Your data matches 30 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000655
(load all 24 compositions to match this statistic)
Mapping
St000655: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 1 = 0 + 1
[1,0,1,0]
 => 1 = 0 + 1
[1,1,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
Description
The length of the minimal rise of a Dyck path. For the length of a maximal rise, see [[St000444]].
Matching statistic: St000657
(load all 31 compositions to match this statistic)
Mapping
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => 1 = 0 + 1
[1,0,1,0]
 => [1,1] => 1 = 0 + 1
[1,1,0,0]
 => [2] => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,1,1] => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,2] => 1 = 0 + 1
[1,1,0,0,1,0]
 => [2,1] => 1 = 0 + 1
[1,1,0,1,0,0]
 => [2,1] => 1 = 0 + 1
[1,1,1,0,0,0]
 => [3] => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [1,2,1] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [1,3] => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [2,2] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [2,1,1] => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [2,1,1] => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [2,2] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [3,1] => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [3,1] => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => [3,1] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [4] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => 2 = 1 + 1
Description
The smallest part of an integer composition.
Matching statistic: St000210
Mapping
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000210: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [[1]]
 => [1] => 0
[1,0,1,0]
 => [1,0,1,0]
 => [[1,0],[0,1]]
 => [1,2] => 0
[1,1,0,0]
 => [1,1,0,0]
 => [[0,1],[1,0]]
 => [2,1] => 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [1,2,3] => 0
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [2,1,3] => 0
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [1,3,2] => 0
[1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [1,3,2] => 0
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [3,1,2] => 2
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [2,1,3,4] => 0
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 0
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 0
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [3,1,2,4] => 0
[1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,2,4,3] => 0
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [2,1,4,3] => 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 0
[1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 0
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [2,1,4,3] => 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 0
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 0
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 0
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => 3
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [2,1,3,4,5] => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => 0
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [3,1,2,4,5] => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [2,1,4,3,5] => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 0
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [2,1,4,3,5] => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => 0
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [4,1,2,3,5] => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 0
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [3,1,2,5,4] => 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 0
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => 0
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 0
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => 0
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 0
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 0
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [3,1,2,5,4] => 1
Description
Minimum over maximum difference of elements in cycles. Given a cycle $C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is $\max \{ a_i-a_j | a_i, a_j \in C \}$. The statistic is then the minimum of this value over all cycles in the permutation. For example, all permutations with a fixed-point has statistic value 0, and all permutations of $[n]$ with only one cycle, has statistic value $n-1$.
Matching statistic: St000685
(load all 2 compositions to match this statistic)
Mapping
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000685: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1,0]
 => 1 = 0 + 1
[1,0,1,0]
 => [1,1] => [2] => [1,1,0,0]
 => 1 = 0 + 1
[1,1,0,0]
 => [2] => [1,1] => [1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,2] => [1,2] => [1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [2,1] => [2,1] => [1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [2,1] => [2,1] => [1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
Description
The dominant dimension of the LNakayama algebra associated to a Dyck path. To every Dyck path there is an LNakayama algebra associated as described in [[St000684]].
Matching statistic: St000700
(load all 2 compositions to match this statistic)
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000700: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [.,.]
 => [[]]
 => 1 = 0 + 1
[1,0,1,0]
 => [2,1] => [[.,.],.]
 => [[],[]]
 => 1 = 0 + 1
[1,1,0,0]
 => [1,2] => [.,[.,.]]
 => [[[]]]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [3,2,1] => [[[.,.],.],.]
 => [[],[],[]]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [2,3,1] => [[.,[.,.]],.]
 => [[[]],[]]
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [3,1,2] => [[.,.],[.,.]]
 => [[],[[]]]
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [2,1,3] => [[.,.],[.,.]]
 => [[],[[]]]
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [.,[.,[.,.]]]
 => [[[[]]]]
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => [[],[],[],[]]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [[[.,[.,.]],.],.]
 => [[[]],[],[]]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [[[.,.],[.,.]],.]
 => [[],[[]],[]]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [[[.,.],[.,.]],.]
 => [[],[[]],[]]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [[.,[.,[.,.]]],.]
 => [[[[]]],[]]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [[[.,.],.],[.,.]]
 => [[],[],[[]]]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => [[[]],[[]]]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [[[.,.],.],[.,.]]
 => [[],[],[[]]]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => [[],[],[[]]]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [[.,[.,.]],[.,.]]
 => [[[]],[[]]]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [[.,.],[.,[.,.]]]
 => [[],[[[]]]]
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [[.,.],[.,[.,.]]]
 => [[],[[[]]]]
 => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => [[],[[[]]]]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [[[[[]]]]]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => [[],[],[],[],[]]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [[[[.,[.,.]],.],.],.]
 => [[[]],[],[],[]]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
 => [[],[[]],[],[]]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [[[[.,.],[.,.]],.],.]
 => [[],[[]],[],[]]
 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
 => [[[[]]],[],[]]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [[[[.,.],.],[.,.]],.]
 => [[],[],[[]],[]]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [[[.,[.,.]],[.,.]],.]
 => [[[]],[[]],[]]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
 => [[],[],[[]],[]]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
 => [[],[],[[]],[]]
 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
 => [[[]],[[]],[]]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
 => [[],[[[]]],[]]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [[[.,.],[.,[.,.]]],.]
 => [[],[[[]]],[]]
 => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
 => [[],[[[]]],[]]
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
 => [[[[[]]]],[]]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
 => [[],[],[],[[]]]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
 => [[[]],[],[[]]]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
 => [[],[[]],[[]]]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
 => [[],[[]],[[]]]
 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
 => [[[[]]],[[]]]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
 => [[],[],[],[[]]]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [[[.,[.,.]],.],[.,.]]
 => [[[]],[],[[]]]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
 => [[],[],[],[[]]]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => [[],[],[],[[]]]
 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
 => [[[]],[],[[]]]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [[[.,.],[.,.]],[.,.]]
 => [[],[[]],[[]]]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
 => [[],[[]],[[]]]
 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
 => [[],[[]],[[]]]
 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
 => [[[[]]],[[]]]
 => 2 = 1 + 1
Description
The protection number of an ordered tree. This is the minimal distance from the root to a leaf.
Matching statistic: St001075
(load all 12 compositions to match this statistic)
Mapping
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001075: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => {{1}}
 => ? = 0 + 1
[1,0,1,0]
 => [1,0,1,0]
 => {{1},{2}}
 => 1 = 0 + 1
[1,1,0,0]
 => [1,1,0,0]
 => {{1,2}}
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => {{1,3},{2}}
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => {{1,3},{2},{4}}
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => {{1},{2,4},{3}}
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => {{1,4},{2},{3}}
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => {{1,4},{2,3}}
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => {{1,3,4},{2}}
 => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => {{1,2,4},{3}}
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => {{1,3},{2},{4},{5}}
 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => {{1},{2,4},{3},{5}}
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => {{1,4},{2},{3},{5}}
 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => {{1,4},{2,3},{5}}
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => {{1,3,4},{2},{5}}
 => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => {{1,2,4},{3},{5}}
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => {{1,3},{2},{4,5}}
 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => {{1},{2},{3,5},{4}}
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => {{1,2},{3,5},{4}}
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => {{1},{2,5},{3},{4}}
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => {{1,5},{2},{3},{4}}
 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => {{1,5},{2,3},{4}}
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => {{1},{2,5},{3,4}}
 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => {{1,5},{2},{3,4}}
 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => {{1,5},{2,4},{3}}
 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => {{1,5},{2,3,4}}
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 1 = 0 + 1
Description
The minimal size of a block of a set partition.
Matching statistic: St000487
(load all 5 compositions to match this statistic)
Mapping
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000487: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,0]
 => [[1]]
 => [1] => ? = 0 + 1
[1,0,1,0]
 => [1,0,1,0]
 => [[1,0],[0,1]]
 => [1,2] => 1 = 0 + 1
[1,1,0,0]
 => [1,1,0,0]
 => [[0,1],[1,0]]
 => [2,1] => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [[1,0,0],[0,1,0],[0,0,1]]
 => [1,2,3] => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => [[0,1,0],[1,0,0],[0,0,1]]
 => [2,1,3] => 1 = 0 + 1
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => [[1,0,0],[0,0,1],[0,1,0]]
 => [1,3,2] => 1 = 0 + 1
[1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => [[0,1,0],[1,-1,1],[0,1,0]]
 => [1,3,2] => 1 = 0 + 1
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [[0,0,1],[1,0,0],[0,1,0]]
 => [3,1,2] => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [2,1,3,4] => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
 => [1,3,2,4] => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [3,1,2,4] => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [1,2,4,3] => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [2,1,4,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
 => [1,2,4,3] => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
 => [2,1,4,3] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
 => [1,4,2,3] => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [4,1,2,3] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,2,3,4,5] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [2,1,3,4,5] => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [1,3,2,4,5] => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [3,1,2,4,5] => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [2,1,4,3,5] => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,2,4,3,5] => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [2,1,4,3,5] => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [1,4,2,3,5] => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [4,1,2,3,5] => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [3,1,2,5,4] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,2,3,5,4] => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [2,1,3,5,4] => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [1,3,2,5,4] => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
 => [3,1,2,5,4] => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [1,2,5,3,4] => 1 = 0 + 1
Description
The length of the shortest cycle of a permutation.
Matching statistic: St000993
Mapping
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1]
 => [1]
 => ? = 0 + 1
[1,0,1,0]
 => [1,1] => [1,1]
 => [2]
 => 1 = 0 + 1
[1,1,0,0]
 => [2] => [2]
 => [1,1]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,1,1] => [1,1,1]
 => [3]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,2] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [2,1] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [2,1] => [2,1]
 => [2,1]
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => [3] => [3]
 => [1,1,1]
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,1,1,1]
 => [4]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [2,1,1]
 => [3,1]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,1,1]
 => [3,1]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [1,2,1] => [2,1,1]
 => [3,1]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [3,1]
 => [2,1,1]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [2,1,1]
 => [3,1]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [2,2] => [2,2]
 => [2,2]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [2,1,1] => [2,1,1]
 => [3,1]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [2,1,1] => [2,1,1]
 => [3,1]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [2,2] => [2,2]
 => [2,2]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [3,1] => [3,1]
 => [2,1,1]
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [3,1] => [3,1]
 => [2,1,1]
 => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => [3,1] => [3,1]
 => [2,1,1]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [4] => [4]
 => [1,1,1,1]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [5]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [2,1,1,1]
 => [4,1]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,1,1,1]
 => [4,1]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => [2,1,1,1]
 => [4,1]
 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [3,1,1]
 => [3,1,1]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [2,1,1,1]
 => [4,1]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [2,2,1]
 => [3,2]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => [2,1,1,1]
 => [4,1]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [2,1,1,1]
 => [4,1]
 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [2,2,1]
 => [3,2]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [3,1,1]
 => [3,1,1]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [3,1,1]
 => [3,1,1]
 => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [3,1,1]
 => [3,1,1]
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [4,1]
 => [2,1,1,1]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [2,1,1,1]
 => [4,1]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [2,2,1]
 => [3,2]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,2,1]
 => [3,2]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [2,2,1]
 => [3,2]
 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [3,2]
 => [2,2,1]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => [2,1,1,1]
 => [4,1]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [2,2,1]
 => [3,2]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [2,1,1,1]
 => [4,1]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [2,1,1,1]
 => [4,1]
 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [2,2,1]
 => [3,2]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [2,2,1]
 => [3,2]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [2,2,1]
 => [3,2]
 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [2,2,1]
 => [3,2]
 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [3,2]
 => [2,2,1]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => [3,1,1]
 => [3,1,1]
 => 1 = 0 + 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St001038
(load all 2 compositions to match this statistic)
Mapping
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1,0]
 => [1,0]
 => ? = 0 + 1
[1,0,1,0]
 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[1,1,0,0]
 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 0 + 1
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000667
Mapping
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 99%●distinct values known / distinct values provided: 50%
Values
[1,0]
 => [1] => [1]
 => []
 => ? = 0 + 1
[1,0,1,0]
 => [1,1] => [1,1]
 => [1]
 => 1 = 0 + 1
[1,1,0,0]
 => [2] => [2]
 => []
 => ? = 1 + 1
[1,0,1,0,1,0]
 => [1,1,1] => [1,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,0,1,1,0,0]
 => [1,2] => [2,1]
 => [1]
 => 1 = 0 + 1
[1,1,0,0,1,0]
 => [2,1] => [2,1]
 => [1]
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [2,1] => [2,1]
 => [1]
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => [3] => [3]
 => []
 => ? = 2 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [2,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => [1,2,1] => [2,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [2,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [2,2] => [2,2]
 => [2]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [2,1,1] => [2,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [2,1,1] => [2,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [2,2] => [2,2]
 => [2]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => [3,1] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [4] => [4]
 => []
 => ? = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,1,1,1,1]
 => [1,1,1,1]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [2,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,2,1] => [2,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [3,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [2,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [2,2,1]
 => [2,1]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,2,1,1] => [2,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,2,1,1] => [2,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,2,2] => [2,2,1]
 => [2,1]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [3,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,3,1] => [3,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,3,1] => [3,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [4,1]
 => [1]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [2,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [2,2,1]
 => [2,1]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,2,1]
 => [2,1]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,2,1] => [2,2,1]
 => [2,1]
 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [3,2]
 => [2]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,1,1,1] => [2,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,1,2] => [2,2,1]
 => [2,1]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,1,1,1] => [2,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,1,1,1] => [2,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,1,2] => [2,2,1]
 => [2,1]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,2,1] => [2,2,1]
 => [2,1]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,2,1] => [2,2,1]
 => [2,1]
 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,2,1] => [2,2,1]
 => [2,1]
 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3] => [3,2]
 => [2]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => [3,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,2] => [3,2]
 => [2]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [3,1,1] => [3,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,1,1] => [3,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [5] => [5]
 => []
 => ? = 4 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
 => [6] => [6]
 => []
 => ? = 5 + 1
Description
The greatest common divisor of the parts of the partition.
The following 20 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000264The girth of a graph, which is not a tree. St001498The normalised height of a Nakayama algebra with magnitude 1. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000260The radius of a connected graph. St000259The diameter of a connected graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001545The second Elser number of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000455The second largest eigenvalue of a graph if it is integral. St001846The number of elements which do not have a complement in the lattice. St001820The size of the image of the pop stack sorting operator. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001330The hat guessing number of a graph. St000750The number of occurrences of the pattern 4213 in a permutation.