- FindStat

Your data matches 149 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001526
(load all 23 compositions to match this statistic)

Mapping

St001526: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St000836
(load all 27 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of descents of distance 2 of a permutation. This is,

$\operatorname{des}_2(\pi) = | \{ i : \pi(i) > \pi(i+2) \} |$ .

Matching statistic: St000837
(load all 19 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000837: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of ascents of distance 2 of a permutation. This is,

$\operatorname{asc}_2(\pi) = | \{ i : \pi(i) < \pi(i+2) \} |$ .

Matching statistic: St001315
(load all 4 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001315: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => [2,1] => ([(0,1)],2)
 => 2 = 0 + 2
[1,1,0,0]
 => [1,2] => ([],2)
 => 2 = 0 + 2
[1,0,1,0,1,0]
 => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 0 + 2
[1,0,1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[1,1,0,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[1,1,0,1,0,0]
 => [2,1,3] => ([(1,2)],3)
 => 3 = 1 + 2
[1,1,1,0,0,0]
 => [1,2,3] => ([],3)
 => 3 = 1 + 2
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 0 + 2
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 0 + 2
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 0 + 2
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 1 + 2
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 1 + 2
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 0 + 2
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 0 + 2
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 1 + 2
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 1 + 2
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => 3 = 1 + 2
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 3 = 1 + 2
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => 3 = 1 + 2
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => ([(2,3)],4)
 => 4 = 2 + 2
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([],4)
 => 4 = 2 + 2
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 2 + 2
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 2 + 2
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2 = 0 + 2
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3 = 1 + 2
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 3 = 1 + 2
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3 = 1 + 2
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 2 + 2
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => 4 = 2 + 2
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 1 + 2

Description

The dissociation number of a graph.

Matching statistic: St001960

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of descents of a permutation minus one if its first entry is not one. This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].

Matching statistic: St000628
(load all 6 compositions to match this statistic)

Mapping

Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000628: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => 1010 => 1101 => 0010 => 1 = 0 + 1
[1,1,0,0]
 => 1100 => 1101 => 0010 => 1 = 0 + 1
[1,0,1,0,1,0]
 => 101010 => 110101 => 001010 => 1 = 0 + 1
[1,0,1,1,0,0]
 => 101100 => 110101 => 001010 => 1 = 0 + 1
[1,1,0,0,1,0]
 => 110010 => 110101 => 001010 => 1 = 0 + 1
[1,1,0,1,0,0]
 => 110100 => 111001 => 000110 => 2 = 1 + 1
[1,1,1,0,0,0]
 => 111000 => 111001 => 000110 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => 10101010 => 11010101 => 00101010 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => 10101100 => 11010101 => 00101010 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => 10110010 => 11010101 => 00101010 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => 10110100 => 11011001 => 00100110 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => 10111000 => 11011001 => 00100110 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => 11001010 => 11010101 => 00101010 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => 11001100 => 11010101 => 00101010 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => 11010010 => 11100101 => 00011010 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => 11010100 => 11101001 => 00010110 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => 11011000 => 11101001 => 00010110 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => 11100010 => 11100101 => 00011010 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => 11100100 => 11101001 => 00010110 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => 11101000 => 11110001 => 00001110 => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
 => 11110000 => 11110001 => 00001110 => 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => 1101010101 => 0010101010 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 1101010101 => 0010101010 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 1101010101 => 0010101010 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => 1101011001 => 0010100110 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 1101011001 => 0010100110 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 1101010101 => 0010101010 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 1101010101 => 0010101010 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => 1101100101 => 0010011010 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => 1101101001 => 0010010110 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => 1101101001 => 0010010110 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 1101100101 => 0010011010 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => 1101101001 => 0010010110 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => 1101110001 => 0010001110 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 1101110001 => 0010001110 => 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 1101010101 => 0010101010 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 1101010101 => 0010101010 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 1101010101 => 0010101010 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => 1101011001 => 0010100110 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 1101011001 => 0010100110 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => 1110010101 => 0001101010 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => 1110010101 => 0001101010 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => 1110100101 => 0001011010 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => 1110101001 => 0001010110 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => 1110101001 => 0001010110 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => 1110100101 => 0001011010 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => 1110101001 => 0001010110 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => 1110110001 => 0001001110 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => 1110110001 => 0001001110 => 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 1110010101 => 0001101010 => 2 = 1 + 1

Description

The balance of a binary word. The balance of a word is the smallest number

$q$ such that the word is

$q$ -balanced [1]. A binary word

$w$ is

$q$ -balanced if for any two factors

$u$ ,

$v$ of

$w$ of the same length, the difference between the number of ones in

$u$ and

$v$ is at most

$q$ .

Matching statistic: St000982
(load all 3 compositions to match this statistic)

Mapping

Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => 1010 => 0101 => 1010 => 1 = 0 + 1
[1,1,0,0]
 => 1100 => 1001 => 1010 => 1 = 0 + 1
[1,0,1,0,1,0]
 => 101010 => 010101 => 101010 => 1 = 0 + 1
[1,0,1,1,0,0]
 => 101100 => 011001 => 101010 => 1 = 0 + 1
[1,1,0,0,1,0]
 => 110010 => 100101 => 101010 => 1 = 0 + 1
[1,1,0,1,0,0]
 => 110100 => 101001 => 110010 => 2 = 1 + 1
[1,1,1,0,0,0]
 => 111000 => 110001 => 110010 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => 10101010 => 01010101 => 10101010 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => 10101100 => 01011001 => 10101010 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => 10110010 => 01100101 => 10101010 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => 10110100 => 01101001 => 10110010 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => 10111000 => 01110001 => 10110010 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => 11001010 => 10010101 => 10101010 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => 11001100 => 10011001 => 10101010 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => 11010010 => 10100101 => 11001010 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => 11010100 => 10101001 => 11010010 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => 11011000 => 10110001 => 11010010 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => 11100010 => 11000101 => 11001010 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => 11100100 => 11001001 => 11010010 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => 11101000 => 11010001 => 11100010 => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
 => 11110000 => 11100001 => 11100010 => 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => 0101010101 => 1010101010 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 0101011001 => 1010101010 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 0101100101 => 1010101010 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => 0101101001 => 1010110010 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 0101110001 => 1010110010 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 0110010101 => 1010101010 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 0110011001 => 1010101010 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => 0110100101 => 1011001010 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => 0110101001 => 1011010010 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => 0110110001 => 1011010010 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 0111000101 => 1011001010 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => 0111001001 => 1011010010 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => 0111010001 => 1011100010 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 0111100001 => 1011100010 => 3 = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 1001010101 => 1010101010 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 1001011001 => 1010101010 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 1001100101 => 1010101010 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => 1001101001 => 1010110010 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 1001110001 => 1010110010 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => 1010010101 => 1100101010 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => 1010011001 => 1100101010 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => 1010100101 => 1101001010 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => 1010101001 => 1101010010 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => 1010110001 => 1101010010 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => 1011000101 => 1101001010 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => 1011001001 => 1101010010 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => 1011010001 => 1101100010 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => 1011100001 => 1101100010 => 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 1100010101 => 1100101010 => 2 = 1 + 1

Description

The length of the longest constant subword.

Matching statistic: St000877
(load all 2 compositions to match this statistic)

Mapping

Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000877: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => 1010 => 1101 => 0010 => 2 = 0 + 2
[1,1,0,0]
 => 1100 => 1101 => 0010 => 2 = 0 + 2
[1,0,1,0,1,0]
 => 101010 => 110101 => 001010 => 2 = 0 + 2
[1,0,1,1,0,0]
 => 101100 => 110101 => 001010 => 2 = 0 + 2
[1,1,0,0,1,0]
 => 110010 => 110101 => 001010 => 2 = 0 + 2
[1,1,0,1,0,0]
 => 110100 => 111001 => 000110 => 3 = 1 + 2
[1,1,1,0,0,0]
 => 111000 => 111001 => 000110 => 3 = 1 + 2
[1,0,1,0,1,0,1,0]
 => 10101010 => 11010101 => 00101010 => 2 = 0 + 2
[1,0,1,0,1,1,0,0]
 => 10101100 => 11010101 => 00101010 => 2 = 0 + 2
[1,0,1,1,0,0,1,0]
 => 10110010 => 11010101 => 00101010 => 2 = 0 + 2
[1,0,1,1,0,1,0,0]
 => 10110100 => 11011001 => 00100110 => 3 = 1 + 2
[1,0,1,1,1,0,0,0]
 => 10111000 => 11011001 => 00100110 => 3 = 1 + 2
[1,1,0,0,1,0,1,0]
 => 11001010 => 11010101 => 00101010 => 2 = 0 + 2
[1,1,0,0,1,1,0,0]
 => 11001100 => 11010101 => 00101010 => 2 = 0 + 2
[1,1,0,1,0,0,1,0]
 => 11010010 => 11100101 => 00011010 => 3 = 1 + 2
[1,1,0,1,0,1,0,0]
 => 11010100 => 11101001 => 00010110 => 3 = 1 + 2
[1,1,0,1,1,0,0,0]
 => 11011000 => 11101001 => 00010110 => 3 = 1 + 2
[1,1,1,0,0,0,1,0]
 => 11100010 => 11100101 => 00011010 => 3 = 1 + 2
[1,1,1,0,0,1,0,0]
 => 11100100 => 11101001 => 00010110 => 3 = 1 + 2
[1,1,1,0,1,0,0,0]
 => 11101000 => 11110001 => 00001110 => 4 = 2 + 2
[1,1,1,1,0,0,0,0]
 => 11110000 => 11110001 => 00001110 => 4 = 2 + 2
[1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => 1101010101 => 0010101010 => 2 = 0 + 2
[1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 1101010101 => 0010101010 => 2 = 0 + 2
[1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 1101010101 => 0010101010 => 2 = 0 + 2
[1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => 1101011001 => 0010100110 => 3 = 1 + 2
[1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 1101011001 => 0010100110 => 3 = 1 + 2
[1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 1101010101 => 0010101010 => 2 = 0 + 2
[1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 1101010101 => 0010101010 => 2 = 0 + 2
[1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => 1101100101 => 0010011010 => 3 = 1 + 2
[1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => 1101101001 => 0010010110 => 3 = 1 + 2
[1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => 1101101001 => 0010010110 => 3 = 1 + 2
[1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 1101100101 => 0010011010 => 3 = 1 + 2
[1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => 1101101001 => 0010010110 => 3 = 1 + 2
[1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => 1101110001 => 0010001110 => 4 = 2 + 2
[1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 1101110001 => 0010001110 => 4 = 2 + 2
[1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 1101010101 => 0010101010 => 2 = 0 + 2
[1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 1101010101 => 0010101010 => 2 = 0 + 2
[1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 1101010101 => 0010101010 => 2 = 0 + 2
[1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => 1101011001 => 0010100110 => 3 = 1 + 2
[1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 1101011001 => 0010100110 => 3 = 1 + 2
[1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => 1110010101 => 0001101010 => 3 = 1 + 2
[1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => 1110010101 => 0001101010 => 3 = 1 + 2
[1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => 1110100101 => 0001011010 => 3 = 1 + 2
[1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => 1110101001 => 0001010110 => 3 = 1 + 2
[1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => 1110101001 => 0001010110 => 3 = 1 + 2
[1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => 1110100101 => 0001011010 => 3 = 1 + 2
[1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => 1110101001 => 0001010110 => 3 = 1 + 2
[1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => 1110110001 => 0001001110 => 4 = 2 + 2
[1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => 1110110001 => 0001001110 => 4 = 2 + 2
[1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 1110010101 => 0001101010 => 3 = 1 + 2

Description

The depth of the binary word interpreted as a path. This is the maximal value of the number of zeros minus the number of ones occurring in a prefix of the binary word, see [1, sec.9.1.2]. The number of binary words of length

$n$ with depth

$k$ is

$\binom{n}{\lfloor\frac{(n+1) - (-1)^{n-k}(k+1)}{2}\rfloor}$ , see [2].

Matching statistic: St000292

Mapping

Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => 1010 => 1101 => 1000 => 0
[1,1,0,0]
 => 1100 => 1101 => 1000 => 0
[1,0,1,0,1,0]
 => 101010 => 110101 => 100000 => 0
[1,0,1,1,0,0]
 => 101100 => 110101 => 100000 => 0
[1,1,0,0,1,0]
 => 110010 => 110101 => 100000 => 0
[1,1,0,1,0,0]
 => 110100 => 111001 => 101100 => 1
[1,1,1,0,0,0]
 => 111000 => 111001 => 101100 => 1
[1,0,1,0,1,0,1,0]
 => 10101010 => 11010101 => 10000000 => 0
[1,0,1,0,1,1,0,0]
 => 10101100 => 11010101 => 10000000 => 0
[1,0,1,1,0,0,1,0]
 => 10110010 => 11010101 => 10000000 => 0
[1,0,1,1,0,1,0,0]
 => 10110100 => 11011001 => 10001100 => 1
[1,0,1,1,1,0,0,0]
 => 10111000 => 11011001 => 10001100 => 1
[1,1,0,0,1,0,1,0]
 => 11001010 => 11010101 => 10000000 => 0
[1,1,0,0,1,1,0,0]
 => 11001100 => 11010101 => 10000000 => 0
[1,1,0,1,0,0,1,0]
 => 11010010 => 11100101 => 10110000 => 1
[1,1,0,1,0,1,0,0]
 => 11010100 => 11101001 => 10111100 => 1
[1,1,0,1,1,0,0,0]
 => 11011000 => 11101001 => 10111100 => 1
[1,1,1,0,0,0,1,0]
 => 11100010 => 11100101 => 10110000 => 1
[1,1,1,0,0,1,0,0]
 => 11100100 => 11101001 => 10111100 => 1
[1,1,1,0,1,0,0,0]
 => 11101000 => 11110001 => 10100100 => 2
[1,1,1,1,0,0,0,0]
 => 11110000 => 11110001 => 10100100 => 2
[1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => 1101010101 => 1000000000 => 0
[1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 1101010101 => 1000000000 => 0
[1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 1101010101 => 1000000000 => 0
[1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => 1101011001 => 1000001100 => ? = 1
[1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 1101011001 => 1000001100 => ? = 1
[1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 1101010101 => 1000000000 => 0
[1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 1101010101 => 1000000000 => 0
[1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => 1101100101 => 1000110000 => ? = 1
[1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => 1101101001 => 1000111100 => 1
[1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => 1101101001 => 1000111100 => 1
[1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 1101100101 => 1000110000 => ? = 1
[1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => 1101101001 => 1000111100 => 1
[1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => 1101110001 => 1000100100 => ? = 2
[1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 1101110001 => 1000100100 => ? = 2
[1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 1101010101 => 1000000000 => 0
[1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 1101010101 => 1000000000 => 0
[1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 1101010101 => 1000000000 => 0
[1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => 1101011001 => 1000001100 => ? = 1
[1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 1101011001 => 1000001100 => ? = 1
[1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => 1110010101 => 1011000000 => ? = 1
[1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => 1110010101 => 1011000000 => ? = 1
[1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => 1110100101 => 1011110000 => 1
[1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => 1110101001 => 1011111100 => ? = 1
[1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => 1110101001 => 1011111100 => ? = 1
[1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => 1110100101 => 1011110000 => 1
[1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => 1110101001 => 1011111100 => ? = 1
[1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => 1110110001 => 1011100100 => 2
[1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => 1110110001 => 1011100100 => 2
[1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 1110010101 => 1011000000 => ? = 1
[1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 1110010101 => 1011000000 => ? = 1
[1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => 1110100101 => 1011110000 => 1
[1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => 1110101001 => 1011111100 => ? = 1
[1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => 1110101001 => 1011111100 => ? = 1
[1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => 1111000101 => 1010010000 => ? = 2
[1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => 1111001001 => 1010011100 => ? = 2
[1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => 1111010001 => 1010000100 => ? = 2
[1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => 1111010001 => 1010000100 => ? = 2
[1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => 1111000101 => 1010010000 => ? = 2
[1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => 1111001001 => 1010011100 => ? = 2
[1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => 1111010001 => 1010000100 => ? = 2
[1,1,1,1,0,1,0,0,0,0]
 => 1111010000 => 1111100001 => 1010110100 => 3
[1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 1111100001 => 1010110100 => 3

Description

The number of ascents of a binary word.

Matching statistic: St000291

Mapping

Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0]
 => 1010 => 1101 => 1000 => 1 = 0 + 1
[1,1,0,0]
 => 1100 => 1101 => 1000 => 1 = 0 + 1
[1,0,1,0,1,0]
 => 101010 => 110101 => 100000 => 1 = 0 + 1
[1,0,1,1,0,0]
 => 101100 => 110101 => 100000 => 1 = 0 + 1
[1,1,0,0,1,0]
 => 110010 => 110101 => 100000 => 1 = 0 + 1
[1,1,0,1,0,0]
 => 110100 => 111001 => 101100 => 2 = 1 + 1
[1,1,1,0,0,0]
 => 111000 => 111001 => 101100 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => 10101010 => 11010101 => 10000000 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
 => 10101100 => 11010101 => 10000000 => 1 = 0 + 1
[1,0,1,1,0,0,1,0]
 => 10110010 => 11010101 => 10000000 => 1 = 0 + 1
[1,0,1,1,0,1,0,0]
 => 10110100 => 11011001 => 10001100 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => 10111000 => 11011001 => 10001100 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => 11001010 => 11010101 => 10000000 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => 11001100 => 11010101 => 10000000 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => 11010010 => 11100101 => 10110000 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => 11010100 => 11101001 => 10111100 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => 11011000 => 11101001 => 10111100 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => 11100010 => 11100101 => 10110000 => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => 11100100 => 11101001 => 10111100 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => 11101000 => 11110001 => 10100100 => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
 => 11110000 => 11110001 => 10100100 => 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => 1101010101 => 1000000000 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 1101010101 => 1000000000 => 1 = 0 + 1
[1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 1101010101 => 1000000000 => 1 = 0 + 1
[1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => 1101011001 => 1000001100 => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 1101011001 => 1000001100 => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 1101010101 => 1000000000 => 1 = 0 + 1
[1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 1101010101 => 1000000000 => 1 = 0 + 1
[1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => 1101100101 => 1000110000 => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => 1101101001 => 1000111100 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => 1101101001 => 1000111100 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 1101100101 => 1000110000 => ? = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => 1101101001 => 1000111100 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => 1101110001 => 1000100100 => ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 1101110001 => 1000100100 => ? = 2 + 1
[1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 1101010101 => 1000000000 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 1101010101 => 1000000000 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 1101010101 => 1000000000 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => 1101011001 => 1000001100 => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 1101011001 => 1000001100 => ? = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => 1110010101 => 1011000000 => ? = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => 1110010101 => 1011000000 => ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => 1110100101 => 1011110000 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => 1110101001 => 1011111100 => ? = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => 1110101001 => 1011111100 => ? = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => 1110100101 => 1011110000 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => 1110101001 => 1011111100 => ? = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => 1110110001 => 1011100100 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => 1110110001 => 1011100100 => 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
 => 1110001010 => 1110010101 => 1011000000 => ? = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
 => 1110001100 => 1110010101 => 1011000000 => ? = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => 1110010010 => 1110100101 => 1011110000 => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => 1110101001 => 1011111100 => ? = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => 1110101001 => 1011111100 => ? = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
 => 1110100010 => 1111000101 => 1010010000 => ? = 2 + 1
[1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => 1111001001 => 1010011100 => ? = 2 + 1
[1,1,1,0,1,0,1,0,0,0]
 => 1110101000 => 1111010001 => 1010000100 => ? = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
 => 1110110000 => 1111010001 => 1010000100 => ? = 2 + 1
[1,1,1,1,0,0,0,0,1,0]
 => 1111000010 => 1111000101 => 1010010000 => ? = 2 + 1
[1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => 1111001001 => 1010011100 => ? = 2 + 1
[1,1,1,1,0,0,1,0,0,0]
 => 1111001000 => 1111010001 => 1010000100 => ? = 2 + 1
[1,1,1,1,0,1,0,0,0,0]
 => 1111010000 => 1111100001 => 1010110100 => 4 = 3 + 1
[1,1,1,1,1,0,0,0,0,0]
 => 1111100000 => 1111100001 => 1010110100 => 4 = 3 + 1

Description

The number of descents of a binary word.

The following 139 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000390The number of runs of ones in a binary word. St001118The acyclic chromatic index of a graph. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St000381The largest part of an integer composition. St000904The maximal number of repetitions of an integer composition. St000455The second largest eigenvalue of a graph if it is integral. St001095The number of non-isomorphic posets with precisely one further covering relation. St001394The genus of a permutation. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. 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

$eA$ . 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

$eA$ . 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

$eA$ . St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000352The Elizalde-Pak rank of a permutation. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001280The number of parts of an integer partition that are at least two. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001924The number of cells in an integer partition whose arm and leg length coincide. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000934The 2-degree of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001175The size of a partition minus the hook length of the base cell. St001587Half of the largest even part of an integer partition. St001623The number of doubly irreducible elements of a lattice. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000264The girth of a graph, which is not a tree. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000217The number of occurrences of the pattern 312 in a permutation. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000365The number of double ascents of a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000516The number of stretching pairs of a permutation. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000664The number of right ropes of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000872The number of very big descents of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001470The cyclic holeyness of a permutation. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001871The number of triconnected components of a graph. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000089The absolute variation of a composition. St000091The descent variation of a composition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000255The number of reduced Kogan faces with the permutation as type. St000461The rix statistic of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000570The Edelman-Greene number of a permutation. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000654The first descent of a permutation. St000711The number of big exceedences of a permutation. St000873The aix statistic of a permutation. St001162The minimum jump of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001344The neighbouring number of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001520The number of strict 3-descents. St001569The maximal modular displacement of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001905The number of preferred parking spots in a parking function less than the index of the car. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000921The number of internal inversions of a binary word. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001316The domatic number of a graph. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001517The length of a longest pair of twins in a permutation. St001691The number of kings in a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St000297The number of leading ones in a binary word. St000392The length of the longest run of ones in a binary word. St000519The largest length of a factor maximising the subword complexity. St000917The open packing number of a graph. St000922The minimal number such that all substrings of this length are unique. St001415The length of the longest palindromic prefix of a binary word. St001372The length of a longest cyclic run of ones of a binary word. St001416The length of a longest palindromic factor of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001301The first Betti number of the order complex associated with the poset. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001631The number of simple modules

$S$ with

$dim Ext^1(S,A)=1$ in the incidence algebra

$A$ of the poset. St000908The length of the shortest maximal antichain in a poset. St000911The number of maximal antichains of maximal size in a poset. St000914The sum of the values of the Möbius function of a poset. St001487The number of inner corners of a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000454The largest eigenvalue of a graph if it is integral. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000907The number of maximal antichains of minimal length in a poset. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset.