- FindStat

Your data matches 16 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000651

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000651: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal size of a rise in a permutation. This is

$\max_i \sigma_{i+1}-\sigma_i$ , except for the permutations without rises, where it is

$0$ .

Matching statistic: St000381

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 82% ●values known / values provided: 96%●distinct values known / distinct values provided: 82%

Values

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

Description

The largest part of an integer composition.

Matching statistic: St000392
(load all 2 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 91% ●values known / values provided: 94%●distinct values known / distinct values provided: 91%

Values

[1,0]
 => [1] => [1] =>  => ? = 0
[1,0,1,0]
 => [2,1] => [2,1] => 1 => 1
[1,1,0,0]
 => [1,2] => [1,2] => 0 => 0
[1,0,1,0,1,0]
 => [2,3,1] => [3,1,2] => 10 => 1
[1,0,1,1,0,0]
 => [2,1,3] => [2,1,3] => 10 => 1
[1,1,0,0,1,0]
 => [1,3,2] => [1,3,2] => 01 => 1
[1,1,0,1,0,0]
 => [3,1,2] => [3,2,1] => 11 => 2
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => 00 => 0
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [4,1,2,3] => 100 => 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [3,1,2,4] => 100 => 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,1,4,3] => 101 => 1
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [4,3,1,2] => 101 => 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => 100 => 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [1,4,2,3] => 010 => 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [1,3,2,4] => 010 => 1
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [4,2,1,3] => 110 => 2
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [3,1,4,2] => 110 => 2
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [3,2,1,4] => 110 => 2
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [1,2,4,3] => 001 => 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,4,3,2] => 011 => 2
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [4,3,2,1] => 111 => 3
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => 1000 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => 1000 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [3,1,2,5,4] => 1001 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [5,4,1,2,3] => 1001 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [3,1,2,4,5] => 1000 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [2,1,5,3,4] => 1010 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => 1010 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [5,3,1,2,4] => 1010 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [4,1,2,5,3] => 1010 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [4,3,1,2,5] => 1010 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => 1001 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [2,1,5,4,3] => 1011 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [5,4,3,1,2] => 1011 => 2
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => 1000 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [1,5,2,3,4] => 0100 => 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [1,4,2,3,5] => 0100 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => 0101 => 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [1,5,4,2,3] => 0101 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => 0100 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [5,2,1,3,4] => 1100 => 2
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [4,2,1,3,5] => 1100 => 2
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [3,1,5,2,4] => 1100 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [5,2,4,1,3] => 1100 => 2
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [3,1,4,2,5] => 1100 => 2
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [3,2,1,5,4] => 1101 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [5,4,2,1,3] => 1101 => 2
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [3,1,5,4,2] => 1101 => 2
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [3,2,1,4,5] => 1100 => 2
[1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => [1,2,5,3,4] => 0010 => 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [2,3,4,5,1,7,6,8] => [5,1,2,3,4,7,6,8] => ? => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,6,1,5,7,8] => [6,5,1,2,3,4,7,8] => ? => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,3,1,5,6,4,7,8] => [3,1,2,6,4,5,7,8] => ? => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,3,1,5,4,6,7,8] => [3,1,2,5,4,6,7,8] => ? => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [2,3,5,6,7,1,8,4] => [8,4,6,1,2,3,5,7] => ? => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [2,1,4,3,6,7,5,8] => [2,1,4,3,7,5,6,8] => ? => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
 => [2,1,4,6,3,7,5,8] => [2,1,7,5,3,4,6,8] => ? => ? = 1
[1,0,1,1,0,0,1,1,0,1,1,1,0,0,0,0]
 => [2,1,4,6,3,5,7,8] => [2,1,6,5,3,4,7,8] => ? => ? = 1
[1,0,1,1,0,1,1,0,0,1,1,1,0,0,0,0]
 => [2,4,1,6,3,5,7,8] => [6,5,3,1,2,4,7,8] => ? => ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,1,0,0]
 => [2,4,6,1,3,7,5,8] => [4,1,2,7,5,3,6,8] => ? => ? = 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0,1,0]
 => [2,4,6,7,1,3,8,5] => [6,3,8,5,1,2,4,7] => ? => ? = 1
[1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [2,4,6,7,1,3,5,8] => [6,3,7,5,1,2,4,8] => ? => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,3,4,6,7,2,5,8] => ? => ? => ? = 1
[1,1,0,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,6,2,5,7,8] => [1,6,5,2,3,4,7,8] => ? => ? = 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [1,3,5,6,7,2,8,4] => [1,8,4,6,2,3,5,7] => ? => ? = 1
[1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,3,2,4,6,7,5,8] => [1,3,2,4,7,5,6,8] => ? => ? = 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => [3,4,5,1,6,7,2,8] => [7,2,4,1,3,5,6,8] => ? => ? = 2
[1,1,1,0,0,1,1,1,0,1,0,0,1,0,0,0]
 => [1,4,7,2,8,3,5,6] => [1,4,2,8,6,3,7,5] => ? => ? = 2
[1,1,1,0,0,1,1,1,0,1,1,0,0,0,0,0]
 => [1,4,7,2,3,5,6,8] => [1,4,2,7,6,5,3,8] => ? => ? = 2
[1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
 => [5,1,2,6,7,3,8,4] => [8,4,6,3,2,1,5,7] => ? => ? = 4
[1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
 => [5,1,2,6,7,8,3,4] => [7,3,2,1,5,8,4,6] => ? => ? = 4
[1,1,1,1,0,1,0,0,0,1,0,1,1,0,0,0]
 => [5,1,2,6,7,3,4,8] => [7,4,6,3,2,1,5,8] => ? => ? = 4
[1,1,1,1,0,1,0,0,1,0,0,0,1,1,0,0]
 => [5,1,6,2,3,7,4,8] => [7,4,2,1,5,3,6,8] => ? => ? = 4
[1,1,1,1,0,1,0,0,1,0,0,1,1,0,0,0]
 => [5,1,6,2,7,3,4,8] => [6,3,7,4,2,1,5,8] => ? => ? = 4
[1,1,1,1,0,1,0,0,1,0,1,0,0,1,0,0]
 => [5,1,6,7,2,8,3,4] => [5,2,1,8,4,7,3,6] => ? => ? = 4
[1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0]
 => [5,6,1,7,8,2,3,4] => [6,2,8,4,7,3,1,5] => ? => ? = 4
[1,1,1,1,0,1,0,1,0,0,1,1,0,0,0,0]
 => [5,6,1,7,2,3,4,8] => [6,3,1,5,2,7,4,8] => ? => ? = 4
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => [11,1,2,3,4,5,6,7,8,9,10] => [11,10,9,8,7,6,5,4,3,2,1] => 1111111111 => ? = 10
[1,0,1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => [2,1,4,6,3,8,5,10,7,9] => [2,1,10,9,7,5,3,4,6,8] => ? => ? = 1

Description

The length of the longest run of ones in a binary word.

Matching statistic: St000444
(load all 2 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 72%●distinct values known / distinct values provided: 64%

Values

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

Description

The length of the maximal rise of a Dyck path.

Matching statistic: St001418

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St001418: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 64%

Values

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

Description

Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. The stable Auslander algebra is by definition the stable endomorphism ring of the direct sum of all indecomposable modules.

Matching statistic: St000485
(load all 2 compositions to match this statistic)

Mapping

Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 55%

Values

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

Description

The length of the longest cycle of a permutation.

Matching statistic: St001372

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 64%

Values

[1,0]
 => [1] => [1,0]
 => 10 => 1 = 0 + 1
[1,0,1,0]
 => [2,1] => [1,1,0,0]
 => 1100 => 2 = 1 + 1
[1,1,0,0]
 => [1,2] => [1,0,1,0]
 => 1010 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [2,3,1] => [1,1,0,1,0,0]
 => 110100 => 2 = 1 + 1
[1,0,1,1,0,0]
 => [2,1,3] => [1,1,0,0,1,0]
 => 110010 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [1,3,2] => [1,0,1,1,0,0]
 => 101100 => 2 = 1 + 1
[1,1,0,1,0,0]
 => [3,1,2] => [1,1,1,0,0,0]
 => 111000 => 3 = 2 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 11010100 => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 11010010 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 11001100 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 11011000 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 11001010 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 10110100 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 10110010 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 11100100 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 11101000 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 11100010 => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 10101100 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 10111000 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 11110000 => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 10101010 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 11010101010100 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,5,6,1,7] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => 11010101010010 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,3,4,5,1,6,7] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => 11010101001010 => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,1,5,6,7] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => 11010100101010 => ? = 1 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,1,4,5,6,7] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => 11010010101010 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,1,4,5,6,7,3] => [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
 => 11001101010100 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,1,4,5,6,3,7] => [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
 => 11001101010010 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,1,4,5,3,6,7] => [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => 11001101001010 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,6,7,5] => [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
 => 11001100110100 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,6,5,7] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => 11001100110010 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [2,1,4,6,3,7,5] => [1,1,0,0,1,1,0,1,1,0,0,1,0,0]
 => 11001101100100 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,1,4,6,7,3,5] => [1,1,0,0,1,1,0,1,1,0,1,0,0,0]
 => 11001101101000 => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [2,1,4,6,3,5,7] => [1,1,0,0,1,1,0,1,1,0,0,0,1,0]
 => 11001101100010 => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,1,4,3,5,6,7] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => 11001100101010 => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [2,4,1,5,6,7,3] => [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
 => 11011001010100 => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [2,4,1,5,6,3,7] => [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
 => 11011001010010 => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,3,6,7] => [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
 => 11011001001010 => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
 => [2,4,5,1,6,7,3] => [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
 => 11011010010100 => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
 => [2,4,5,1,6,3,7] => [1,1,0,1,1,0,1,0,0,1,0,0,1,0]
 => 11011010010010 => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
 => [2,4,5,6,1,7,3] => [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
 => 11011010100100 => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,4,5,6,7,1,3] => [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
 => 11011010101000 => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,4,5,6,1,3,7] => [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
 => 11011010100010 => ? = 1 + 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,4,5,1,3,6,7] => [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
 => 11011010001010 => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => [2,4,1,3,6,7,5] => [1,1,0,1,1,0,0,0,1,1,0,1,0,0]
 => 11011000110100 => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
 => [2,4,1,3,6,5,7] => [1,1,0,1,1,0,0,0,1,1,0,0,1,0]
 => 11011000110010 => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
 => [2,4,1,6,3,7,5] => [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
 => 11011001100100 => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,4,1,6,7,3,5] => [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => 11011001101000 => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
 => [2,4,1,6,3,5,7] => [1,1,0,1,1,0,0,1,1,0,0,0,1,0]
 => 11011001100010 => ? = 1 + 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
 => [2,4,6,1,3,7,5] => [1,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => 11011011000100 => ? = 1 + 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,4,6,1,7,3,5] => [1,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => 11011011001000 => ? = 1 + 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,4,6,7,1,3,5] => [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => 11011011010000 => ? = 1 + 1
[1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,4,6,1,3,5,7] => [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
 => 11011011000010 => ? = 1 + 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,4,1,3,5,6,7] => [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
 => 11011000101010 => ? = 1 + 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,1,3,4,5,6,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => 11001010101010 => ? = 1 + 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 11111110000000 => ? = 6 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => 10101010101010 => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => 1101010101010100 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,5,6,7,1,8] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => 1101010101010010 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [2,3,4,5,6,1,8,7] => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => 1101010101001100 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,3,4,5,6,8,1,7] => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => 1101010101011000 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,3,4,5,6,1,7,8] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => 1101010101001010 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [2,3,4,5,1,7,8,6] => [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => 1101010100110100 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [2,3,4,5,1,7,6,8] => [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => 1101010100110010 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [2,3,4,5,7,1,8,6] => [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => 1101010101100100 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,3,4,5,7,8,1,6] => [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => 1101010101101000 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [2,3,4,5,7,1,6,8] => [1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
 => 1101010101100010 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [2,3,4,5,1,6,8,7] => [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => 1101010100101100 => ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1,6,7,8] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => 1101010100101010 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,3,4,1,6,7,8,5] => [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => 1101010011010100 => ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [2,3,4,1,6,7,5,8] => [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => 1101010011010010 => ? = 1 + 1

Description

The length of a longest cyclic run of ones of a binary word. Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.

Matching statistic: St000308

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 64%

Values

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

Description

The height of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices

$\{0,1,2,\ldots,n\}$ and root

$0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1]. The statistic is given by the height of this tree. See also [[St000325]] for the width of this tree.

Matching statistic: St001192
(load all 4 compositions to match this statistic)

Mapping

St001192: Dyck paths ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 55%

Values

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

Description

The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ .

Matching statistic: St001239
(load all 4 compositions to match this statistic)

Mapping

Mp00028: Dyck paths —reverse⟶ Dyck paths
St001239: Dyck paths ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 55%

Values

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

Description

The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra.

The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000451The length of the longest pattern of the form k 1 2. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001624The breadth of a lattice.