- FindStat

Your data matches 27 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000884
(load all 41 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of isolated descents of a permutation. A descent

$i$ is isolated if neither

$i+1$ nor

$i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].

Matching statistic: St001657

Mapping

Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 83% ●values known / values provided: 94%●distinct values known / distinct values provided: 83%

Values

[1,0]
 => {{1}}
 => [1]
 => 0
[1,0,1,0]
 => {{1},{2}}
 => [1,1]
 => 0
[1,1,0,0]
 => {{1,2}}
 => [2]
 => 1
[1,0,1,0,1,0]
 => {{1},{2},{3}}
 => [1,1,1]
 => 0
[1,0,1,1,0,0]
 => {{1},{2,3}}
 => [2,1]
 => 1
[1,1,0,0,1,0]
 => {{1,2},{3}}
 => [2,1]
 => 1
[1,1,0,1,0,0]
 => {{1,3},{2}}
 => [2,1]
 => 1
[1,1,1,0,0,0]
 => {{1,2,3}}
 => [3]
 => 0
[1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => [1,1,1,1]
 => 0
[1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => [2,1,1]
 => 1
[1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => [2,1,1]
 => 1
[1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => [2,1,1]
 => 1
[1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => [3,1]
 => 0
[1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => [2,1,1]
 => 1
[1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => [2,2]
 => 2
[1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => [2,1,1]
 => 1
[1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => [2,1,1]
 => 1
[1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => [3,1]
 => 0
[1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => [3,1]
 => 0
[1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => [2,2]
 => 2
[1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => [3,1]
 => 0
[1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => [4]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => [3,1,1]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => [2,2,1]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => {{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => {{1},{2,4,5},{3}}
 => [3,1,1]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => [3,1,1]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => [2,2,1]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => {{1},{2,3,5},{4}}
 => [3,1,1]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => [4,1]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => [2,2,1]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => [2,2,1]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => [2,2,1]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => [3,2]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => {{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => [2,2,1]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => {{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => {{1,4,5},{2},{3}}
 => [3,1,1]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => {{1,3,4},{2},{5}}
 => [3,1,1]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => {{1,5},{2},{3,4}}
 => [2,2,1]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => {{1,3,5},{2},{4}}
 => [3,1,1]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => {{1,3,4,5},{2}}
 => [4,1]
 => 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3},{4},{5},{6,7},{8}}
 => ?
 => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4,5},{6},{7},{8}}
 => ?
 => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => {{1},{2},{3},{4,5},{6,7},{8}}
 => ?
 => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => {{1},{2},{3,4},{5,6},{7},{8}}
 => ?
 => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => {{1},{2,3},{4},{5},{6},{7},{8}}
 => ?
 => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4},{5},{6,7},{8}}
 => ?
 => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => {{1},{2,3},{4,5},{6},{7},{8}}
 => ?
 => ? = 2
[1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => {{1},{2,4},{3},{5,7},{6},{8}}
 => ?
 => ? = 2
[1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5,6,7},{8}}
 => ?
 => ? = 0
[1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
 => {{1},{2,7,8},{3,4,5,6}}
 => ?
 => ? = 0
[1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => {{1,4,7,8},{2},{3},{5},{6}}
 => ?
 => ? = 0
[1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => {{1,3,4,5,7,8},{2},{6}}
 => ?
 => ? = 0
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => {{1,2,3},{4},{5},{6,7,8}}
 => ?
 => ? = 0
[1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
 => {{1,2,3},{4,7,8},{5,6}}
 => ?
 => ? = 1
[1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
 => {{1,4,5},{2,3},{6,7,8}}
 => ?
 => ? = 1
[1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
 => {{1,7,8},{2,3},{4,5,6}}
 => ?
 => ? = 1
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
 => {{1,2,7,8},{3,4},{5,6}}
 => ?
 => ? = 2
[1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
 => {{1,6,7},{2,3,4,5},{8}}
 => ?
 => ? = 0
[1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
 => {{1,8},{2,7},{3,4,5,6}}
 => ?
 => ? = 2
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => {{1,2,3,4,7,8},{5,6}}
 => ?
 => ? = 1
[]
 => {}
 => ?
 => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => {{1,2},{3,4},{5,6},{7,8},{9,10}}
 => ?
 => ? = 5
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => {{1,2},{3,4,5,6},{7,8,9,10}}
 => ?
 => ? = 1
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => {{1,2},{3,4,5,6,7,8},{9,10}}
 => ?
 => ? = 2
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => {{1,2,3,4},{5,6},{7,8},{9,10}}
 => ?
 => ? = 3
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => {{1,2,3,4},{5,6},{7,8,9,10}}
 => ?
 => ? = 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => {{1,2,3,4},{5,6,7,8},{9,10}}
 => ?
 => ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
 => {{1,2,3,4,5,6,7,8,9,10},{11,12}}
 => ?
 => ? = 1

Description

The number of twos in an integer partition. The total number of twos in all partitions of

$n$ is equal to the total number of singletons [[St001484]] in all partitions of

$n-1$ , see [1].

Matching statistic: St000658
(load all 37 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000658: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 84%●distinct values known / distinct values provided: 83%

Values

[1,0]
 => [1] => [.,.]
 => [1,0]
 => ? = 0
[1,0,1,0]
 => [2,1] => [[.,.],.]
 => [1,0,1,0]
 => 0
[1,1,0,0]
 => [1,2] => [.,[.,.]]
 => [1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => [3,2,1] => [[[.,.],.],.]
 => [1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0]
 => [2,3,1] => [[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => [3,1,2] => [[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => [2,1,3] => [[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
 => [1,0,1,1,1,0,0,0,1,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [[[.,.],[.,[.,.]]],.]
 => [1,0,1,1,1,0,0,0,1,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,0,0,1,0,1,0]
 => 0
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [7,5,6,3,4,8,1,2] => ?
 => ?
 => ? = 4
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,4,1,2,3] => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [6,7,8,5,4,1,2,3] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 0
[1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
 => [6,7,4,5,8,1,2,3] => [[[.,[.,[.,.]]],[.,.]],[.,[.,.]]]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 1
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,8,1,2,3] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
 => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [7,8,5,3,4,1,2,6] => ?
 => ?
 => ? = 4
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [7,5,6,3,4,1,2,8] => ?
 => ?
 => ? = 4
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [7,5,3,4,6,1,2,8] => ?
 => ?
 => ? = 4
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,1,2,3,4] => [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [7,8,5,6,1,2,3,4] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,1,2,3,4] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 0
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => [7,6,5,4,1,2,3,8] => [[[[[.,[.,[.,.]]],.],.],.],[.,.]]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [7,5,6,3,1,2,4,8] => ?
 => ?
 => ? = 4
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [7,5,3,4,1,2,6,8] => ?
 => ?
 => ? = 4
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [8,7,6,1,2,3,4,5] => [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 0
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [6,7,8,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 0
[1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
 => [8,5,6,1,2,3,4,7] => [[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [8,7,1,2,3,4,5,6] => [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [7,8,1,2,3,4,5,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
 => [7,5,1,2,3,4,6,8] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [8,4,1,2,3,5,6,7] => [[[.,[.,[.,.]]],[.,[.,[.,.]]]],.]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [7,1,2,3,4,5,6,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [5,1,2,3,4,6,7,8] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 0
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [10,1,2,3,4,5,6,7,8,9] => [[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 0
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [9,2,1,3,4,5,6,7,8] => [[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]],.]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,1] => [[[.,.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 0
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [11,1,2,3,4,5,6,7,8,9,10] => [[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 0
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [10,2,1,3,4,5,6,7,8,9] => [[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
 => ?
 => ? = 0
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [9,3,1,2,4,5,6,7,8] => [[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]],.]
 => ?
 => ? = 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [9,2,3,1,4,5,6,7,8] => [[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]],.]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,1] => [[[.,.],[.,.]],[.,[.,[.,[.,[.,.]]]]]]
 => ?
 => ? = 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,1] => [[[.,.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,10,1] => [[[.,.],.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,11,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
 => ?
 => ? = 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => [9,8,1,2,3,4,5,6,7] => [[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
 => [9,10,1,2,3,4,5,6,7,8] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],[.,.]]
 => ?
 => ? = 1
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [3,4,5,6,7,8,9,10,1,2] => [[.,[.,.]],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ?
 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
 => [10,9,1,2,3,4,5,6,7,8] => [[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.],.]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 0
[]
 => [] => .
 => ?
 => ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [8,7,6,5,4,3,2,1,9] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [8,7,6,5,4,3,1,2,9] => [[[[[[[.,[.,.]],.],.],.],.],.],[.,.]]
 => ?
 => ? = 2
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [8,7,6,5,4,2,1,3,9] => [[[[[[[.,.],[.,.]],.],.],.],.],[.,.]]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [9,8,7,6,5,4,3,2,1] => [[[[[[[[[.,.],.],.],.],.],.],.],.],.]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0

Description

The number of rises of length 2 of a Dyck path. This is also the number of

$(1,1)$ steps of the associated Łukasiewicz path, see [1]. A related statistic is the number of double rises in a Dyck path, [[St000024]].

Matching statistic: St001139
(load all 3 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001139: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 84%●distinct values known / distinct values provided: 83%

Values

[1,0]
 => [1] => [.,.]
 => [1,0]
 => ? = 0
[1,0,1,0]
 => [2,1] => [[.,.],.]
 => [1,0,1,0]
 => 0
[1,1,0,0]
 => [1,2] => [.,[.,.]]
 => [1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => [3,2,1] => [[[.,.],.],.]
 => [1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0]
 => [2,3,1] => [[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => [3,1,2] => [[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => [2,1,3] => [[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
 => [1,0,1,1,1,0,0,0,1,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
 => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [[[.,.],[.,[.,.]]],.]
 => [1,0,1,1,1,0,0,0,1,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,0,0,1,0,1,0]
 => 0
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
 => [7,5,6,3,4,8,1,2] => ?
 => ?
 => ? = 4
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,4,1,2,3] => [[[[[[.,[.,[.,.]]],.],.],.],.],.]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => [6,7,8,5,4,1,2,3] => [[[[.,[.,[.,.]]],.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 0
[1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
 => [6,7,4,5,8,1,2,3] => [[[.,[.,[.,.]]],[.,.]],[.,[.,.]]]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 1
[1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => [4,5,6,7,8,1,2,3] => [[.,[.,[.,.]]],[.,[.,[.,[.,.]]]]]
 => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
 => [7,8,5,3,4,1,2,6] => ?
 => ?
 => ? = 4
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
 => [7,5,6,3,4,1,2,8] => ?
 => ?
 => ? = 4
[1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
 => [7,5,3,4,6,1,2,8] => ?
 => ?
 => ? = 4
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [8,7,6,5,1,2,3,4] => [[[[[.,[.,[.,[.,.]]]],.],.],.],.]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 0
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => [7,8,5,6,1,2,3,4] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => [5,6,7,8,1,2,3,4] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 0
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => [7,6,5,4,1,2,3,8] => [[[[[.,[.,[.,.]]],.],.],.],[.,.]]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,1,1,1,0,0,1,0,0,1,1,0,0,1,0,0]
 => [7,5,6,3,1,2,4,8] => ?
 => ?
 => ? = 4
[1,1,1,1,0,0,1,1,0,0,1,0,0,1,0,0]
 => [7,5,3,4,1,2,6,8] => ?
 => ?
 => ? = 4
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [8,7,6,1,2,3,4,5] => [[[[.,[.,[.,[.,[.,.]]]]],.],.],.]
 => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 0
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => [6,7,8,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],[.,[.,.]]]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 0
[1,1,1,1,1,0,0,0,0,1,1,0,0,0,1,0]
 => [8,5,6,1,2,3,4,7] => [[[.,[.,[.,[.,.]]]],[.,[.,.]]],.]
 => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => [8,7,1,2,3,4,5,6] => [[[.,[.,[.,[.,[.,[.,.]]]]]],.],.]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 0
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => [7,8,1,2,3,4,5,6] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 1
[1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
 => [7,5,1,2,3,4,6,8] => [[[.,[.,[.,[.,.]]]],[.,.]],[.,.]]
 => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => [8,4,1,2,3,5,6,7] => [[[.,[.,[.,.]]],[.,[.,[.,.]]]],.]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [8,1,2,3,4,5,6,7] => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => [7,1,2,3,4,5,6,8] => [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 1
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
 => [5,1,2,3,4,6,7,8] => [[.,[.,[.,[.,.]]]],[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7,8] => [.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [9,1,2,3,4,5,6,7,8] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 0
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [10,1,2,3,4,5,6,7,8,9] => [[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 0
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => [9,2,1,3,4,5,6,7,8] => [[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]],.]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,1] => [[[.,.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 0
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => [11,1,2,3,4,5,6,7,8,9,10] => [[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 0
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => [10,2,1,3,4,5,6,7,8,9] => [[[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
 => ?
 => ? = 0
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => [9,3,1,2,4,5,6,7,8] => [[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]],.]
 => ?
 => ? = 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => [9,2,3,1,4,5,6,7,8] => [[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]],.]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 0
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,1] => [[[.,.],[.,.]],[.,[.,[.,[.,[.,.]]]]]]
 => ?
 => ? = 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,1] => [[[.,.],.],[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,10,1] => [[[.,.],.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
 => [2,3,4,5,6,7,8,9,10,11,1] => [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
 => ?
 => ? = 0
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => [9,8,1,2,3,4,5,6,7] => [[[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.],.]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
 => [9,10,1,2,3,4,5,6,7,8] => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],[.,.]]
 => ?
 => ? = 1
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [3,4,5,6,7,8,9,10,1,2] => [[.,[.,.]],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => ?
 => ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
 => [10,9,1,2,3,4,5,6,7,8] => [[[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.],.]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
 => ? = 0
[]
 => [] => .
 => ?
 => ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [8,7,6,5,4,3,2,1,9] => [[[[[[[[.,.],.],.],.],.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [8,7,6,5,4,3,1,2,9] => [[[[[[[.,[.,.]],.],.],.],.],.],[.,.]]
 => ?
 => ? = 2
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [8,7,6,5,4,2,1,3,9] => [[[[[[[.,.],[.,.]],.],.],.],.],[.,.]]
 => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 2
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7,8,9] => [.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [9,8,7,6,5,4,3,2,1] => [[[[[[[[[.,.],.],.],.],.],.],.],.],.]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0

Description

The number of occurrences of hills of size 2 in a Dyck path. A hill of size two is a subpath beginning at height zero, consisting of two up steps followed by two down steps.

Matching statistic: St001465
(load all 5 compositions to match this statistic)

Mapping

Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 77%●distinct values known / distinct values provided: 67%

Values

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

Description

The number of adjacent transpositions in the cycle decomposition of a permutation.

Matching statistic: St001125

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001125: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 69%●distinct values known / distinct values provided: 67%

Values

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

Description

The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra.

Matching statistic: St001276

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001276: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 69%●distinct values known / distinct values provided: 67%

Values

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

Description

The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. Generalising the notion of k-regular modules from simple to arbitrary indecomposable modules, we call an indecomposable module

$M$ over an algebra

$A$ k-regular in case it has projective dimension k and

$Ext_A^i(M,A)=0$ for

$i \neq k$ and

$Ext_A^k(M,A)$ is 1-dimensional. The number of Dyck paths where the statistic returns 0 might be given by [[OEIS:A035929]] .

Matching statistic: St000502

Mapping

Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 67% ●values known / values provided: 69%●distinct values known / distinct values provided: 67%

Values

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

Description

The number of successions of a set partitions. This is the number of indices

$i$ such that

$i$ and

$i+1$ belonging to the same block.

Matching statistic: St001479

Mapping

Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001479: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 69%●distinct values known / distinct values provided: 67%

Values

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

Description

The number of bridges of a graph. A bridge is an edge whose removal increases the number of connected components of the graph.

Matching statistic: St001466
(load all 4 compositions to match this statistic)

Mapping

Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001466: Permutations ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 67%

Values

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

Description

The number of transpositions swapping cyclically adjacent numbers in a permutation. Put differently, this is the number of adjacent two-cycles in the chord diagram of a permutation.

The following 17 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001061The number of indices that are both descents and recoils of a permutation. St000214The number of adjacencies of a permutation. St000237The number of small exceedances. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St000542The number of left-to-right-minima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001461The number of topologically connected components of the chord diagram of a permutation. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001274The number of indecomposable injective modules with projective dimension equal to two. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000640The rank of the largest boolean interval in a poset. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.