- FindStat

Your data matches 47 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001004
(load all 14 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001004: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of indices that are either left-to-right maxima or right-to-left minima. The (bivariate) generating function for this statistic is (essentially) given in [1], the mid points of a

$321$ pattern in the permutation are those elements which are neither left-to-right maxima nor a right-to-left minima, see [[St000371]] and [[St000372]].

Matching statistic: St000203

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000203: Binary trees ⟶ ℤResult quality: 58% ●values known / values provided: 68%●distinct values known / distinct values provided: 58%

Values

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

Description

The number of external nodes of a binary tree. That is, the number of nodes that can be reached from the root by only left steps or only right steps, plus

$1$ for the root node itself. A counting formula for the number of external node in all binary trees of size

$n$ can be found in [1].

Matching statistic: St001068

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001068: Dyck paths ⟶ ℤResult quality: 58% ●values known / values provided: 68%●distinct values known / distinct values provided: 58%

Values

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

Description

Number of torsionless simple modules in the corresponding Nakayama algebra.

Matching statistic: St000053

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 58% ●values known / values provided: 68%●distinct values known / distinct values provided: 58%

Values

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

Description

The number of valleys of the Dyck path.

Matching statistic: St000998

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 58% ●values known / values provided: 68%●distinct values known / distinct values provided: 58%

Values

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

Description

Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001012

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001012: Dyck paths ⟶ ℤResult quality: 58% ●values known / values provided: 68%●distinct values known / distinct values provided: 58%

Values

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

Description

Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St001499

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 67%●distinct values known / distinct values provided: 50%

Values

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

Description

The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.

Matching statistic: St001461
(load all 4 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St001461: Permutations ⟶ ℤResult quality: 58% ●values known / values provided: 67%●distinct values known / distinct values provided: 58%

Values

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

Description

The number of topologically connected components of the chord diagram of a permutation. The chord diagram of a permutation

$\pi\in\mathfrak S_n$ is obtained by placing labels

$1,\dots,n$ in cyclic order on a cycle and drawing a (straight) arc from

$i$ to

$\pi(i)$ for every label

$i$ . This statistic records the number of topologically connected components in the chord diagram. In particular, if two arcs cross, all four labels connected by the two arcs are in the same component. The permutation

$\pi\in\mathfrak S_n$ stabilizes an interval

$I=\{a,a+1,\dots,b\}$ if

$\pi(I)=I$ . It is stabilized-interval-free, if the only interval

$\pi$ stablizes is

$\{1,\dots,n\}$ . Thus, this statistic is

$1$ if

$\pi$ is stabilized-interval-free.

Matching statistic: St001654
(load all 2 compositions to match this statistic)

Mapping

Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001654: Graphs ⟶ ℤResult quality: 58% ●values known / values provided: 65%●distinct values known / distinct values provided: 58%

Values

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

Description

The monophonic hull number of a graph. The monophonic hull of a set of vertices

$M$ of a graph

$G$ is the set of vertices that lie on at least one induced path between vertices in

$M$ . The monophonic hull number is the size of the smallest set

$M$ such that the monophonic hull of

$M$ is all of

$G$ . For example, the monophonic hull number of a graph

$G$ with

$n$ vertices is

$n$ if and only if

$G$ is a disjoint union of complete graphs.

Matching statistic: St000011
(load all 2 compositions to match this statistic)

Mapping

Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 92%

Values

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

Description

The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.

The following 37 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000439The position of the first down step of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000160The multiplicity of the smallest part of a partition. St000475The number of parts equal to 1 in a partition. St000007The number of saliances of the permutation. St000678The number of up steps after the last double rise of a Dyck path. St000025The number of initial rises of a Dyck path. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000314The number of left-to-right-maxima of a permutation. St000015The number of peaks of a Dyck path. St000542The number of left-to-right-minima of a permutation. St000991The number of right-to-left minima of a permutation. St000331The number of upper interactions of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000469The distinguishing number of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001691The number of kings in a graph. St000907The number of maximal antichains of minimal length in a poset. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000315The number of isolated vertices of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St000717The number of ordinal summands of a poset. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000942The number of critical left to right maxima of the parking functions.