15),(5,18),(5,22),(6,8),(6,10),(6,11),(6,12),(6,22),(8,20),(8,25),(9,19),(9,25),(10,20),(10,21),(10,25),(10,26),(11,17),(11,25),(11,26),(12,17),(12,20),(12,26),(13,21),(13,26),(14,19),(14,26),(15,19),(15,21),(15,25),(16,17),(16,26),(17,24),(18,19),(18,25),(18,26),(19,23),(20,23),(20,24),(21,23),(21,24),(22,20),(22,21),(22,25),(22,26),(23,7),(24,7),(25,23),(25,24),(26,23),(26,24)],27)
=> ? = 6 - 1
[4,6,3,1,5,2] => [4,6,3,1,5,2] => [6,3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,19),(2,10),(2,13),(2,19),(2,20),(3,9),(3,13),(3,18),(3,20),(4,12),(4,14),(4,18),(4,19),(4,20),(5,11),(5,14),(5,18),(5,19),(5,20),(6,8),(6,9),(6,10),(6,11),(6,12),(8,21),(8,22),(9,15),(9,21),(9,25),(10,15),(10,22),(10,25),(11,16),(11,21),(11,22),(11,25),(12,16),(12,21),(12,22),(12,25),(13,15),(13,25),(14,16),(14,17),(14,25),(15,24),(16,23),(16,24),(17,23),(18,17),(18,21),(18,25),(19,17),(19,22),(19,25),(20,17),(20,25),(21,23),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24)],26)
=> ? = 6 - 1
[4,6,3,2,5,1] => [4,6,3,2,5,1] => [4,3,1,5,2,6] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,7),(2,18),(2,22),(3,9),(3,11),(3,19),(4,10),(4,12),(4,17),(4,19),(5,11),(5,12),(5,13),(5,19),(6,2),(6,9),(6,10),(6,13),(6,17),(7,20),(7,21),(9,16),(9,18),(9,22),(10,15),(10,18),(10,22),(11,14),(11,16),(12,14),(12,15),(12,22),(13,7),(13,15),(13,16),(13,22),(14,21),(15,20),(15,21),(16,20),(16,21),(17,18),(17,22),(18,20),(19,14),(19,22),(20,8),(21,8),(22,20),(22,21)],23)
=> ? = 6 - 1
[4,6,5,1,2,3] => [4,6,5,1,3,2] => [6,5,1,3,2,4] => ([(0,1),(0,2),(0,3),(0,5),(1,11),(1,14),(2,10),(2,13),(2,14),(3,10),(3,12),(3,14),(4,7),(4,8),(4,9),(5,4),(5,11),(5,12),(5,13),(7,17),(8,17),(8,18),(9,17),(9,18),(10,15),(11,7),(11,16),(12,8),(12,15),(12,16),(13,9),(13,15),(13,16),(14,15),(14,16),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
=> ? = 6 - 1
[4,6,5,1,3,2] => [4,6,5,1,3,2] => [6,5,1,3,2,4] => ([(0,1),(0,2),(0,3),(0,5),(1,11),(1,14),(2,10),(2,13),(2,14),(3,10),(3,12),(3,14),(4,7),(4,8),(4,9),(5,4),(5,11),(5,12),(5,13),(7,17),(8,17),(8,18),(9,17),(9,18),(10,15),(11,7),(11,16),(12,8),(12,15),(12,16),(13,9),(13,15),(13,16),(14,15),(14,16),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
=> ? = 6 - 1
[4,6,5,2,1,3] => [4,6,5,2,1,3] => [4,6,1,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(1,8),(1,18),(2,11),(2,12),(2,14),(2,18),(3,10),(3,13),(3,14),(3,18),(4,7),(4,9),(4,12),(4,13),(5,8),(5,9),(5,10),(5,11),(7,16),(7,23),(8,16),(8,19),(9,16),(9,17),(9,20),(9,23),(10,19),(10,20),(11,19),(11,20),(11,23),(12,15),(12,23),(13,15),(13,20),(13,23),(14,15),(14,17),(14,20),(15,22),(16,21),(16,22),(17,21),(17,22),(18,17),(18,19),(18,23),(19,21),(20,21),(20,22),(21,6),(22,6),(23,21),(23,22)],24)
=> ? = 6 - 1
[4,6,5,2,3,1] => [4,6,5,2,3,1] => [4,5,1,3,2,6] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,15),(1,18),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,9),(4,10),(4,13),(5,1),(5,8),(5,11),(5,13),(6,19),(6,20),(8,14),(8,18),(9,14),(9,16),(10,16),(10,17),(11,15),(11,17),(11,18),(12,15),(12,16),(12,18),(13,6),(13,14),(13,17),(14,19),(15,20),(16,19),(16,20),(17,19),(17,20),(18,19),(18,20),(19,7),(20,7)],21)
=> ? = 6 - 1
[4,6,5,3,1,2] => [4,6,5,3,1,2] => [6,4,1,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
=> ? = 6 - 1
[4,6,5,3,2,1] => [4,6,5,3,2,1] => [5,4,1,3,2,6] => ([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
=> ? = 6 - 1
[5,1,3,6,4,2] => [5,1,6,4,3,2] => [6,5,4,1,3,2] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 6 - 1
[5,1,4,2,3,6] => [5,1,6,4,3,2] => [6,5,4,1,3,2] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 6 - 1
[5,1,6,4,3,2] => [5,1,6,4,3,2] => [6,5,4,1,3,2] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 6 - 1
[5,2,3,6,4,1] => [5,2,6,4,3,1] => [2,5,4,1,3,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,19),(1,23),(2,8),(2,16),(2,17),(3,10),(3,11),(3,17),(4,12),(4,13),(4,16),(4,17),(5,1),(5,9),(5,11),(5,12),(5,16),(6,8),(6,9),(6,10),(6,13),(8,18),(8,20),(9,14),(9,15),(9,20),(9,23),(10,14),(10,18),(11,14),(11,19),(11,23),(12,15),(12,19),(12,23),(13,15),(13,18),(13,20),(14,21),(14,22),(15,21),(15,22),(16,19),(16,20),(16,23),(17,18),(17,23),(18,21),(19,22),(20,21),(20,22),(21,7),(22,7),(23,21),(23,22)],24)
=> ? = 6 - 1
[5,2,4,1,3,6] => [5,2,6,1,4,3] => [2,6,5,1,3,4] => ([(0,3),(0,4),(0,5),(0,6),(1,11),(1,16),(1,18),(2,12),(2,15),(3,9),(3,14),(4,8),(4,10),(4,14),(5,1),(5,8),(5,13),(5,14),(6,2),(6,9),(6,10),(6,13),(8,16),(8,18),(9,15),(9,17),(10,15),(10,17),(10,18),(11,19),(11,20),(12,19),(12,20),(13,11),(13,12),(13,17),(13,18),(14,16),(14,17),(15,19),(16,20),(17,19),(17,20),(18,19),(18,20),(19,7),(20,7)],21)
=> ? = 6 - 1
[5,2,6,4,3,1] => [5,2,6,4,3,1] => [2,5,4,1,3,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,19),(1,23),(2,8),(2,16),(2,17),(3,10),(3,11),(3,17),(4,12),(4,13),(4,16),(4,17),(5,1),(5,9),(5,11),(5,12),(5,16),(6,8),(6,9),(6,10),(6,13),(8,18),(8,20),(9,14),(9,15),(9,20),(9,23),(10,14),(10,18),(11,14),(11,19),(11,23),(12,15),(12,19),(12,23),(13,15),(13,18),(13,20),(14,21),(14,22),(15,21),(15,22),(16,19),(16,20),(16,23),(17,18),(17,23),(18,21),(19,22),(20,21),(20,22),(21,7),(22,7),(23,21),(23,22)],24)
=> ? = 6 - 1
[5,3,1,6,4,2] => [5,3,1,6,4,2] => [6,2,5,1,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,21),(2,11),(2,12),(2,19),(3,9),(3,13),(3,19),(4,10),(4,13),(4,19),(5,10),(5,12),(5,14),(5,19),(6,1),(6,9),(6,11),(6,14),(6,19),(8,17),(8,20),(9,18),(9,21),(10,16),(10,21),(11,15),(11,18),(11,21),(12,15),(12,16),(13,21),(14,8),(14,15),(14,16),(14,18),(15,17),(15,20),(16,17),(16,20),(17,7),(18,17),(18,20),(19,16),(19,18),(19,21),(20,7),(21,20)],22)
=> ? = 6 - 1
[1,4,2,5,3,6,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,2,5,3,7,6] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,2,6,5,3,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,2,6,7,3,5] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,2,7,5,6,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,2,7,6,5,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,3,5,2,6,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,3,5,2,7,6] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,3,6,5,2,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,3,6,7,2,5] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,3,7,5,6,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,3,7,6,5,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,5,6,2,3,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,5,6,3,2,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,5,6,7,2,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,5,6,7,3,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,5,7,2,6,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,5,7,3,6,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,5,7,6,2,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,5,7,6,3,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,6,5,2,3,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,6,5,2,7,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,6,5,3,2,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,6,5,3,7,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,6,7,2,5,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,6,7,3,5,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,6,7,5,2,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,6,7,5,3,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,7,5,2,3,6] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,7,5,2,6,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,7,5,3,2,6] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,7,5,3,6,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,7,6,2,3,5] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,7,6,3,2,5] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,7,6,5,2,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,4,7,6,5,3,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,5,2,3,4,6,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(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] => [7,6,5,4,3,2,1] => ([(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] => [7,6,5,4,3,2,1] => ([(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] => [7,6,5,4,3,2,1] => ([(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] => [7,6,5,4,3,2,1] => ([(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] => [7,6,5,4,3,2,1] => ([(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] => [7,6,5,4,3,2,1] => ([(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] => [7,6,5,4,3,2,1] => ([(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] => [7,6,5,4,3,2,1] => ([(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] => [7,6,5,4,3,2,1] => ([(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] => [7,6,5,4,3,2,1] => ([(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] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,5,4,2,6,3,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7 = 8 - 1
[1,5,4,2,7,6,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(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 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 25% ●values known / values provided: 33%●distinct values known / distinct values provided: 25%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 25% ●values known / values provided: 33%●distinct values known / distinct values provided: 25%
Values
[1] => [1] => [1] => ([],1)
=> ? = 0 - 2
[1,2] => [1,2] => [2,1] => ([(0,1)],2)
=> ? = 2 - 2
[2,1] => [2,1] => [1,2] => ([(0,1)],2)
=> ? = 2 - 2
[3,5,1,2,4,6] => [3,6,1,5,4,2] => [6,1,5,4,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ? = 6 - 2
[3,5,1,4,2,6] => [3,6,1,5,4,2] => [6,1,5,4,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ? = 6 - 2
[3,5,1,6,2,4] => [3,6,1,5,4,2] => [6,1,5,4,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ? = 6 - 2
[3,5,1,6,4,2] => [3,6,1,5,4,2] => [6,1,5,4,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ? = 6 - 2
[3,5,2,1,4,6] => [3,6,2,1,5,4] => [3,1,6,5,2,4] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(1,16),(1,18),(2,7),(2,17),(2,19),(3,11),(3,12),(3,16),(3,18),(4,10),(4,13),(4,18),(5,9),(5,10),(5,12),(5,18),(6,2),(6,9),(6,11),(6,13),(6,16),(7,20),(9,15),(9,17),(9,19),(9,22),(10,14),(10,19),(11,15),(11,17),(11,19),(11,22),(12,14),(12,15),(12,22),(13,7),(13,19),(13,22),(14,21),(15,20),(15,21),(16,17),(16,22),(17,20),(17,21),(18,14),(18,22),(19,20),(19,21),(20,8),(21,8),(22,20),(22,21)],23)
=> ? = 6 - 2
[3,5,2,4,1,6] => [3,6,2,5,1,4] => [3,1,6,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,17),(1,20),(2,13),(2,15),(2,17),(2,18),(2,20),(3,12),(3,14),(3,17),(3,18),(4,8),(4,10),(4,12),(4,20),(5,8),(5,11),(5,14),(5,15),(5,20),(6,9),(6,10),(6,11),(6,13),(6,18),(8,16),(8,22),(8,26),(9,21),(9,26),(10,21),(10,25),(10,26),(11,19),(11,21),(11,26),(12,16),(12,25),(13,21),(13,22),(13,26),(14,16),(14,19),(14,25),(15,19),(15,22),(15,25),(15,26),(16,23),(17,25),(17,26),(18,19),(18,22),(18,25),(18,26),(19,23),(19,24),(20,21),(20,22),(20,25),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24),(26,23),(26,24)],27)
=> ? = 6 - 2
[3,5,2,6,1,4] => [3,6,2,5,1,4] => [3,1,6,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,17),(1,20),(2,13),(2,15),(2,17),(2,18),(2,20),(3,12),(3,14),(3,17),(3,18),(4,8),(4,10),(4,12),(4,20),(5,8),(5,11),(5,14),(5,15),(5,20),(6,9),(6,10),(6,11),(6,13),(6,18),(8,16),(8,22),(8,26),(9,21),(9,26),(10,21),(10,25),(10,26),(11,19),(11,21),(11,26),(12,16),(12,25),(13,21),(13,22),(13,26),(14,16),(14,19),(14,25),(15,19),(15,22),(15,25),(15,26),(16,23),(17,25),(17,26),(18,19),(18,22),(18,25),(18,26),(19,23),(19,24),(20,21),(20,22),(20,25),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24),(26,23),(26,24)],27)
=> ? = 6 - 2
[3,5,2,6,4,1] => [3,6,2,5,4,1] => [3,1,5,4,2,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,18),(1,23),(2,13),(2,14),(2,19),(3,11),(3,12),(3,19),(4,9),(4,10),(4,14),(4,19),(5,1),(5,9),(5,12),(5,15),(5,19),(6,10),(6,11),(6,13),(6,15),(7,21),(7,22),(9,17),(9,18),(9,23),(10,16),(10,17),(10,20),(11,20),(11,23),(12,18),(12,23),(13,16),(13,20),(14,16),(14,23),(15,7),(15,17),(15,20),(15,23),(16,22),(17,21),(17,22),(18,21),(19,18),(19,20),(19,23),(20,21),(20,22),(21,8),(22,8),(23,21),(23,22)],24)
=> ? = 6 - 2
[3,6,1,2,5,4] => [3,6,1,5,4,2] => [6,1,5,4,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ? = 6 - 2
[3,6,1,4,5,2] => [3,6,1,5,4,2] => [6,1,5,4,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ? = 6 - 2
[3,6,1,5,2,4] => [3,6,1,5,4,2] => [6,1,5,4,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ? = 6 - 2
[3,6,1,5,4,2] => [3,6,1,5,4,2] => [6,1,5,4,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,10),(1,17),(2,7),(2,13),(3,9),(3,11),(3,13),(4,1),(4,8),(4,11),(4,13),(5,7),(5,8),(5,9),(7,16),(8,12),(8,16),(8,17),(9,12),(9,16),(10,14),(11,10),(11,12),(11,17),(12,14),(12,15),(13,16),(13,17),(14,6),(15,6),(16,15),(17,14),(17,15)],18)
=> ? = 6 - 2
[3,6,2,1,5,4] => [3,6,2,1,5,4] => [3,1,6,5,2,4] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(1,16),(1,18),(2,7),(2,17),(2,19),(3,11),(3,12),(3,16),(3,18),(4,10),(4,13),(4,18),(5,9),(5,10),(5,12),(5,18),(6,2),(6,9),(6,11),(6,13),(6,16),(7,20),(9,15),(9,17),(9,19),(9,22),(10,14),(10,19),(11,15),(11,17),(11,19),(11,22),(12,14),(12,15),(12,22),(13,7),(13,19),(13,22),(14,21),(15,20),(15,21),(16,17),(16,22),(17,20),(17,21),(18,14),(18,22),(19,20),(19,21),(20,8),(21,8),(22,20),(22,21)],23)
=> ? = 6 - 2
[3,6,2,4,5,1] => [3,6,2,5,4,1] => [3,1,5,4,2,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,18),(1,23),(2,13),(2,14),(2,19),(3,11),(3,12),(3,19),(4,9),(4,10),(4,14),(4,19),(5,1),(5,9),(5,12),(5,15),(5,19),(6,10),(6,11),(6,13),(6,15),(7,21),(7,22),(9,17),(9,18),(9,23),(10,16),(10,17),(10,20),(11,20),(11,23),(12,18),(12,23),(13,16),(13,20),(14,16),(14,23),(15,7),(15,17),(15,20),(15,23),(16,22),(17,21),(17,22),(18,21),(19,18),(19,20),(19,23),(20,21),(20,22),(21,8),(22,8),(23,21),(23,22)],24)
=> ? = 6 - 2
[3,6,2,5,1,4] => [3,6,2,5,1,4] => [3,1,6,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,17),(1,20),(2,13),(2,15),(2,17),(2,18),(2,20),(3,12),(3,14),(3,17),(3,18),(4,8),(4,10),(4,12),(4,20),(5,8),(5,11),(5,14),(5,15),(5,20),(6,9),(6,10),(6,11),(6,13),(6,18),(8,16),(8,22),(8,26),(9,21),(9,26),(10,21),(10,25),(10,26),(11,19),(11,21),(11,26),(12,16),(12,25),(13,21),(13,22),(13,26),(14,16),(14,19),(14,25),(15,19),(15,22),(15,25),(15,26),(16,23),(17,25),(17,26),(18,19),(18,22),(18,25),(18,26),(19,23),(19,24),(20,21),(20,22),(20,25),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24),(26,23),(26,24)],27)
=> ? = 6 - 2
[3,6,2,5,4,1] => [3,6,2,5,4,1] => [3,1,5,4,2,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,18),(1,23),(2,13),(2,14),(2,19),(3,11),(3,12),(3,19),(4,9),(4,10),(4,14),(4,19),(5,1),(5,9),(5,12),(5,15),(5,19),(6,10),(6,11),(6,13),(6,15),(7,21),(7,22),(9,17),(9,18),(9,23),(10,16),(10,17),(10,20),(11,20),(11,23),(12,18),(12,23),(13,16),(13,20),(14,16),(14,23),(15,7),(15,17),(15,20),(15,23),(16,22),(17,21),(17,22),(18,21),(19,18),(19,20),(19,23),(20,21),(20,22),(21,8),(22,8),(23,21),(23,22)],24)
=> ? = 6 - 2
[4,1,5,3,2,6] => [4,1,6,5,3,2] => [6,5,1,4,3,2] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 6 - 2
[4,1,6,3,5,2] => [4,1,6,5,3,2] => [6,5,1,4,3,2] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 6 - 2
[4,2,5,3,1,6] => [4,2,6,5,1,3] => [2,6,1,4,3,5] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,21),(1,22),(2,8),(2,10),(2,17),(3,12),(3,13),(3,15),(3,17),(4,11),(4,14),(4,15),(4,17),(5,8),(5,9),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(8,19),(8,23),(9,1),(9,19),(9,20),(9,23),(10,18),(10,19),(11,18),(11,20),(12,18),(12,20),(12,23),(13,16),(13,23),(14,16),(14,20),(14,23),(15,16),(15,19),(15,20),(16,22),(17,18),(17,19),(17,23),(18,21),(19,21),(19,22),(20,21),(20,22),(21,7),(22,7),(23,21),(23,22)],24)
=> ? = 6 - 2
[4,2,6,3,5,1] => [4,2,6,5,3,1] => [2,5,1,4,3,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,18),(1,23),(2,13),(2,14),(2,19),(3,11),(3,12),(3,19),(4,9),(4,10),(4,14),(4,19),(5,1),(5,9),(5,12),(5,15),(5,19),(6,10),(6,11),(6,13),(6,15),(7,21),(7,22),(9,17),(9,18),(9,23),(10,16),(10,17),(10,20),(11,20),(11,23),(12,18),(12,23),(13,16),(13,20),(14,16),(14,23),(15,7),(15,17),(15,20),(15,23),(16,22),(17,21),(17,22),(18,21),(19,18),(19,20),(19,23),(20,21),(20,22),(21,8),(22,8),(23,21),(23,22)],24)
=> ? = 6 - 2
[4,5,1,2,3,6] => [4,6,1,5,3,2] => [6,5,1,4,2,3] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ? = 6 - 2
[4,5,2,1,3,6] => [4,6,2,1,5,3] => [3,6,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,9),(1,18),(1,22),(2,11),(2,14),(2,16),(2,18),(3,9),(3,14),(3,15),(3,22),(4,12),(4,13),(4,16),(4,22),(5,10),(5,13),(5,15),(5,18),(5,22),(6,8),(6,10),(6,11),(6,12),(6,22),(8,20),(8,25),(9,19),(9,25),(10,20),(10,21),(10,25),(10,26),(11,17),(11,25),(11,26),(12,17),(12,20),(12,26),(13,21),(13,26),(14,19),(14,26),(15,19),(15,21),(15,25),(16,17),(16,26),(17,24),(18,19),(18,25),(18,26),(19,23),(20,23),(20,24),(21,23),(21,24),(22,20),(22,21),(22,25),(22,26),(23,7),(24,7),(25,23),(25,24),(26,23),(26,24)],27)
=> ? = 6 - 2
[4,5,3,1,2,6] => [4,6,3,1,5,2] => [6,3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,19),(2,10),(2,13),(2,19),(2,20),(3,9),(3,13),(3,18),(3,20),(4,12),(4,14),(4,18),(4,19),(4,20),(5,11),(5,14),(5,18),(5,19),(5,20),(6,8),(6,9),(6,10),(6,11),(6,12),(8,21),(8,22),(9,15),(9,21),(9,25),(10,15),(10,22),(10,25),(11,16),(11,21),(11,22),(11,25),(12,16),(12,21),(12,22),(12,25),(13,15),(13,25),(14,16),(14,17),(14,25),(15,24),(16,23),(16,24),(17,23),(18,17),(18,21),(18,25),(19,17),(19,22),(19,25),(20,17),(20,25),(21,23),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24)],26)
=> ? = 6 - 2
[4,5,3,2,1,6] => [4,6,3,2,1,5] => [4,3,1,6,2,5] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(1,16),(1,18),(2,7),(2,17),(2,19),(3,11),(3,12),(3,16),(3,18),(4,10),(4,13),(4,18),(5,9),(5,10),(5,12),(5,18),(6,2),(6,9),(6,11),(6,13),(6,16),(7,20),(9,15),(9,17),(9,19),(9,22),(10,14),(10,19),(11,15),(11,17),(11,19),(11,22),(12,14),(12,15),(12,22),(13,7),(13,19),(13,22),(14,21),(15,20),(15,21),(16,17),(16,22),(17,20),(17,21),(18,14),(18,22),(19,20),(19,21),(20,8),(21,8),(22,20),(22,21)],23)
=> ? = 6 - 2
[4,5,6,1,2,3] => [4,6,5,1,3,2] => [6,5,1,3,2,4] => ([(0,1),(0,2),(0,3),(0,5),(1,11),(1,14),(2,10),(2,13),(2,14),(3,10),(3,12),(3,14),(4,7),(4,8),(4,9),(5,4),(5,11),(5,12),(5,13),(7,17),(8,17),(8,18),(9,17),(9,18),(10,15),(11,7),(11,16),(12,8),(12,15),(12,16),(13,9),(13,15),(13,16),(14,15),(14,16),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
=> ? = 6 - 2
[4,5,6,1,3,2] => [4,6,5,1,3,2] => [6,5,1,3,2,4] => ([(0,1),(0,2),(0,3),(0,5),(1,11),(1,14),(2,10),(2,13),(2,14),(3,10),(3,12),(3,14),(4,7),(4,8),(4,9),(5,4),(5,11),(5,12),(5,13),(7,17),(8,17),(8,18),(9,17),(9,18),(10,15),(11,7),(11,16),(12,8),(12,15),(12,16),(13,9),(13,15),(13,16),(14,15),(14,16),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
=> ? = 6 - 2
[4,5,6,2,1,3] => [4,6,5,2,1,3] => [4,6,1,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(1,8),(1,18),(2,11),(2,12),(2,14),(2,18),(3,10),(3,13),(3,14),(3,18),(4,7),(4,9),(4,12),(4,13),(5,8),(5,9),(5,10),(5,11),(7,16),(7,23),(8,16),(8,19),(9,16),(9,17),(9,20),(9,23),(10,19),(10,20),(11,19),(11,20),(11,23),(12,15),(12,23),(13,15),(13,20),(13,23),(14,15),(14,17),(14,20),(15,22),(16,21),(16,22),(17,21),(17,22),(18,17),(18,19),(18,23),(19,21),(20,21),(20,22),(21,6),(22,6),(23,21),(23,22)],24)
=> ? = 6 - 2
[4,5,6,2,3,1] => [4,6,5,2,3,1] => [4,5,1,3,2,6] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,15),(1,18),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,9),(4,10),(4,13),(5,1),(5,8),(5,11),(5,13),(6,19),(6,20),(8,14),(8,18),(9,14),(9,16),(10,16),(10,17),(11,15),(11,17),(11,18),(12,15),(12,16),(12,18),(13,6),(13,14),(13,17),(14,19),(15,20),(16,19),(16,20),(17,19),(17,20),(18,19),(18,20),(19,7),(20,7)],21)
=> ? = 6 - 2
[4,5,6,3,1,2] => [4,6,5,3,1,2] => [6,4,1,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
=> ? = 6 - 2
[4,5,6,3,2,1] => [4,6,5,3,2,1] => [5,4,1,3,2,6] => ([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
=> ? = 6 - 2
[4,6,1,2,5,3] => [4,6,1,5,3,2] => [6,5,1,4,2,3] => ([(0,1),(0,2),(0,4),(0,5),(1,9),(1,16),(2,10),(2,16),(3,6),(3,7),(3,15),(4,9),(4,11),(4,16),(5,3),(5,10),(5,11),(5,16),(6,13),(7,13),(7,14),(9,12),(10,6),(10,15),(11,7),(11,12),(11,15),(12,14),(13,8),(14,8),(15,13),(15,14),(16,12),(16,15)],17)
=> ? = 6 - 2
[4,6,2,1,5,3] => [4,6,2,1,5,3] => [3,6,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,9),(1,18),(1,22),(2,11),(2,14),(2,16),(2,18),(3,9),(3,14),(3,15),(3,22),(4,12),(4,13),(4,16),(4,22),(5,10),(5,13),(5,15),(5,18),(5,22),(6,8),(6,10),(6,11),(6,12),(6,22),(8,20),(8,25),(9,19),(9,25),(10,20),(10,21),(10,25),(10,26),(11,17),(11,25),(11,26),(12,17),(12,20),(12,26),(13,21),(13,26),(14,19),(14,26),(15,19),(15,21),(15,25),(16,17),(16,26),(17,24),(18,19),(18,25),(18,26),(19,23),(20,23),(20,24),(21,23),(21,24),(22,20),(22,21),(22,25),(22,26),(23,7),(24,7),(25,23),(25,24),(26,23),(26,24)],27)
=> ? = 6 - 2
[4,6,3,1,5,2] => [4,6,3,1,5,2] => [6,3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,19),(2,10),(2,13),(2,19),(2,20),(3,9),(3,13),(3,18),(3,20),(4,12),(4,14),(4,18),(4,19),(4,20),(5,11),(5,14),(5,18),(5,19),(5,20),(6,8),(6,9),(6,10),(6,11),(6,12),(8,21),(8,22),(9,15),(9,21),(9,25),(10,15),(10,22),(10,25),(11,16),(11,21),(11,22),(11,25),(12,16),(12,21),(12,22),(12,25),(13,15),(13,25),(14,16),(14,17),(14,25),(15,24),(16,23),(16,24),(17,23),(18,17),(18,21),(18,25),(19,17),(19,22),(19,25),(20,17),(20,25),(21,23),(21,24),(22,23),(22,24),(23,7),(24,7),(25,23),(25,24)],26)
=> ? = 6 - 2
[4,6,3,2,5,1] => [4,6,3,2,5,1] => [4,3,1,5,2,6] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(1,17),(1,19),(2,7),(2,18),(2,22),(3,9),(3,11),(3,19),(4,10),(4,12),(4,17),(4,19),(5,11),(5,12),(5,13),(5,19),(6,2),(6,9),(6,10),(6,13),(6,17),(7,20),(7,21),(9,16),(9,18),(9,22),(10,15),(10,18),(10,22),(11,14),(11,16),(12,14),(12,15),(12,22),(13,7),(13,15),(13,16),(13,22),(14,21),(15,20),(15,21),(16,20),(16,21),(17,18),(17,22),(18,20),(19,14),(19,22),(20,8),(21,8),(22,20),(22,21)],23)
=> ? = 6 - 2
[4,6,5,1,2,3] => [4,6,5,1,3,2] => [6,5,1,3,2,4] => ([(0,1),(0,2),(0,3),(0,5),(1,11),(1,14),(2,10),(2,13),(2,14),(3,10),(3,12),(3,14),(4,7),(4,8),(4,9),(5,4),(5,11),(5,12),(5,13),(7,17),(8,17),(8,18),(9,17),(9,18),(10,15),(11,7),(11,16),(12,8),(12,15),(12,16),(13,9),(13,15),(13,16),(14,15),(14,16),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
=> ? = 6 - 2
[4,6,5,1,3,2] => [4,6,5,1,3,2] => [6,5,1,3,2,4] => ([(0,1),(0,2),(0,3),(0,5),(1,11),(1,14),(2,10),(2,13),(2,14),(3,10),(3,12),(3,14),(4,7),(4,8),(4,9),(5,4),(5,11),(5,12),(5,13),(7,17),(8,17),(8,18),(9,17),(9,18),(10,15),(11,7),(11,16),(12,8),(12,15),(12,16),(13,9),(13,15),(13,16),(14,15),(14,16),(15,18),(16,17),(16,18),(17,6),(18,6)],19)
=> ? = 6 - 2
[4,6,5,2,1,3] => [4,6,5,2,1,3] => [4,6,1,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(1,8),(1,18),(2,11),(2,12),(2,14),(2,18),(3,10),(3,13),(3,14),(3,18),(4,7),(4,9),(4,12),(4,13),(5,8),(5,9),(5,10),(5,11),(7,16),(7,23),(8,16),(8,19),(9,16),(9,17),(9,20),(9,23),(10,19),(10,20),(11,19),(11,20),(11,23),(12,15),(12,23),(13,15),(13,20),(13,23),(14,15),(14,17),(14,20),(15,22),(16,21),(16,22),(17,21),(17,22),(18,17),(18,19),(18,23),(19,21),(20,21),(20,22),(21,6),(22,6),(23,21),(23,22)],24)
=> ? = 6 - 2
[4,6,5,2,3,1] => [4,6,5,2,3,1] => [4,5,1,3,2,6] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,15),(1,18),(2,10),(2,11),(2,12),(3,8),(3,9),(3,12),(4,9),(4,10),(4,13),(5,1),(5,8),(5,11),(5,13),(6,19),(6,20),(8,14),(8,18),(9,14),(9,16),(10,16),(10,17),(11,15),(11,17),(11,18),(12,15),(12,16),(12,18),(13,6),(13,14),(13,17),(14,19),(15,20),(16,19),(16,20),(17,19),(17,20),(18,19),(18,20),(19,7),(20,7)],21)
=> ? = 6 - 2
[4,6,5,3,1,2] => [4,6,5,3,1,2] => [6,4,1,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,15),(2,10),(2,12),(2,16),(3,11),(3,13),(3,15),(3,16),(4,8),(4,10),(4,11),(4,15),(5,8),(5,9),(5,13),(5,16),(6,18),(6,19),(8,14),(8,17),(8,20),(9,17),(9,21),(10,20),(10,21),(11,14),(11,20),(12,21),(13,6),(13,14),(13,17),(14,19),(15,17),(15,20),(15,21),(16,6),(16,20),(16,21),(17,18),(17,19),(18,7),(19,7),(20,18),(20,19),(21,18)],22)
=> ? = 6 - 2
[4,6,5,3,2,1] => [4,6,5,3,2,1] => [5,4,1,3,2,6] => ([(0,1),(0,3),(0,4),(0,5),(1,6),(1,15),(2,7),(2,8),(2,13),(3,10),(3,12),(3,15),(4,2),(4,11),(4,12),(4,15),(5,6),(5,10),(5,11),(6,16),(7,17),(8,17),(8,18),(10,14),(10,16),(11,8),(11,14),(11,16),(12,7),(12,13),(12,14),(13,17),(13,18),(14,17),(14,18),(15,13),(15,16),(16,18),(17,9),(18,9)],19)
=> ? = 6 - 2
[5,1,3,6,4,2] => [5,1,6,4,3,2] => [6,5,4,1,3,2] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 6 - 2
[5,1,4,2,3,6] => [5,1,6,4,3,2] => [6,5,4,1,3,2] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 6 - 2
[5,1,6,4,3,2] => [5,1,6,4,3,2] => [6,5,4,1,3,2] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 6 - 2
[5,2,3,6,4,1] => [5,2,6,4,3,1] => [2,5,4,1,3,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,19),(1,23),(2,8),(2,16),(2,17),(3,10),(3,11),(3,17),(4,12),(4,13),(4,16),(4,17),(5,1),(5,9),(5,11),(5,12),(5,16),(6,8),(6,9),(6,10),(6,13),(8,18),(8,20),(9,14),(9,15),(9,20),(9,23),(10,14),(10,18),(11,14),(11,19),(11,23),(12,15),(12,19),(12,23),(13,15),(13,18),(13,20),(14,21),(14,22),(15,21),(15,22),(16,19),(16,20),(16,23),(17,18),(17,23),(18,21),(19,22),(20,21),(20,22),(21,7),(22,7),(23,21),(23,22)],24)
=> ? = 6 - 2
[5,2,4,1,3,6] => [5,2,6,1,4,3] => [2,6,5,1,3,4] => ([(0,3),(0,4),(0,5),(0,6),(1,11),(1,16),(1,18),(2,12),(2,15),(3,9),(3,14),(4,8),(4,10),(4,14),(5,1),(5,8),(5,13),(5,14),(6,2),(6,9),(6,10),(6,13),(8,16),(8,18),(9,15),(9,17),(10,15),(10,17),(10,18),(11,19),(11,20),(12,19),(12,20),(13,11),(13,12),(13,17),(13,18),(14,16),(14,17),(15,19),(16,20),(17,19),(17,20),(18,19),(18,20),(19,7),(20,7)],21)
=> ? = 6 - 2
[5,2,6,4,3,1] => [5,2,6,4,3,1] => [2,5,4,1,3,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,19),(1,23),(2,8),(2,16),(2,17),(3,10),(3,11),(3,17),(4,12),(4,13),(4,16),(4,17),(5,1),(5,9),(5,11),(5,12),(5,16),(6,8),(6,9),(6,10),(6,13),(8,18),(8,20),(9,14),(9,15),(9,20),(9,23),(10,14),(10,18),(11,14),(11,19),(11,23),(12,15),(12,19),(12,23),(13,15),(13,18),(13,20),(14,21),(14,22),(15,21),(15,22),(16,19),(16,20),(16,23),(17,18),(17,23),(18,21),(19,22),(20,21),(20,22),(21,7),(22,7),(23,21),(23,22)],24)
=> ? = 6 - 2
[5,3,1,6,4,2] => [5,3,1,6,4,2] => [6,2,5,1,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,21),(2,11),(2,12),(2,19),(3,9),(3,13),(3,19),(4,10),(4,13),(4,19),(5,10),(5,12),(5,14),(5,19),(6,1),(6,9),(6,11),(6,14),(6,19),(8,17),(8,20),(9,18),(9,21),(10,16),(10,21),(11,15),(11,18),(11,21),(12,15),(12,16),(13,21),(14,8),(14,15),(14,16),(14,18),(15,17),(15,20),(16,17),(16,20),(17,7),(18,17),(18,20),(19,16),(19,18),(19,21),(20,7),(21,20)],22)
=> ? = 6 - 2
[1,4,2,5,3,6,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,2,5,3,7,6] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,2,6,5,3,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,2,6,7,3,5] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,2,7,5,6,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,2,7,6,5,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,3,5,2,6,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,3,5,2,7,6] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,3,6,5,2,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,3,6,7,2,5] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,3,7,5,6,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,3,7,6,5,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,5,6,2,3,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,5,6,3,2,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,5,6,7,2,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,5,6,7,3,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,5,7,2,6,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,5,7,3,6,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,5,7,6,2,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,5,7,6,3,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,6,5,2,3,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,6,5,2,7,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,6,5,3,2,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,6,5,3,7,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,6,7,2,5,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,6,7,3,5,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,6,7,5,2,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,6,7,5,3,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,7,5,2,3,6] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,7,5,2,6,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,7,5,3,2,6] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,7,5,3,6,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,7,6,2,3,5] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,7,6,3,2,5] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,7,6,5,2,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,4,7,6,5,3,2] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,5,2,3,4,6,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(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] => [7,6,5,4,3,2,1] => ([(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] => [7,6,5,4,3,2,1] => ([(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] => [7,6,5,4,3,2,1] => ([(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] => [7,6,5,4,3,2,1] => ([(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] => [7,6,5,4,3,2,1] => ([(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] => [7,6,5,4,3,2,1] => ([(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] => [7,6,5,4,3,2,1] => ([(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] => [7,6,5,4,3,2,1] => ([(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] => [7,6,5,4,3,2,1] => ([(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] => [7,6,5,4,3,2,1] => ([(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] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,5,4,2,6,3,7] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 6 = 8 - 2
[1,5,4,2,7,6,3] => [1,7,6,5,4,3,2] => [7,6,5,4,3,2,1] => ([(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: St001603
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 32%●distinct values known / distinct values provided: 25%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 32%●distinct values known / distinct values provided: 25%
Values
[1] => [1] => [1]
=> []
=> ? = 0 - 7
[1,2] => [1,2] => [2]
=> []
=> ? = 2 - 7
[2,1] => [1,2] => [2]
=> []
=> ? = 2 - 7
[3,5,1,2,4,6] => [1,2,4,6,3,5] => [4,2]
=> [2]
=> ? = 6 - 7
[3,5,1,4,2,6] => [1,4,2,6,3,5] => [4,2]
=> [2]
=> ? = 6 - 7
[3,5,1,6,2,4] => [1,6,2,4,3,5] => [4,1,1]
=> [1,1]
=> ? = 6 - 7
[3,5,1,6,4,2] => [1,6,2,3,5,4] => [4,1,1]
=> [1,1]
=> ? = 6 - 7
[3,5,2,1,4,6] => [1,4,6,2,3,5] => [4,2]
=> [2]
=> ? = 6 - 7
[3,5,2,4,1,6] => [1,6,2,4,3,5] => [4,1,1]
=> [1,1]
=> ? = 6 - 7
[3,5,2,6,1,4] => [1,4,2,6,3,5] => [4,2]
=> [2]
=> ? = 6 - 7
[3,5,2,6,4,1] => [1,2,6,3,5,4] => [4,1,1]
=> [1,1]
=> ? = 6 - 7
[3,6,1,2,5,4] => [1,2,5,3,6,4] => [4,2]
=> [2]
=> ? = 6 - 7
[3,6,1,4,5,2] => [1,4,5,2,3,6] => [4,2]
=> [2]
=> ? = 6 - 7
[3,6,1,5,2,4] => [1,5,2,4,3,6] => [4,1,1]
=> [1,1]
=> ? = 6 - 7
[3,6,1,5,4,2] => [1,5,2,3,6,4] => [4,2]
=> [2]
=> ? = 6 - 7
[3,6,2,1,5,4] => [1,5,2,3,6,4] => [4,2]
=> [2]
=> ? = 6 - 7
[3,6,2,4,5,1] => [1,2,4,5,3,6] => [5,1]
=> [1]
=> ? = 6 - 7
[3,6,2,5,1,4] => [1,4,2,5,3,6] => [4,2]
=> [2]
=> ? = 6 - 7
[3,6,2,5,4,1] => [1,2,5,3,6,4] => [4,2]
=> [2]
=> ? = 6 - 7
[4,1,5,3,2,6] => [1,5,2,6,3,4] => [4,2]
=> [2]
=> ? = 6 - 7
[4,1,6,3,5,2] => [1,6,2,3,5,4] => [4,1,1]
=> [1,1]
=> ? = 6 - 7
[4,2,5,3,1,6] => [1,6,2,5,3,4] => [4,1,1]
=> [1,1]
=> ? = 6 - 7
[4,2,6,3,5,1] => [1,2,6,3,5,4] => [4,1,1]
=> [1,1]
=> ? = 6 - 7
[4,5,1,2,3,6] => [1,2,3,6,4,5] => [5,1]
=> [1]
=> ? = 6 - 7
[4,5,2,1,3,6] => [1,3,6,2,4,5] => [4,2]
=> [2]
=> ? = 6 - 7
[4,5,3,1,2,6] => [1,2,6,3,4,5] => [5,1]
=> [1]
=> ? = 6 - 7
[4,5,3,2,1,6] => [1,6,2,3,4,5] => [5,1]
=> [1]
=> ? = 6 - 7
[4,5,6,1,2,3] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 6 - 7
[4,5,6,1,3,2] => [1,3,2,4,5,6] => [5,1]
=> [1]
=> ? = 6 - 7
[4,5,6,2,1,3] => [1,3,2,4,5,6] => [5,1]
=> [1]
=> ? = 6 - 7
[4,5,6,2,3,1] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 6 - 7
[4,5,6,3,1,2] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 6 - 7
[4,5,6,3,2,1] => [1,2,3,4,5,6] => [6]
=> []
=> ? = 6 - 7
[4,6,1,2,5,3] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> ? = 6 - 7
[4,6,2,1,5,3] => [1,5,2,3,4,6] => [5,1]
=> [1]
=> ? = 6 - 7
[4,6,3,1,5,2] => [1,5,2,3,4,6] => [5,1]
=> [1]
=> ? = 6 - 7
[4,6,3,2,5,1] => [1,2,5,3,4,6] => [5,1]
=> [1]
=> ? = 6 - 7
[4,6,5,1,2,3] => [1,2,3,4,6,5] => [5,1]
=> [1]
=> ? = 6 - 7
[4,6,5,1,3,2] => [1,3,2,4,6,5] => [4,2]
=> [2]
=> ? = 6 - 7
[4,6,5,2,1,3] => [1,3,2,4,6,5] => [4,2]
=> [2]
=> ? = 6 - 7
[4,6,5,2,3,1] => [1,2,3,4,6,5] => [5,1]
=> [1]
=> ? = 6 - 7
[4,6,5,3,1,2] => [1,2,3,4,6,5] => [5,1]
=> [1]
=> ? = 6 - 7
[4,6,5,3,2,1] => [1,2,3,4,6,5] => [5,1]
=> [1]
=> ? = 6 - 7
[5,1,3,6,4,2] => [1,3,6,2,4,5] => [4,2]
=> [2]
=> ? = 6 - 7
[5,1,4,2,3,6] => [1,4,2,3,6,5] => [4,2]
=> [2]
=> ? = 6 - 7
[5,1,6,4,3,2] => [1,6,2,3,4,5] => [5,1]
=> [1]
=> ? = 6 - 7
[5,2,3,6,4,1] => [1,2,3,6,4,5] => [5,1]
=> [1]
=> ? = 6 - 7
[5,2,4,1,3,6] => [1,3,6,2,4,5] => [4,2]
=> [2]
=> ? = 6 - 7
[5,2,6,4,3,1] => [1,2,6,3,4,5] => [5,1]
=> [1]
=> ? = 6 - 7
[5,3,1,6,4,2] => [1,6,2,3,4,5] => [5,1]
=> [1]
=> ? = 6 - 7
[1,4,2,5,3,7,6] => [1,4,2,5,3,7,6] => [4,3]
=> [3]
=> 1 = 8 - 7
[1,4,2,6,5,3,7] => [1,4,2,6,3,7,5] => [4,3]
=> [3]
=> 1 = 8 - 7
[1,4,2,6,7,3,5] => [1,4,2,6,7,3,5] => [4,3]
=> [3]
=> 1 = 8 - 7
[1,4,3,5,2,6,7] => [1,4,2,6,7,3,5] => [4,3]
=> [3]
=> 1 = 8 - 7
[1,4,3,6,5,2,7] => [1,4,2,7,3,6,5] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,4,5,7,2,6,3] => [1,4,5,7,2,6,3] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,4,5,7,3,6,2] => [1,4,5,7,2,3,6] => [4,3]
=> [3]
=> 1 = 8 - 7
[1,4,5,7,6,2,3] => [1,4,5,7,2,3,6] => [4,3]
=> [3]
=> 1 = 8 - 7
[1,4,5,7,6,3,2] => [1,4,5,7,2,3,6] => [4,3]
=> [3]
=> 1 = 8 - 7
[1,4,6,5,2,3,7] => [1,4,6,2,3,7,5] => [4,3]
=> [3]
=> 1 = 8 - 7
[1,4,6,5,2,7,3] => [1,4,6,2,7,3,5] => [4,3]
=> [3]
=> 1 = 8 - 7
[1,4,6,5,3,2,7] => [1,4,6,2,7,3,5] => [4,3]
=> [3]
=> 1 = 8 - 7
[1,4,6,5,3,7,2] => [1,4,6,2,3,7,5] => [4,3]
=> [3]
=> 1 = 8 - 7
[1,4,6,7,2,5,3] => [1,4,6,7,2,5,3] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,4,6,7,3,5,2] => [1,4,6,7,2,3,5] => [4,3]
=> [3]
=> 1 = 8 - 7
[1,4,6,7,5,2,3] => [1,4,6,7,2,3,5] => [4,3]
=> [3]
=> 1 = 8 - 7
[1,4,6,7,5,3,2] => [1,4,6,7,2,3,5] => [4,3]
=> [3]
=> 1 = 8 - 7
[1,4,7,5,2,3,6] => [1,4,7,2,3,6,5] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,4,7,5,2,6,3] => [1,4,7,2,6,3,5] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,4,7,5,3,2,6] => [1,4,7,2,6,3,5] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,4,7,5,3,6,2] => [1,4,7,2,3,6,5] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,4,7,6,3,2,5] => [1,4,7,2,5,3,6] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,5,2,4,7,6,3] => [1,5,2,4,7,3,6] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,5,2,6,7,4,3] => [1,5,2,6,7,3,4] => [4,3]
=> [3]
=> 1 = 8 - 7
[1,5,3,2,4,7,6] => [1,5,2,4,7,3,6] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,5,3,7,6,2,4] => [1,5,2,4,3,7,6] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,5,4,2,6,3,7] => [1,5,2,6,3,7,4] => [4,3]
=> [3]
=> 1 = 8 - 7
[1,5,4,3,6,2,7] => [1,5,2,7,3,6,4] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,5,6,2,4,3,7] => [1,5,6,2,4,3,7] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,5,6,2,4,7,3] => [1,5,6,2,4,7,3] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,5,6,2,7,4,3] => [1,5,6,2,7,3,4] => [4,3]
=> [3]
=> 1 = 8 - 7
[1,5,6,3,4,2,7] => [1,5,6,2,7,3,4] => [4,3]
=> [3]
=> 1 = 8 - 7
[1,5,6,3,7,4,2] => [1,5,6,2,3,7,4] => [4,3]
=> [3]
=> 1 = 8 - 7
[1,5,7,2,4,3,6] => [1,5,7,2,4,3,6] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,5,7,2,4,6,3] => [1,5,7,2,4,6,3] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,5,7,2,6,3,4] => [1,5,7,2,6,3,4] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,5,7,3,4,2,6] => [1,5,7,2,6,3,4] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,5,7,3,6,2,4] => [1,5,7,2,4,3,6] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,6,2,7,3,5,4] => [1,6,2,7,3,5,4] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,6,3,4,2,5,7] => [1,6,2,5,7,3,4] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,6,3,7,2,5,4] => [1,6,2,5,3,7,4] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,6,4,2,5,7,3] => [1,6,2,5,7,3,4] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,6,4,3,2,5,7] => [1,6,2,5,7,3,4] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,6,5,2,4,7,3] => [1,6,2,4,7,3,5] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,6,5,3,2,4,7] => [1,6,2,4,7,3,5] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,6,7,2,3,5,4] => [1,6,7,2,3,5,4] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,6,7,2,5,4,3] => [1,6,7,2,5,3,4] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,6,7,3,2,4,5] => [1,6,7,2,4,5,3] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,6,7,3,2,5,4] => [1,6,7,2,5,3,4] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
[1,6,7,3,5,4,2] => [1,6,7,2,3,5,4] => [4,2,1]
=> [2,1]
=> 1 = 8 - 7
Description
The number of colourings of a polygon such that the multiplicities of a colour are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St000235
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000235: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 75%
Mp00223: Permutations —runsort⟶ Permutations
St000235: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 75%
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 2
[2,1] => [2,1] => [1,2] => 2
[3,5,1,2,4,6] => [4,5,3,1,2,6] => [1,2,6,3,4,5] => 6
[3,5,1,4,2,6] => [5,4,3,2,1,6] => [1,6,2,3,4,5] => 6
[3,5,1,6,2,4] => [5,6,3,4,1,2] => [1,2,3,4,5,6] => 6
[3,5,1,6,4,2] => [6,5,3,4,2,1] => [1,2,3,4,5,6] => 6
[3,5,2,1,4,6] => [4,5,3,1,2,6] => [1,2,6,3,4,5] => 6
[3,5,2,4,1,6] => [5,4,3,2,1,6] => [1,6,2,3,4,5] => 6
[3,5,2,6,1,4] => [5,6,3,4,1,2] => [1,2,3,4,5,6] => 6
[3,5,2,6,4,1] => [6,5,3,4,2,1] => [1,2,3,4,5,6] => 6
[3,6,1,2,5,4] => [4,6,3,1,5,2] => [1,5,2,3,4,6] => 6
[3,6,1,4,5,2] => [6,5,3,4,2,1] => [1,2,3,4,5,6] => 6
[3,6,1,5,2,4] => [5,6,3,4,1,2] => [1,2,3,4,5,6] => 6
[3,6,1,5,4,2] => [6,5,3,4,2,1] => [1,2,3,4,5,6] => 6
[3,6,2,1,5,4] => [4,6,3,1,5,2] => [1,5,2,3,4,6] => 6
[3,6,2,4,5,1] => [6,5,3,4,2,1] => [1,2,3,4,5,6] => 6
[3,6,2,5,1,4] => [5,6,3,4,1,2] => [1,2,3,4,5,6] => 6
[3,6,2,5,4,1] => [6,5,3,4,2,1] => [1,2,3,4,5,6] => 6
[4,1,5,3,2,6] => [5,2,4,3,1,6] => [1,6,2,4,3,5] => 6
[4,1,6,3,5,2] => [6,2,5,4,3,1] => [1,2,5,3,4,6] => 6
[4,2,5,3,1,6] => [5,4,3,2,1,6] => [1,6,2,3,4,5] => 6
[4,2,6,3,5,1] => [6,4,5,2,3,1] => [1,2,3,4,5,6] => 6
[4,5,1,2,3,6] => [5,4,3,2,1,6] => [1,6,2,3,4,5] => 6
[4,5,2,1,3,6] => [5,4,3,2,1,6] => [1,6,2,3,4,5] => 6
[4,5,3,1,2,6] => [5,4,3,2,1,6] => [1,6,2,3,4,5] => 6
[4,5,3,2,1,6] => [5,4,3,2,1,6] => [1,6,2,3,4,5] => 6
[4,5,6,1,2,3] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[4,5,6,1,3,2] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[4,5,6,2,1,3] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[4,5,6,2,3,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[4,5,6,3,1,2] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[4,5,6,3,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[4,6,1,2,5,3] => [6,5,3,4,2,1] => [1,2,3,4,5,6] => 6
[4,6,2,1,5,3] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[4,6,3,1,5,2] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[4,6,3,2,5,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[4,6,5,1,2,3] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[4,6,5,1,3,2] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[4,6,5,2,1,3] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[4,6,5,2,3,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[4,6,5,3,1,2] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[4,6,5,3,2,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[5,1,3,6,4,2] => [6,2,5,4,3,1] => [1,2,5,3,4,6] => 6
[5,1,4,2,3,6] => [5,2,4,3,1,6] => [1,6,2,4,3,5] => 6
[5,1,6,4,3,2] => [6,2,5,4,3,1] => [1,2,5,3,4,6] => 6
[5,2,3,6,4,1] => [6,5,3,4,2,1] => [1,2,3,4,5,6] => 6
[5,2,4,1,3,6] => [5,4,3,2,1,6] => [1,6,2,3,4,5] => 6
[5,2,6,4,3,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 6
[5,3,1,6,4,2] => [6,5,3,4,2,1] => [1,2,3,4,5,6] => 6
[1,4,2,5,3,6,7] => [1,5,3,4,2,6,7] => [1,5,2,6,7,3,4] => ? = 8
[1,4,2,5,3,7,6] => [1,5,3,4,2,7,6] => [1,5,2,7,3,4,6] => ? = 8
[1,4,2,6,5,3,7] => [1,6,3,5,4,2,7] => [1,6,2,7,3,5,4] => ? = 8
[1,4,2,6,7,3,5] => [1,6,3,7,5,2,4] => [1,6,2,4,3,7,5] => ? = 8
[1,4,2,7,5,6,3] => [1,7,3,6,5,4,2] => [1,7,2,3,6,4,5] => ? = 8
[1,4,2,7,6,5,3] => [1,7,3,6,5,4,2] => [1,7,2,3,6,4,5] => ? = 8
[1,4,3,5,2,6,7] => [1,5,3,4,2,6,7] => [1,5,2,6,7,3,4] => ? = 8
[1,4,3,5,2,7,6] => [1,5,3,4,2,7,6] => [1,5,2,7,3,4,6] => ? = 8
[1,4,3,6,5,2,7] => [1,6,3,5,4,2,7] => [1,6,2,7,3,5,4] => ? = 8
[1,4,3,6,7,2,5] => [1,6,3,7,5,2,4] => [1,6,2,4,3,7,5] => ? = 8
[1,4,3,7,5,6,2] => [1,7,3,6,5,4,2] => [1,7,2,3,6,4,5] => ? = 8
[1,4,3,7,6,5,2] => [1,7,3,6,5,4,2] => [1,7,2,3,6,4,5] => ? = 8
[1,4,5,6,2,3,7] => [1,6,5,4,3,2,7] => [1,6,2,7,3,4,5] => ? = 8
[1,4,5,6,3,2,7] => [1,6,5,4,3,2,7] => [1,6,2,7,3,4,5] => ? = 8
[1,4,5,6,7,2,3] => [1,7,6,4,5,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,4,5,6,7,3,2] => [1,7,6,4,5,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,4,5,7,2,6,3] => [1,7,5,6,3,4,2] => [1,7,2,3,4,5,6] => ? = 8
[1,4,5,7,3,6,2] => [1,7,5,6,3,4,2] => [1,7,2,3,4,5,6] => ? = 8
[1,4,5,7,6,2,3] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,4,5,7,6,3,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,4,6,5,2,3,7] => [1,6,5,4,3,2,7] => [1,6,2,7,3,4,5] => ? = 8
[1,4,6,5,2,7,3] => [1,7,5,4,3,6,2] => [1,7,2,3,6,4,5] => ? = 8
[1,4,6,5,3,2,7] => [1,6,5,4,3,2,7] => [1,6,2,7,3,4,5] => ? = 8
[1,4,6,5,3,7,2] => [1,7,5,4,3,6,2] => [1,7,2,3,6,4,5] => ? = 8
[1,4,6,7,2,5,3] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,4,6,7,3,5,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,4,6,7,5,2,3] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,4,6,7,5,3,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,4,7,5,2,3,6] => [1,6,7,5,4,2,3] => [1,6,7,2,3,4,5] => ? = 8
[1,4,7,5,2,6,3] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,4,7,5,3,2,6] => [1,6,7,5,4,2,3] => [1,6,7,2,3,4,5] => ? = 8
[1,4,7,5,3,6,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,4,7,6,2,3,5] => [1,6,7,5,4,2,3] => [1,6,7,2,3,4,5] => ? = 8
[1,4,7,6,3,2,5] => [1,6,7,5,4,2,3] => [1,6,7,2,3,4,5] => ? = 8
[1,4,7,6,5,2,3] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,4,7,6,5,3,2] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,2,3,4,6,7] => [1,5,3,4,2,6,7] => [1,5,2,6,7,3,4] => ? = 8
[1,5,2,3,4,7,6] => [1,5,3,4,2,7,6] => [1,5,2,7,3,4,6] => ? = 8
[1,5,2,4,6,3,7] => [1,6,3,4,5,2,7] => [1,6,2,7,3,4,5] => ? = 8
[1,5,2,4,7,6,3] => [1,7,3,4,6,5,2] => [1,7,2,3,4,6,5] => ? = 8
[1,5,2,6,7,4,3] => [1,7,3,6,5,4,2] => [1,7,2,3,6,4,5] => ? = 8
[1,5,2,7,6,3,4] => [1,7,3,6,5,4,2] => [1,7,2,3,6,4,5] => ? = 8
[1,5,3,2,4,6,7] => [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] => [1,5,2,7,3,4,6] => ? = 8
[1,5,3,4,6,2,7] => [1,6,4,3,5,2,7] => [1,6,2,7,3,5,4] => ? = 8
[1,5,3,4,7,6,2] => [1,7,4,3,6,5,2] => [1,7,2,3,6,4,5] => ? = 8
[1,5,3,6,7,4,2] => [1,7,6,4,5,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,3,7,6,2,4] => [1,7,6,5,4,3,2] => [1,7,2,3,4,5,6] => ? = 8
[1,5,4,2,6,3,7] => [1,6,4,3,5,2,7] => [1,6,2,7,3,5,4] => ? = 8
[1,5,4,2,7,6,3] => [1,7,4,3,6,5,2] => [1,7,2,3,6,4,5] => ? = 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
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000311: Graphs ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 75%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St000311: Graphs ⟶ ℤResult quality: 23% ●values known / values provided: 23%●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
[3,5,1,2,4,6] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[3,5,1,4,2,6] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(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
[3,5,2,4,1,6] => [1,3,2,5,4,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(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
[3,6,1,2,5,4] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[3,6,1,4,5,2] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[3,6,1,5,2,4] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[3,6,1,5,4,2] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[3,6,2,1,5,4] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[3,6,2,4,5,1] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[3,6,2,5,1,4] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[3,6,2,5,4,1] => [1,3,2,6,4,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,1,5,3,2,6] => [1,4,3,5,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,1,6,3,5,2] => [1,4,3,6,2,5] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,2,5,3,1,6] => [1,4,3,5,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,2,6,3,5,1] => [1,4,3,6,2,5] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,5,1,2,3,6] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,5,2,1,3,6] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,5,3,1,2,6] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,5,3,2,1,6] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,5,6,1,2,3] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,5,6,1,3,2] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,5,6,2,1,3] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,5,6,2,3,1] => [1,4,2,5,3,6] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,5,6,3,1,2] => [1,4,3,6,2,5] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,5,6,3,2,1] => [1,4,3,6,2,5] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,6,1,2,5,3] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,6,2,1,5,3] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,6,3,1,5,2] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,6,3,2,5,1] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,6,5,1,2,3] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,6,5,1,3,2] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,6,5,2,1,3] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,6,5,2,3,1] => [1,4,2,6,3,5] => [.,[[.,.],[[.,.],[.,.]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,6,5,3,1,2] => [1,4,3,5,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[4,6,5,3,2,1] => [1,4,3,5,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[5,1,3,6,4,2] => [1,5,4,6,2,3] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(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
[5,1,6,4,3,2] => [1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[5,2,3,6,4,1] => [1,5,4,6,2,3] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[5,2,4,1,3,6] => [1,5,3,4,2,6] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[5,2,6,4,3,1] => [1,5,3,6,2,4] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[5,3,1,6,4,2] => [1,5,4,6,2,3] => [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6
[1,4,2,5,3,6,7] => [1,2,4,5,3,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,2,5,3,7,6] => [1,2,4,5,3,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,2,6,5,3,7] => [1,2,4,6,3,5,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,2,6,7,3,5] => [1,2,4,6,3,5,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,2,7,5,6,3] => [1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,2,7,6,5,3] => [1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,3,5,2,6,7] => [1,2,4,5,3,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,3,5,2,7,6] => [1,2,4,5,3,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,3,6,5,2,7] => [1,2,4,6,3,5,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,3,6,7,2,5] => [1,2,4,6,3,5,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,3,7,5,6,2] => [1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,3,7,6,5,2] => [1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,5,6,2,3,7] => [1,2,4,6,3,5,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,5,6,3,2,7] => [1,2,4,6,3,5,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,5,6,7,2,3] => [1,2,4,6,3,5,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,5,6,7,3,2] => [1,2,4,6,3,5,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,5,7,2,6,3] => [1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,5,7,3,6,2] => [1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,5,7,6,2,3] => [1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,5,7,6,3,2] => [1,2,4,7,3,5,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,6,5,2,3,7] => [1,2,4,5,3,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,6,5,2,7,3] => [1,2,4,5,3,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,6,5,3,2,7] => [1,2,4,5,3,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,6,5,3,7,2] => [1,2,4,5,3,6,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,6,7,2,5,3] => [1,2,4,7,3,6,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,6,7,3,5,2] => [1,2,4,7,3,6,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,6,7,5,2,3] => [1,2,4,7,3,6,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,6,7,5,3,2] => [1,2,4,7,3,6,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,7,5,2,3,6] => [1,2,4,5,3,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,7,5,2,6,3] => [1,2,4,5,3,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,7,5,3,2,6] => [1,2,4,5,3,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,7,5,3,6,2] => [1,2,4,5,3,7,6] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,7,6,2,3,5] => [1,2,4,6,3,7,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,7,6,3,2,5] => [1,2,4,6,3,7,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,7,6,5,2,3] => [1,2,4,6,3,7,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,4,7,6,5,3,2] => [1,2,4,6,3,7,5] => [.,[.,[[.,[.,.]],[[.,.],.]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[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
[1,5,2,4,6,3,7] => [1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,5,2,4,7,6,3] => [1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,5,2,6,7,4,3] => [1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,5,2,7,6,3,4] => [1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,5,3,2,4,6,7] => [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
[1,5,3,4,6,2,7] => [1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,5,3,4,7,6,2] => [1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,5,3,6,7,4,2] => [1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,5,3,7,6,2,4] => [1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,5,4,2,6,3,7] => [1,2,5,6,3,4,7] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
[1,5,4,2,7,6,3] => [1,2,5,7,3,4,6] => [.,[.,[[.,[.,.]],[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ? = 8
Description
The number of vertices of odd degree in a graph.
Matching statistic: St000456
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 75%
Mp00011: Binary trees —to graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 23% ●values known / values provided: 23%●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
[3,5,1,2,4,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> ([(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
[3,5,1,4,2,6] => [[.,[.,.]],[[.,.],[.,.]]]
=> ([(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
[3,5,1,6,2,4] => [[.,[.,.]],[[.,.],[.,.]]]
=> ([(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
[3,5,2,4,1,6] => [[[.,[.,.]],[.,.]],[.,.]]
=> ([(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
[3,5,2,6,1,4] => [[[.,[.,.]],[.,.]],[.,.]]
=> ([(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
[3,6,1,2,5,4] => [[.,[.,.]],[.,[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> 7 = 6 + 1
[3,6,1,4,5,2] => [[.,[.,.]],[[.,[.,.]],.]]
=> ([(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
[3,6,1,5,2,4] => [[.,[.,.]],[[.,.],[.,.]]]
=> ([(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
[3,6,1,5,4,2] => [[.,[.,.]],[[[.,.],.],.]]
=> ([(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
[3,6,2,1,5,4] => [[[.,[.,.]],.],[[.,.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> 7 = 6 + 1
[3,6,2,4,5,1] => [[[.,[.,.]],[.,[.,.]]],.]
=> ([(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
[3,6,2,5,1,4] => [[[.,[.,.]],[.,.]],[.,.]]
=> ([(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
[3,6,2,5,4,1] => [[[.,[.,.]],[[.,.],.]],.]
=> ([(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
[4,1,5,3,2,6] => [[.,.],[[[.,.],.],[.,.]]]
=> ([(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
[4,1,6,3,5,2] => [[.,.],[[[.,.],[.,.]],.]]
=> ([(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
[4,2,5,3,1,6] => [[[.,.],[[.,.],.]],[.,.]]
=> ([(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
[4,2,6,3,5,1] => [[[.,.],[[.,.],[.,.]]],.]
=> ([(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
[4,5,1,2,3,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> ([(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
[4,5,2,1,3,6] => [[[.,[.,.]],.],[.,[.,.]]]
=> ([(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
[4,5,3,1,2,6] => [[[.,[.,.]],.],[.,[.,.]]]
=> ([(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
[4,5,3,2,1,6] => [[[[.,[.,.]],.],.],[.,.]]
=> ([(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
[4,5,6,1,2,3] => [[.,[.,[.,.]]],[.,[.,.]]]
=> ([(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
[4,5,6,1,3,2] => [[.,[.,[.,.]]],[[.,.],.]]
=> ([(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
[4,5,6,2,1,3] => [[[.,[.,[.,.]]],.],[.,.]]
=> ([(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
[4,5,6,2,3,1] => [[[.,[.,[.,.]]],[.,.]],.]
=> ([(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
[4,5,6,3,1,2] => [[[.,[.,[.,.]]],.],[.,.]]
=> ([(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
[4,5,6,3,2,1] => [[[[.,[.,[.,.]]],.],.],.]
=> ([(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
[4,6,1,2,5,3] => [[.,[.,.]],[.,[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> 7 = 6 + 1
[4,6,2,1,5,3] => [[[.,[.,.]],.],[[.,.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> 7 = 6 + 1
[4,6,3,1,5,2] => [[[.,[.,.]],.],[[.,.],.]]
=> ([(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
[4,6,3,2,5,1] => [[[[.,[.,.]],.],[.,.]],.]
=> ([(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
[4,6,5,1,2,3] => [[.,[[.,.],.]],[.,[.,.]]]
=> ([(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
[4,6,5,1,3,2] => [[.,[[.,.],.]],[[.,.],.]]
=> ([(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
[4,6,5,2,1,3] => [[[.,[[.,.],.]],.],[.,.]]
=> ([(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
[4,6,5,2,3,1] => [[[.,[[.,.],.]],[.,.]],.]
=> ([(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
[4,6,5,3,1,2] => [[[.,[[.,.],.]],.],[.,.]]
=> ([(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
[4,6,5,3,2,1] => [[[[.,[[.,.],.]],.],.],.]
=> ([(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
[5,1,3,6,4,2] => [[.,.],[[.,[[.,.],.]],.]]
=> ([(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
[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
[5,1,6,4,3,2] => [[.,.],[[[[.,.],.],.],.]]
=> ([(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
[5,2,3,6,4,1] => [[[.,.],[.,[[.,.],.]]],.]
=> ([(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
[5,2,4,1,3,6] => [[[.,.],[.,.]],[.,[.,.]]]
=> ([(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
[5,2,6,4,3,1] => [[[.,.],[[[.,.],.],.]],.]
=> ([(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
[5,3,1,6,4,2] => [[[.,.],.],[[[.,.],.],.]]
=> ([(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
[1,4,2,5,3,6,7] => [.,[[.,.],[[.,.],[.,[.,.]]]]]
=> ([(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
[1,4,2,5,3,7,6] => [.,[[.,.],[[.,.],[[.,.],.]]]]
=> ([(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
[1,4,2,6,5,3,7] => [.,[[.,.],[[[.,.],.],[.,.]]]]
=> ([(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
[1,4,2,6,7,3,5] => [.,[[.,.],[[.,[.,.]],[.,.]]]]
=> ([(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
[1,4,2,7,5,6,3] => [.,[[.,.],[[[.,.],[.,.]],.]]]
=> ([(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
[1,4,2,7,6,5,3] => [.,[[.,.],[[[[.,.],.],.],.]]]
=> ([(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
[1,4,3,5,2,6,7] => [.,[[[.,.],[.,.]],[.,[.,.]]]]
=> ([(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
[1,4,3,5,2,7,6] => [.,[[[.,.],[.,.]],[[.,.],.]]]
=> ([(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
[1,4,3,6,5,2,7] => [.,[[[.,.],[[.,.],.]],[.,.]]]
=> ([(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
[1,4,3,6,7,2,5] => [.,[[[.,.],[.,[.,.]]],[.,.]]]
=> ([(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
[1,4,3,7,5,6,2] => [.,[[[.,.],[[.,.],[.,.]]],.]]
=> ([(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
[1,4,3,7,6,5,2] => [.,[[[.,.],[[[.,.],.],.]],.]]
=> ([(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
[1,4,5,6,2,3,7] => [.,[[.,[.,[.,.]]],[.,[.,.]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 + 1
[1,4,5,6,3,2,7] => [.,[[[.,[.,[.,.]]],.],[.,.]]]
=> ([(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
[1,4,5,6,7,2,3] => [.,[[.,[.,[.,[.,.]]]],[.,.]]]
=> ([(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
[1,4,5,6,7,3,2] => [.,[[[.,[.,[.,[.,.]]]],.],.]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 8 + 1
[1,4,5,7,2,6,3] => [.,[[.,[.,[.,.]]],[[.,.],.]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 + 1
[1,4,5,7,3,6,2] => [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> ([(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
[1,4,5,7,6,2,3] => [.,[[.,[.,[[.,.],.]]],[.,.]]]
=> ([(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
[1,4,5,7,6,3,2] => [.,[[[.,[.,[[.,.],.]]],.],.]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 8 + 1
[1,4,6,5,2,3,7] => [.,[[.,[[.,.],.]],[.,[.,.]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 + 1
[1,4,6,5,2,7,3] => [.,[[.,[[.,.],.]],[[.,.],.]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 + 1
[1,4,6,5,3,2,7] => [.,[[[.,[[.,.],.]],.],[.,.]]]
=> ([(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
[1,4,6,5,3,7,2] => [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ([(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
[1,4,6,7,2,5,3] => [.,[[.,[.,[.,.]]],[[.,.],.]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 + 1
[1,4,6,7,3,5,2] => [.,[[[.,[.,[.,.]]],[.,.]],.]]
=> ([(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
[1,4,6,7,5,2,3] => [.,[[.,[[.,[.,.]],.]],[.,.]]]
=> ([(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
[1,4,6,7,5,3,2] => [.,[[[.,[[.,[.,.]],.]],.],.]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 8 + 1
[1,4,7,5,2,3,6] => [.,[[.,[[.,.],.]],[.,[.,.]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 + 1
[1,4,7,5,2,6,3] => [.,[[.,[[.,.],.]],[[.,.],.]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 + 1
[1,4,7,5,3,2,6] => [.,[[[.,[[.,.],.]],.],[.,.]]]
=> ([(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
[1,4,7,5,3,6,2] => [.,[[[.,[[.,.],.]],[.,.]],.]]
=> ([(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
[1,4,7,6,2,3,5] => [.,[[.,[[.,.],.]],[.,[.,.]]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 + 1
[1,4,7,6,3,2,5] => [.,[[[.,[[.,.],.]],.],[.,.]]]
=> ([(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
[1,4,7,6,5,2,3] => [.,[[.,[[[.,.],.],.]],[.,.]]]
=> ([(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
[1,4,7,6,5,3,2] => [.,[[[.,[[[.,.],.],.]],.],.]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,7),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 8 + 1
[1,5,2,3,4,6,7] => [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> ([(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
[1,5,2,4,6,3,7] => [.,[[.,.],[[.,[.,.]],[.,.]]]]
=> ([(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
[1,5,2,4,7,6,3] => [.,[[.,.],[[.,[[.,.],.]],.]]]
=> ([(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
[1,5,2,6,7,4,3] => [.,[[.,.],[[[.,[.,.]],.],.]]]
=> ([(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
[1,5,2,7,6,3,4] => [.,[[.,.],[[[.,.],.],[.,.]]]]
=> ([(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
[1,5,3,2,4,6,7] => [.,[[[.,.],.],[.,[.,[.,.]]]]]
=> ([(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
[1,5,3,4,6,2,7] => [.,[[[.,.],[.,[.,.]]],[.,.]]]
=> ([(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
[1,5,3,4,7,6,2] => [.,[[[.,.],[.,[[.,.],.]]],.]]
=> ([(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
[1,5,3,6,7,4,2] => [.,[[[.,.],[[.,[.,.]],.]],.]]
=> ([(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
[1,5,3,7,6,2,4] => [.,[[[.,.],[[.,.],.]],[.,.]]]
=> ([(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
[1,5,4,2,6,3,7] => [.,[[[.,.],.],[[.,.],[.,.]]]]
=> ([(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
[1,5,4,2,7,6,3] => [.,[[[.,.],.],[[[.,.],.],.]]]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,7),(2,3),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 8 + 1
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
The following 92 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001468The smallest fixpoint of a permutation. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St000625The sum of the minimal distances to a greater element. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001074The number of inversions of the cyclic embedding of a permutation. St000197The number of entries equal to positive one in the alternating sign matrix. St000619The number of cyclic descents of a permutation. St000653The last descent of a permutation. St000956The maximal displacement of a permutation. St000064The number of one-box pattern of a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000673The number of non-fixed points of a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000060The greater neighbor of the maximum. St000216The absolute length of a permutation. St000530The number of permutations with the same descent word as the given 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. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. 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. 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. St000219The number of occurrences of the pattern 231 in a permutation. St000890The number of nonzero entries in an alternating sign matrix. St000327The number of cover relations in a poset. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. 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. 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. 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. St000656The number of cuts of a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000189The number of elements in the 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. 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. 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 $eA$. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St001332The number of steps on the non-negative side of the walk associated with the permutation. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001645The pebbling number of a connected graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000401The size of the symmetry class of a permutation. St001875The number of simple modules with projective dimension at most 1.
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