- FindStat

Your data matches 48 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000940

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000940: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2,4,3] => [3,1] => [[3,3],[2]]
 => [2]
 => 0
[1,3,4,2] => [3,1] => [[3,3],[2]]
 => [2]
 => 0
[1,4,3,2] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
[2,3,4,1] => [3,1] => [[3,3],[2]]
 => [2]
 => 0
[2,4,3,1] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
[3,4,2,1] => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
[1,2,3,5,4] => [4,1] => [[4,4],[3]]
 => [3]
 => 0
[1,2,4,3,5] => [3,2] => [[4,3],[2]]
 => [2]
 => 0
[1,2,4,5,3] => [4,1] => [[4,4],[3]]
 => [3]
 => 0
[1,2,5,3,4] => [3,2] => [[4,3],[2]]
 => [2]
 => 0
[1,2,5,4,3] => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 0
[1,3,2,5,4] => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
[1,3,4,2,5] => [3,2] => [[4,3],[2]]
 => [2]
 => 0
[1,3,4,5,2] => [4,1] => [[4,4],[3]]
 => [3]
 => 0
[1,3,5,2,4] => [3,2] => [[4,3],[2]]
 => [2]
 => 0
[1,3,5,4,2] => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 0
[1,4,2,5,3] => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
[1,4,3,2,5] => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 0
[1,4,3,5,2] => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
[1,4,5,2,3] => [3,2] => [[4,3],[2]]
 => [2]
 => 0
[1,4,5,3,2] => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 0
[1,5,2,4,3] => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
[1,5,3,2,4] => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 0
[1,5,3,4,2] => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
[1,5,4,2,3] => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 0
[1,5,4,3,2] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
[2,1,3,5,4] => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 0
[2,1,4,5,3] => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 0
[2,1,5,4,3] => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 0
[2,3,1,5,4] => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
[2,3,4,1,5] => [3,2] => [[4,3],[2]]
 => [2]
 => 0
[2,3,4,5,1] => [4,1] => [[4,4],[3]]
 => [3]
 => 0
[2,3,5,1,4] => [3,2] => [[4,3],[2]]
 => [2]
 => 0
[2,3,5,4,1] => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 0
[2,4,1,5,3] => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
[2,4,3,1,5] => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 0
[2,4,3,5,1] => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
[2,4,5,1,3] => [3,2] => [[4,3],[2]]
 => [2]
 => 0
[2,4,5,3,1] => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 0
[2,5,1,4,3] => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
[2,5,3,1,4] => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 0
[2,5,3,4,1] => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
[2,5,4,1,3] => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 0
[2,5,4,3,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
[3,1,2,5,4] => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 0
[3,1,4,5,2] => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 0
[3,1,5,4,2] => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 0
[3,2,4,5,1] => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 0
[3,2,5,4,1] => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 0
[3,4,1,5,2] => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1

Description

The number of characters of the symmetric group whose value on the partition is zero. The maximal value for any given size is recorded in [2].

Matching statistic: St000479

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00147: Graphs —square⟶ Graphs
St000479: Graphs ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 6%

Values

[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 18 = 0 + 18
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 18 = 0 + 18
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 18 = 0 + 18
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 18 = 0 + 18
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 18 = 0 + 18
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 18 = 0 + 18
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 18 = 0 + 18
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 18 = 0 + 18
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 18
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 18 = 0 + 18
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 18 = 0 + 18
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 18
[1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 18 = 0 + 18
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 18
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 18 = 0 + 18
[1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 18
[1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 18 = 0 + 18
[1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 18
[1,5,4,2,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 18 = 0 + 18
[1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 18
[2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 18 = 0 + 18
[2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 18 = 0 + 18
[2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 18
[2,4,3,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 18 = 0 + 18
[2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 18
[2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 18 = 0 + 18
[2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[2,5,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 18
[2,5,3,1,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 18 = 0 + 18
[2,5,3,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 18
[2,5,4,1,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 18 = 0 + 18
[2,5,4,3,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[3,1,2,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[3,1,5,4,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[3,2,4,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[3,2,5,4,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[3,4,1,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 18
[3,4,2,1,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 18 = 0 + 18
[3,4,2,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 18
[3,4,5,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 18 = 0 + 18
[3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[3,5,1,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 18
[3,5,2,1,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 18 = 0 + 18
[3,5,2,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 18
[3,5,4,1,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 18 = 0 + 18
[3,5,4,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[4,1,2,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[4,1,3,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[4,1,5,3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[4,2,3,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[4,2,5,3,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[4,3,5,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[4,5,1,3,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 18
[4,5,2,1,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 18 = 0 + 18
[4,5,2,3,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 18
[4,5,3,1,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 18 = 0 + 18
[4,5,3,2,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[5,1,2,4,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[5,1,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[5,1,4,3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[5,2,3,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[5,2,4,3,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[5,3,4,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 18
[1,2,4,3,5,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 18 = 0 + 18
[1,2,5,3,4,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 18 = 0 + 18
[1,2,6,3,4,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 18 = 0 + 18
[1,3,4,2,5,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 18 = 0 + 18
[1,3,5,2,4,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 18 = 0 + 18
[1,3,6,2,4,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 18 = 0 + 18
[1,4,3,2,5,6] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 18 = 0 + 18
[1,4,5,2,3,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 18 = 0 + 18
[1,4,6,2,3,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 18 = 0 + 18
[1,5,3,2,4,6] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 18 = 0 + 18
[1,5,4,2,3,6] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 18 = 0 + 18
[1,5,6,2,3,4] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 18 = 0 + 18
[1,6,3,2,4,5] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 18 = 0 + 18
[1,6,4,2,3,5] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 18 = 0 + 18
[1,6,5,2,3,4] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 18 = 0 + 18
[2,3,4,1,5,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 18 = 0 + 18
[2,3,5,1,4,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 18 = 0 + 18
[2,3,6,1,4,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 18 = 0 + 18
[2,4,3,1,5,6] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 18 = 0 + 18
[2,4,5,1,3,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 18 = 0 + 18
[2,4,6,1,3,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 18 = 0 + 18
[2,5,3,1,4,6] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 18 = 0 + 18
[2,5,4,1,3,6] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 18 = 0 + 18
[2,5,6,1,3,4] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 18 = 0 + 18

Description

The Ramsey number of a graph. This is the smallest integer

$n$ such that every two-colouring of the edges of the complete graph

$K_n$ contains a (not necessarily induced) monochromatic copy of the given graph. [1] Thus, the Ramsey number of the complete graph

$K_n$ is the ordinary Ramsey number

$R(n,n)$ . Very few of these numbers are known, in particular, it is only known that

$43\leq R(5,5)\leq 48$ . [2,3,4,5]

Matching statistic: St000282

Mapping

Mp00109: Permutations —descent word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000282: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 11%

Values

[1,2,4,3] => 001 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 0
[1,3,4,2] => 001 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 0
[1,4,3,2] => 011 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 0
[2,3,4,1] => 001 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 0
[2,4,3,1] => 011 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 0
[3,4,2,1] => 011 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 0
[1,2,3,5,4] => 0001 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 0
[1,2,4,3,5] => 0010 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[1,2,4,5,3] => 0001 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 0
[1,2,5,3,4] => 0010 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[1,2,5,4,3] => 0011 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 0
[1,3,2,5,4] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,3,4,2,5] => 0010 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[1,3,4,5,2] => 0001 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 0
[1,3,5,2,4] => 0010 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[1,3,5,4,2] => 0011 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 0
[1,4,2,5,3] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,4,3,2,5] => 0110 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[1,4,3,5,2] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,4,5,2,3] => 0010 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[1,4,5,3,2] => 0011 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 0
[1,5,2,4,3] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,5,3,2,4] => 0110 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[1,5,3,4,2] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,5,4,2,3] => 0110 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[1,5,4,3,2] => 0111 => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 0
[2,1,3,5,4] => 1001 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[2,1,4,5,3] => 1001 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[2,1,5,4,3] => 1011 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[2,3,1,5,4] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,3,4,1,5] => 0010 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[2,3,4,5,1] => 0001 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 0
[2,3,5,1,4] => 0010 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[2,3,5,4,1] => 0011 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 0
[2,4,1,5,3] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,4,3,1,5] => 0110 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[2,4,3,5,1] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,4,5,1,3] => 0010 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[2,4,5,3,1] => 0011 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 0
[2,5,1,4,3] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,5,3,1,4] => 0110 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[2,5,3,4,1] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[2,5,4,1,3] => 0110 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[2,5,4,3,1] => 0111 => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 0
[3,1,2,5,4] => 1001 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[3,1,4,5,2] => 1001 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[3,1,5,4,2] => 1011 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[3,2,4,5,1] => 1001 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[3,2,5,4,1] => 1011 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[3,4,1,5,2] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[3,4,2,1,5] => 0110 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[3,4,2,5,1] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[3,4,5,1,2] => 0010 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[3,4,5,2,1] => 0011 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 0
[3,5,1,4,2] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[3,5,2,1,4] => 0110 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[3,5,2,4,1] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[3,5,4,1,2] => 0110 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[3,5,4,2,1] => 0111 => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 0
[4,1,2,5,3] => 1001 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[4,1,3,5,2] => 1001 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[4,1,5,3,2] => 1011 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[4,2,3,5,1] => 1001 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[4,2,5,3,1] => 1011 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[4,3,5,2,1] => 1011 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0
[4,5,1,3,2] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,5,2,1,3] => 0110 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[4,5,2,3,1] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[4,5,3,1,2] => 0110 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[4,5,3,2,1] => 0111 => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 0
[5,1,2,4,3] => 1001 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[5,1,3,4,2] => 1001 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0
[1,3,2,5,4,6] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,3,2,6,4,5] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,4,2,5,3,6] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,4,2,6,3,5] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,4,3,5,2,6] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,4,3,6,2,5] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,5,2,4,3,6] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,5,2,6,3,4] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,5,3,4,2,6] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,5,3,6,2,4] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,5,4,6,2,3] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,6,2,4,3,5] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,6,2,5,3,4] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,6,3,4,2,5] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,6,3,5,2,4] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[1,6,4,5,2,3] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,1,4,3,6,5] => 10101 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,1,5,3,6,4] => 10101 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,1,5,4,6,3] => 10101 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,1,6,3,5,4] => 10101 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,1,6,4,5,3] => 10101 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,3,1,5,4,6] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,3,1,6,4,5] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,4,1,5,3,6] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,4,1,6,3,5] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,4,3,5,1,6] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,4,3,6,1,5] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1
[2,5,1,4,3,6] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 1

Description

The size of the preimage of the map 'to poset' from Ordered trees to Posets.

Matching statistic: St001964

Mapping

Mp00109: Permutations —descent word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St001964: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 6%

Values

[1,2,4,3] => 001 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 - 1
[1,3,4,2] => 001 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 - 1
[1,4,3,2] => 011 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 - 1
[2,3,4,1] => 001 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 - 1
[2,4,3,1] => 011 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 - 1
[3,4,2,1] => 011 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 - 1
[1,2,3,5,4] => 0001 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 0 - 1
[1,2,4,3,5] => 0010 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 - 1
[1,2,4,5,3] => 0001 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 0 - 1
[1,2,5,3,4] => 0010 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 - 1
[1,2,5,4,3] => 0011 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 0 - 1
[1,3,2,5,4] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,3,4,2,5] => 0010 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 - 1
[1,3,4,5,2] => 0001 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 0 - 1
[1,3,5,2,4] => 0010 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 - 1
[1,3,5,4,2] => 0011 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 0 - 1
[1,4,2,5,3] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,4,3,2,5] => 0110 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 - 1
[1,4,3,5,2] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,4,5,2,3] => 0010 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 - 1
[1,4,5,3,2] => 0011 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 0 - 1
[1,5,2,4,3] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,5,3,2,4] => 0110 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 - 1
[1,5,3,4,2] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,5,4,2,3] => 0110 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 - 1
[1,5,4,3,2] => 0111 => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 0 - 1
[2,1,3,5,4] => 1001 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 - 1
[2,1,4,5,3] => 1001 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 - 1
[2,1,5,4,3] => 1011 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 - 1
[2,3,1,5,4] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[2,3,4,1,5] => 0010 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 - 1
[2,3,4,5,1] => 0001 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 0 - 1
[2,3,5,1,4] => 0010 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 - 1
[2,3,5,4,1] => 0011 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 0 - 1
[2,4,1,5,3] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[2,4,3,1,5] => 0110 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 - 1
[2,4,3,5,1] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[2,4,5,1,3] => 0010 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 - 1
[2,4,5,3,1] => 0011 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 0 - 1
[2,5,1,4,3] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[2,5,3,1,4] => 0110 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 - 1
[2,5,3,4,1] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[2,5,4,1,3] => 0110 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 - 1
[2,5,4,3,1] => 0111 => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 0 - 1
[3,1,2,5,4] => 1001 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 - 1
[3,1,4,5,2] => 1001 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 - 1
[3,1,5,4,2] => 1011 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 - 1
[3,2,4,5,1] => 1001 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 - 1
[3,2,5,4,1] => 1011 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 - 1
[3,4,1,5,2] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[3,4,2,1,5] => 0110 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 - 1
[3,4,2,5,1] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[3,4,5,1,2] => 0010 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 - 1
[3,4,5,2,1] => 0011 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
 => ? = 0 - 1
[3,5,1,4,2] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[3,5,2,1,4] => 0110 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 - 1
[3,5,2,4,1] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[3,5,4,1,2] => 0110 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 - 1
[3,5,4,2,1] => 0111 => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
 => ? = 0 - 1
[4,1,2,5,3] => 1001 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 - 1
[4,1,3,5,2] => 1001 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 - 1
[4,1,5,3,2] => 1011 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 - 1
[4,2,3,5,1] => 1001 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 0 - 1
[4,2,5,3,1] => 1011 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 - 1
[4,5,1,3,2] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[4,5,2,3,1] => 0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0 = 1 - 1
[1,3,2,5,4,6] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,3,2,6,4,5] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,4,2,5,3,6] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,4,2,6,3,5] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,4,3,5,2,6] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,4,3,6,2,5] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,5,2,4,3,6] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,5,2,6,3,4] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,5,3,4,2,6] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,5,3,6,2,4] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,5,4,6,2,3] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,6,2,4,3,5] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,6,2,5,3,4] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,6,3,4,2,5] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,6,3,5,2,4] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[1,6,4,5,2,3] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[2,1,4,3,6,5] => 10101 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[2,1,5,3,6,4] => 10101 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[2,1,5,4,6,3] => 10101 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[2,1,6,3,5,4] => 10101 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[2,1,6,4,5,3] => 10101 => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[2,3,1,5,4,6] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[2,3,1,6,4,5] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[2,4,1,5,3,6] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[2,4,1,6,3,5] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[2,4,3,5,1,6] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[2,4,3,6,1,5] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[2,5,1,4,3,6] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[2,5,1,6,3,4] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[2,5,3,4,1,6] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[2,5,3,6,1,4] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[2,5,4,6,1,3] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[2,6,1,4,3,5] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1
[2,6,1,5,3,4] => 01010 => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0 = 1 - 1

Description

The interval resolution global dimension of a poset. This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.

Matching statistic: St001645
(load all 2 compositions to match this statistic)

Mapping

Mp00069: Permutations —complement⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 6%

Values

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

Description

The pebbling number of a connected graph.

Matching statistic: St001487
(load all 6 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 11%

Values

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

Description

The number of inner corners of a skew partition.

Matching statistic: St000689

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St000689: Dyck paths ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 11%

Values

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

Description

The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. The correspondence between LNakayama algebras and Dyck paths is explained in [[St000684]]. A module

$M$ is

$n$ -rigid, if

$\operatorname{Ext}^i(M,M)=0$ for

$1\leq i\leq n$ . This statistic gives the maximal

$n$ such that the minimal generator-cogenerator module

$A \oplus D(A)$ of the LNakayama algebra

$A$ corresponding to a Dyck path is

$n$ -rigid. An application is to check for maximal

$n$ -orthogonal objects in the module category in the sense of [2].

Matching statistic: St001001
(load all 3 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 11%

Values

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

Description

The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001556

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001556: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 11%

Values

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

Description

The number of inversions of the third entry of a permutation. This is, for a permutation

$\pi$ of length

$n$ ,

$\# \{3 < k \leq n \mid \pi(3) > \pi(k)\}.$ The number of inversions of the first entry is [[St000054]] and the number of inversions of the second entry is [[St001557]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.

Matching statistic: St000872

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000872: Permutations ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 11%

Values

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

Description

The number of very big descents of a permutation. A very big descent of a permutation

$\pi$ is an index

$i$ such that

$\pi_i - \pi_{i+1} > 2$ . For the number of descents, see [[St000021]] and for the number of big descents, see [[St000647]]. General

$r$ -descents were for example be studied in [1, Section 2].

The following 38 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001960The number of descents of a permutation minus one if its first entry is not one. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000354The number of recoils of a permutation. 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

$eA$ . St001489The maximum of the number of descents and the number of inverse descents. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001330The hat guessing number of a graph. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001618The cardinality of the Frattini sublattice of a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001619The number of non-isomorphic sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001833The number of linear intervals in a lattice. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001620The number of sublattices of a lattice. St001679The number of subsets of a lattice whose meet is the bottom element. St001568The smallest positive integer that does not appear twice in the partition. St000068The number of minimal elements in a poset. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001769The reflection length of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St001545The second Elser number of a connected graph. St001862The number of crossings of a signed permutation.