- FindStat

Your data matches 15 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000442
(load all 4 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal area to the right of an up step of a Dyck path.

Matching statistic: St000013
(load all 4 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 88% ●values known / values provided: 88%●distinct values known / distinct values provided: 100%

Values

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

Description

The height of a Dyck path. The height of a Dyck path

$D$ of semilength

$n$ is defined as the maximal height of a peak of

$D$ . The height of

$D$ at position

$i$ is the number of up-steps minus the number of down-steps before position

$i$ .

Matching statistic: St001039

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal height of a column in the parallelogram polyomino associated with a Dyck path.

Matching statistic: St000730
(load all 2 compositions to match this statistic)

Mapping

St000730: Set partitions ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal arc length of a set partition. The arcs of a set partition are those

$i < j$ that are consecutive elements in the blocks. If there are no arcs, the maximal arc length is

$0$ .

Matching statistic: St001203

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001203: Dyck paths ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 100%

Values

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

Description

We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: In the list

$L$ delete the first entry

$c_0$ and substract from all other entries

$n-1$ and then append the last element 1 (this was suggested by Christian Stump). The result is a Kupisch series of an LNakayama algebra. Example: [5,6,6,6,6] goes into [2,2,2,2,1]. Now associate to the CNakayama algebra with the above properties the Dyck path corresponding to the Kupisch series of the LNakayama algebra. The statistic return the global dimension of the CNakayama algebra divided by 2.

Matching statistic: St000306

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 100%

Values

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

Description

The bounce count of a Dyck path. For a Dyck path

$D$ of length

$2n$ , this is the number of points

$(i,i)$ for

$1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of

$D$ .

Matching statistic: St000662

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 100%

Values

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

Description

The staircase size of the code of a permutation. The code

$c(\pi)$ of a permutation

$\pi$ of length

$n$ is given by the sequence

$(c_1,\ldots,c_{n})$ with

$c_i = |\{j > i : \pi(j) < \pi(i)\}|$ . This is a bijection between permutations and all sequences

$(c_1,\ldots,c_n)$ with

$0 \leq c_i \leq n-i$ . The staircase size of the code is the maximal

$k$ such that there exists a subsequence

$(c_{i_k},\ldots,c_{i_1})$ of

$c(\pi)$ with

$c_{i_j} \geq j$ . This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.

Matching statistic: St000141
(load all 5 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximum drop size of a permutation. The maximum drop size of a permutation

$\pi$ of

$[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of

$i-\pi(i)$ .

Matching statistic: St001046

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001046: Perfect matchings ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 100%

Values

{{1,2}}
 => [2,1] => [1,1,0,0]
 => [(1,4),(2,3)]
 => 1
{{1},{2}}
 => [1,2] => [1,0,1,0]
 => [(1,2),(3,4)]
 => 0
{{1,2,3}}
 => [2,3,1] => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => 1
{{1,2},{3}}
 => [2,1,3] => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 1
{{1,3},{2}}
 => [3,2,1] => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => 2
{{1},{2,3}}
 => [1,3,2] => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1
{{1},{2},{3}}
 => [1,2,3] => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 0
{{1,2,3,4}}
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => 1
{{1,2,3},{4}}
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => 1
{{1,2,4},{3}}
 => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => 2
{{1,2},{3,4}}
 => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => 1
{{1,3,4},{2}}
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => 2
{{1,3},{2,4}}
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => 2
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => 2
{{1,4},{2,3}}
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => 3
{{1},{2,3,4}}
 => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => 3
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => 2
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => 0
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,10)]
 => 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8)]
 => 2
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [(1,6),(2,3),(4,5),(7,10),(8,9)]
 => 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10)]
 => 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => [(1,10),(2,3),(4,7),(5,6),(8,9)]
 => 2
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => [(1,10),(2,3),(4,9),(5,6),(7,8)]
 => 2
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => [(1,8),(2,3),(4,7),(5,6),(9,10)]
 => 2
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => [(1,10),(2,3),(4,9),(5,8),(6,7)]
 => 3
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [(1,4),(2,3),(5,10),(6,7),(8,9)]
 => 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => [(1,10),(2,3),(4,9),(5,8),(6,7)]
 => 3
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => 2
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10)]
 => 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => 2
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => 3
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [(1,8),(2,5),(3,4),(6,7),(9,10)]
 => 2
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8)]
 => 2
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => 2
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => [(1,8),(2,7),(3,4),(5,6),(9,10)]
 => 2
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => [(1,10),(2,5),(3,4),(6,9),(7,8)]
 => 2
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => 3
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9)]
 => 2
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10)]
 => 2
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => 3
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [(1,10),(2,9),(3,6),(4,5),(7,8)]
 => 3
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => 3
{{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
 => ? = 1
{{1,2,3,4,5,6},{7}}
 => [2,3,4,5,6,1,7] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11),(13,14)]
 => ? = 1
{{1,2,3,4,5,7},{6}}
 => [2,3,4,5,7,6,1] => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,9),(10,13),(11,12)]
 => ? = 2
{{1,2,3,4,5},{6,7}}
 => [2,3,4,5,1,7,6] => [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9),(11,14),(12,13)]
 => ? = 1
{{1,2,3,4,5},{6},{7}}
 => [2,3,4,5,1,6,7] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9),(11,12),(13,14)]
 => ? = 1
{{1,2,3,4,6,7},{5}}
 => [2,3,4,6,5,7,1] => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,11),(9,10),(12,13)]
 => ? = 2
{{1,2,3,4,6},{5,7}}
 => [2,3,4,6,7,1,5] => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,13),(9,10),(11,12)]
 => ? = 2
{{1,2,3,4,6},{5},{7}}
 => [2,3,4,6,5,1,7] => [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
 => [(1,12),(2,3),(4,5),(6,7),(8,11),(9,10),(13,14)]
 => ? = 2
{{1,2,3,4,7},{5,6}}
 => [2,3,4,7,6,5,1] => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,13),(9,12),(10,11)]
 => ? = 3
{{1,2,3,4},{5,6,7}}
 => [2,3,4,1,6,7,5] => [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,14),(10,11),(12,13)]
 => ? = 1
{{1,2,3,4},{5,6},{7}}
 => [2,3,4,1,6,5,7] => [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,12),(10,11),(13,14)]
 => ? = 1
{{1,2,3,4,7},{5},{6}}
 => [2,3,4,7,5,6,1] => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,13),(9,12),(10,11)]
 => ? = 3
{{1,2,3,4},{5,7},{6}}
 => [2,3,4,1,7,6,5] => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,14),(10,13),(11,12)]
 => ? = 2
{{1,2,3,4},{5},{6,7}}
 => [2,3,4,1,5,7,6] => [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,10),(11,14),(12,13)]
 => ? = 1
{{1,2,3,4},{5},{6},{7}}
 => [2,3,4,1,5,6,7] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,10),(11,12),(13,14)]
 => ? = 1
{{1,2,3,5,6,7},{4}}
 => [2,3,5,4,6,7,1] => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,9),(7,8),(10,11),(12,13)]
 => ? = 2
{{1,2,3,5,6},{4,7}}
 => [2,3,5,7,6,1,4] => [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,8),(9,12),(10,11)]
 => ? = 3
{{1,2,3,5,6},{4},{7}}
 => [2,3,5,4,6,1,7] => [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
 => [(1,12),(2,3),(4,5),(6,9),(7,8),(10,11),(13,14)]
 => ? = 2
{{1,2,3,5,7},{4,6}}
 => [2,3,5,6,7,4,1] => [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,8),(9,10),(11,12)]
 => ? = 2
{{1,2,3,5},{4,6,7}}
 => [2,3,5,6,1,7,4] => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,11),(7,8),(9,10),(12,13)]
 => ? = 2
{{1,2,3,5},{4,6},{7}}
 => [2,3,5,6,1,4,7] => [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
 => [(1,12),(2,3),(4,5),(6,11),(7,8),(9,10),(13,14)]
 => ? = 2
{{1,2,3,5,7},{4},{6}}
 => [2,3,5,4,7,6,1] => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [(1,14),(2,3),(4,5),(6,9),(7,8),(10,13),(11,12)]
 => ? = 2
{{1,2,3,5},{4,7},{6}}
 => [2,3,5,7,1,6,4] => [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,8),(9,12),(10,11)]
 => ? = 3
{{1,2,3,5},{4},{6,7}}
 => [2,3,5,4,1,7,6] => [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8),(11,14),(12,13)]
 => ? = 2
{{1,2,3,5},{4},{6},{7}}
 => [2,3,5,4,1,6,7] => [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8),(11,12),(13,14)]
 => ? = 2
{{1,2,3,6,7},{4,5}}
 => [2,3,6,5,4,7,1] => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,11),(7,10),(8,9),(12,13)]
 => ? = 3
{{1,2,3,6},{4,5,7}}
 => [2,3,6,5,7,1,4] => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,10),(8,9),(11,12)]
 => ? = 3
{{1,2,3,6},{4,5},{7}}
 => [2,3,6,5,4,1,7] => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [(1,12),(2,3),(4,5),(6,11),(7,10),(8,9),(13,14)]
 => ? = 3
{{1,2,3,7},{4,5,6}}
 => [2,3,7,5,6,4,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,12),(8,11),(9,10)]
 => ? = 4
{{1,2,3},{4,5,6,7}}
 => [2,3,1,5,6,7,4] => [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [(1,6),(2,3),(4,5),(7,14),(8,9),(10,11),(12,13)]
 => ? = 1
{{1,2,3},{4,5,6},{7}}
 => [2,3,1,5,6,4,7] => [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,12),(8,9),(10,11),(13,14)]
 => ? = 1
{{1,2,3,7},{4,5},{6}}
 => [2,3,7,5,4,6,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,12),(8,11),(9,10)]
 => ? = 4
{{1,2,3},{4,5,7},{6}}
 => [2,3,1,5,7,6,4] => [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => [(1,6),(2,3),(4,5),(7,14),(8,9),(10,13),(11,12)]
 => ? = 2
{{1,2,3},{4,5},{6,7}}
 => [2,3,1,5,4,7,6] => [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => [(1,6),(2,3),(4,5),(7,10),(8,9),(11,14),(12,13)]
 => ? = 1
{{1,2,3},{4,5},{6},{7}}
 => [2,3,1,5,4,6,7] => [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,10),(8,9),(11,12),(13,14)]
 => ? = 1
{{1,2,3,6,7},{4},{5}}
 => [2,3,6,4,5,7,1] => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,11),(7,10),(8,9),(12,13)]
 => ? = 3
{{1,2,3,6},{4,7},{5}}
 => [2,3,6,7,5,1,4] => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,12),(8,9),(10,11)]
 => ? = 3
{{1,2,3,6},{4},{5,7}}
 => [2,3,6,4,7,1,5] => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,10),(8,9),(11,12)]
 => ? = 3
{{1,2,3,6},{4},{5},{7}}
 => [2,3,6,4,5,1,7] => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [(1,12),(2,3),(4,5),(6,11),(7,10),(8,9),(13,14)]
 => ? = 3
{{1,2,3,7},{4,6},{5}}
 => [2,3,7,6,5,4,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,12),(8,11),(9,10)]
 => ? = 4
{{1,2,3},{4,6,7},{5}}
 => [2,3,1,6,5,7,4] => [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
 => [(1,6),(2,3),(4,5),(7,14),(8,11),(9,10),(12,13)]
 => ? = 2
{{1,2,3},{4,6},{5,7}}
 => [2,3,1,6,7,4,5] => [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
 => [(1,6),(2,3),(4,5),(7,14),(8,13),(9,10),(11,12)]
 => ? = 2
{{1,2,3},{4,6},{5},{7}}
 => [2,3,1,6,5,4,7] => [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10),(13,14)]
 => ? = 2
{{1,2,3,7},{4},{5,6}}
 => [2,3,7,4,6,5,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,12),(8,11),(9,10)]
 => ? = 4
{{1,2,3},{4,7},{5,6}}
 => [2,3,1,7,6,5,4] => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [(1,6),(2,3),(4,5),(7,14),(8,13),(9,12),(10,11)]
 => ? = 3
{{1,2,3},{4},{5,6,7}}
 => [2,3,1,4,6,7,5] => [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,14),(10,11),(12,13)]
 => ? = 1
{{1,2,3},{4},{5,6},{7}}
 => [2,3,1,4,6,5,7] => [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,12),(10,11),(13,14)]
 => ? = 1
{{1,2,3,7},{4},{5},{6}}
 => [2,3,7,4,5,6,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,12),(8,11),(9,10)]
 => ? = 4
{{1,2,3},{4,7},{5},{6}}
 => [2,3,1,7,5,6,4] => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [(1,6),(2,3),(4,5),(7,14),(8,13),(9,12),(10,11)]
 => ? = 3
{{1,2,3},{4},{5,7},{6}}
 => [2,3,1,4,7,6,5] => [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,14),(10,13),(11,12)]
 => ? = 2

Description

The maximal number of arcs nesting a given arc of a perfect matching. This is also the largest weight of a down step in the histoire d'Hermite corresponding to the perfect matching.

Matching statistic: St000720

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000720: Perfect matchings ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 100%

Values

{{1,2}}
 => [2,1] => [1,1,0,0]
 => [(1,4),(2,3)]
 => 2 = 1 + 1
{{1},{2}}
 => [1,2] => [1,0,1,0]
 => [(1,2),(3,4)]
 => 1 = 0 + 1
{{1,2,3}}
 => [2,3,1] => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => 2 = 1 + 1
{{1,2},{3}}
 => [2,1,3] => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 2 = 1 + 1
{{1,3},{2}}
 => [3,2,1] => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => 3 = 2 + 1
{{1},{2,3}}
 => [1,3,2] => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 2 = 1 + 1
{{1},{2},{3}}
 => [1,2,3] => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
{{1,2,3,4}}
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => 2 = 1 + 1
{{1,2,3},{4}}
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => 2 = 1 + 1
{{1,2,4},{3}}
 => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [(1,8),(2,3),(4,7),(5,6)]
 => 3 = 2 + 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => 2 = 1 + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => 2 = 1 + 1
{{1,3,4},{2}}
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => 3 = 2 + 1
{{1,3},{2,4}}
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => 3 = 2 + 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => 3 = 2 + 1
{{1,4},{2,3}}
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => 4 = 3 + 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => 2 = 1 + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => 2 = 1 + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => 4 = 3 + 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => 3 = 2 + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => 2 = 1 + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => 1 = 0 + 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => 2 = 1 + 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,10)]
 => 2 = 1 + 1
{{1,2,3,5},{4}}
 => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8)]
 => 3 = 2 + 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [(1,6),(2,3),(4,5),(7,10),(8,9)]
 => 2 = 1 + 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,10)]
 => 2 = 1 + 1
{{1,2,4,5},{3}}
 => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
 => [(1,10),(2,3),(4,7),(5,6),(8,9)]
 => 3 = 2 + 1
{{1,2,4},{3,5}}
 => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
 => [(1,10),(2,3),(4,9),(5,6),(7,8)]
 => 3 = 2 + 1
{{1,2,4},{3},{5}}
 => [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
 => [(1,8),(2,3),(4,7),(5,6),(9,10)]
 => 3 = 2 + 1
{{1,2,5},{3,4}}
 => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
 => [(1,10),(2,3),(4,9),(5,8),(6,7)]
 => 4 = 3 + 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [(1,4),(2,3),(5,10),(6,7),(8,9)]
 => 2 = 1 + 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => 2 = 1 + 1
{{1,2,5},{3},{4}}
 => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
 => [(1,10),(2,3),(4,9),(5,8),(6,7)]
 => 4 = 3 + 1
{{1,2},{3,5},{4}}
 => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => 3 = 2 + 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => 2 = 1 + 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10)]
 => 2 = 1 + 1
{{1,3,4,5},{2}}
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => 3 = 2 + 1
{{1,3,4},{2,5}}
 => [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => 4 = 3 + 1
{{1,3,4},{2},{5}}
 => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
 => [(1,8),(2,5),(3,4),(6,7),(9,10)]
 => 3 = 2 + 1
{{1,3,5},{2,4}}
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [(1,10),(2,9),(3,4),(5,6),(7,8)]
 => 3 = 2 + 1
{{1,3},{2,4,5}}
 => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
 => [(1,10),(2,7),(3,4),(5,6),(8,9)]
 => 3 = 2 + 1
{{1,3},{2,4},{5}}
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => [(1,8),(2,7),(3,4),(5,6),(9,10)]
 => 3 = 2 + 1
{{1,3,5},{2},{4}}
 => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
 => [(1,10),(2,5),(3,4),(6,9),(7,8)]
 => 3 = 2 + 1
{{1,3},{2,5},{4}}
 => [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
 => [(1,10),(2,9),(3,4),(5,8),(6,7)]
 => 4 = 3 + 1
{{1,3},{2},{4,5}}
 => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9)]
 => 3 = 2 + 1
{{1,3},{2},{4},{5}}
 => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10)]
 => 3 = 2 + 1
{{1,4,5},{2,3}}
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => 4 = 3 + 1
{{1,4},{2,3,5}}
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [(1,10),(2,9),(3,6),(4,5),(7,8)]
 => 4 = 3 + 1
{{1,4},{2,3},{5}}
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => 4 = 3 + 1
{{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
 => ? = 1 + 1
{{1,2,3,4,5,6},{7}}
 => [2,3,4,5,6,1,7] => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11),(13,14)]
 => ? = 1 + 1
{{1,2,3,4,5,7},{6}}
 => [2,3,4,5,7,6,1] => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,9),(10,13),(11,12)]
 => ? = 2 + 1
{{1,2,3,4,5},{6,7}}
 => [2,3,4,5,1,7,6] => [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9),(11,14),(12,13)]
 => ? = 1 + 1
{{1,2,3,4,5},{6},{7}}
 => [2,3,4,5,1,6,7] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9),(11,12),(13,14)]
 => ? = 1 + 1
{{1,2,3,4,6,7},{5}}
 => [2,3,4,6,5,7,1] => [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,11),(9,10),(12,13)]
 => ? = 2 + 1
{{1,2,3,4,6},{5,7}}
 => [2,3,4,6,7,1,5] => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,13),(9,10),(11,12)]
 => ? = 2 + 1
{{1,2,3,4,6},{5},{7}}
 => [2,3,4,6,5,1,7] => [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
 => [(1,12),(2,3),(4,5),(6,7),(8,11),(9,10),(13,14)]
 => ? = 2 + 1
{{1,2,3,4,7},{5,6}}
 => [2,3,4,7,6,5,1] => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,13),(9,12),(10,11)]
 => ? = 3 + 1
{{1,2,3,4},{5,6,7}}
 => [2,3,4,1,6,7,5] => [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,14),(10,11),(12,13)]
 => ? = 1 + 1
{{1,2,3,4},{5,6},{7}}
 => [2,3,4,1,6,5,7] => [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,12),(10,11),(13,14)]
 => ? = 1 + 1
{{1,2,3,4,7},{5},{6}}
 => [2,3,4,7,5,6,1] => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,13),(9,12),(10,11)]
 => ? = 3 + 1
{{1,2,3,4},{5,7},{6}}
 => [2,3,4,1,7,6,5] => [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,14),(10,13),(11,12)]
 => ? = 2 + 1
{{1,2,3,4},{5},{6,7}}
 => [2,3,4,1,5,7,6] => [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,10),(11,14),(12,13)]
 => ? = 1 + 1
{{1,2,3,4},{5},{6},{7}}
 => [2,3,4,1,5,6,7] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => [(1,8),(2,3),(4,5),(6,7),(9,10),(11,12),(13,14)]
 => ? = 1 + 1
{{1,2,3,5,6,7},{4}}
 => [2,3,5,4,6,7,1] => [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,9),(7,8),(10,11),(12,13)]
 => ? = 2 + 1
{{1,2,3,5,6},{4,7}}
 => [2,3,5,7,6,1,4] => [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,8),(9,12),(10,11)]
 => ? = 3 + 1
{{1,2,3,5,6},{4},{7}}
 => [2,3,5,4,6,1,7] => [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
 => [(1,12),(2,3),(4,5),(6,9),(7,8),(10,11),(13,14)]
 => ? = 2 + 1
{{1,2,3,5,7},{4,6}}
 => [2,3,5,6,7,4,1] => [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,8),(9,10),(11,12)]
 => ? = 2 + 1
{{1,2,3,5},{4,6,7}}
 => [2,3,5,6,1,7,4] => [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,11),(7,8),(9,10),(12,13)]
 => ? = 2 + 1
{{1,2,3,5},{4,6},{7}}
 => [2,3,5,6,1,4,7] => [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
 => [(1,12),(2,3),(4,5),(6,11),(7,8),(9,10),(13,14)]
 => ? = 2 + 1
{{1,2,3,5,7},{4},{6}}
 => [2,3,5,4,7,6,1] => [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
 => [(1,14),(2,3),(4,5),(6,9),(7,8),(10,13),(11,12)]
 => ? = 2 + 1
{{1,2,3,5},{4,7},{6}}
 => [2,3,5,7,1,6,4] => [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,8),(9,12),(10,11)]
 => ? = 3 + 1
{{1,2,3,5},{4},{6,7}}
 => [2,3,5,4,1,7,6] => [1,1,0,1,0,1,1,0,0,0,1,1,0,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8),(11,14),(12,13)]
 => ? = 2 + 1
{{1,2,3,5},{4},{6},{7}}
 => [2,3,5,4,1,6,7] => [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => [(1,10),(2,3),(4,5),(6,9),(7,8),(11,12),(13,14)]
 => ? = 2 + 1
{{1,2,3,6,7},{4,5}}
 => [2,3,6,5,4,7,1] => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,11),(7,10),(8,9),(12,13)]
 => ? = 3 + 1
{{1,2,3,6},{4,5,7}}
 => [2,3,6,5,7,1,4] => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,10),(8,9),(11,12)]
 => ? = 3 + 1
{{1,2,3,6},{4,5},{7}}
 => [2,3,6,5,4,1,7] => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [(1,12),(2,3),(4,5),(6,11),(7,10),(8,9),(13,14)]
 => ? = 3 + 1
{{1,2,3,7},{4,5,6}}
 => [2,3,7,5,6,4,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,12),(8,11),(9,10)]
 => ? = 4 + 1
{{1,2,3},{4,5,6,7}}
 => [2,3,1,5,6,7,4] => [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [(1,6),(2,3),(4,5),(7,14),(8,9),(10,11),(12,13)]
 => ? = 1 + 1
{{1,2,3},{4,5,6},{7}}
 => [2,3,1,5,6,4,7] => [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,12),(8,9),(10,11),(13,14)]
 => ? = 1 + 1
{{1,2,3,7},{4,5},{6}}
 => [2,3,7,5,4,6,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,12),(8,11),(9,10)]
 => ? = 4 + 1
{{1,2,3},{4,5,7},{6}}
 => [2,3,1,5,7,6,4] => [1,1,0,1,0,0,1,1,0,1,1,0,0,0]
 => [(1,6),(2,3),(4,5),(7,14),(8,9),(10,13),(11,12)]
 => ? = 2 + 1
{{1,2,3},{4,5},{6,7}}
 => [2,3,1,5,4,7,6] => [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
 => [(1,6),(2,3),(4,5),(7,10),(8,9),(11,14),(12,13)]
 => ? = 1 + 1
{{1,2,3},{4,5},{6},{7}}
 => [2,3,1,5,4,6,7] => [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => [(1,6),(2,3),(4,5),(7,10),(8,9),(11,12),(13,14)]
 => ? = 1 + 1
{{1,2,3,6,7},{4},{5}}
 => [2,3,6,4,5,7,1] => [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,11),(7,10),(8,9),(12,13)]
 => ? = 3 + 1
{{1,2,3,6},{4,7},{5}}
 => [2,3,6,7,5,1,4] => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,12),(8,9),(10,11)]
 => ? = 3 + 1
{{1,2,3,6},{4},{5,7}}
 => [2,3,6,4,7,1,5] => [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,10),(8,9),(11,12)]
 => ? = 3 + 1
{{1,2,3,6},{4},{5},{7}}
 => [2,3,6,4,5,1,7] => [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [(1,12),(2,3),(4,5),(6,11),(7,10),(8,9),(13,14)]
 => ? = 3 + 1
{{1,2,3,7},{4,6},{5}}
 => [2,3,7,6,5,4,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,12),(8,11),(9,10)]
 => ? = 4 + 1
{{1,2,3},{4,6,7},{5}}
 => [2,3,1,6,5,7,4] => [1,1,0,1,0,0,1,1,1,0,0,1,0,0]
 => [(1,6),(2,3),(4,5),(7,14),(8,11),(9,10),(12,13)]
 => ? = 2 + 1
{{1,2,3},{4,6},{5,7}}
 => [2,3,1,6,7,4,5] => [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
 => [(1,6),(2,3),(4,5),(7,14),(8,13),(9,10),(11,12)]
 => ? = 2 + 1
{{1,2,3},{4,6},{5},{7}}
 => [2,3,1,6,5,4,7] => [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10),(13,14)]
 => ? = 2 + 1
{{1,2,3,7},{4},{5,6}}
 => [2,3,7,4,6,5,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,12),(8,11),(9,10)]
 => ? = 4 + 1
{{1,2,3},{4,7},{5,6}}
 => [2,3,1,7,6,5,4] => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [(1,6),(2,3),(4,5),(7,14),(8,13),(9,12),(10,11)]
 => ? = 3 + 1
{{1,2,3},{4},{5,6,7}}
 => [2,3,1,4,6,7,5] => [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,14),(10,11),(12,13)]
 => ? = 1 + 1
{{1,2,3},{4},{5,6},{7}}
 => [2,3,1,4,6,5,7] => [1,1,0,1,0,0,1,0,1,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,12),(10,11),(13,14)]
 => ? = 1 + 1
{{1,2,3,7},{4},{5},{6}}
 => [2,3,7,4,5,6,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [(1,14),(2,3),(4,5),(6,13),(7,12),(8,11),(9,10)]
 => ? = 4 + 1
{{1,2,3},{4,7},{5},{6}}
 => [2,3,1,7,5,6,4] => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => [(1,6),(2,3),(4,5),(7,14),(8,13),(9,12),(10,11)]
 => ? = 3 + 1
{{1,2,3},{4},{5,7},{6}}
 => [2,3,1,4,7,6,5] => [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
 => [(1,6),(2,3),(4,5),(7,8),(9,14),(10,13),(11,12)]
 => ? = 2 + 1

Description

The size of the largest partition in the oscillating tableau corresponding to the perfect matching. Equivalently, this is the maximal number of crosses in the corresponding triangular rook filling that can be covered by a rectangle.

The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000094The depth of an ordered tree. St001330The hat guessing number of a graph. St001589The nesting number of a perfect matching.