- FindStat

Your data matches 13 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000117
(load all 12 compositions to match this statistic)

Mapping

Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St000117: 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]
 => [1,1,0,0]
 => 2
[1,1,0,0]
 => [1,0,1,0]
 => 0
[1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[1,0,1,1,0,0]
 => [1,1,1,0,0,0]
 => 3
[1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 0
[1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1
[1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0
[1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 3

Description

The number of centered tunnels of a Dyck path. A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b==n then the tunnel is called centered.

Matching statistic: St000241
(load all 13 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of cyclical small excedances. A cyclical small excedance is an index

$i$ such that

$\pi_i = i+1$ considered cyclically.

Matching statistic: St000022
(load all 21 compositions to match this statistic)

Mapping

Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of fixed points of a permutation.

Matching statistic: St000221
(load all 23 compositions to match this statistic)

Mapping

Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000221: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of strong fixed points of a permutation.

$i$ is called a strong fixed point of

$\pi$ if 1.

$j < i$ implies

$\pi_j < \pi_i$ , and 2.

$j > i$ implies

$\pi_j > \pi_i$ This can be described as an occurrence of the mesh pattern ([1], {(0,1),(1,0)}), i.e., the upper left and the lower right quadrants are shaded, see [3]. The generating function for the joint-distribution (RLmin, LRmax, strong fixed points) has a continued fraction expression as given in [4, Lemma 3.2], for LRmax see [[St000314]].

Matching statistic: St000215
(load all 3 compositions to match this statistic)

Mapping

Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000215: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of adjacencies of a permutation, zero appended. An adjacency is a descent of the form

$(e+1,e)$ in the word corresponding to the permutation in one-line notation. This statistic,

$\operatorname{adj_0}$ , counts adjacencies in the word with a zero appended.

$(\operatorname{adj_0}, \operatorname{des})$ and

$(\operatorname{fix}, \operatorname{exc})$ are equidistributed, see [1].

Matching statistic: St000315

Mapping

Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000315: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of isolated vertices of a graph.

Matching statistic: St000475
(load all 3 compositions to match this statistic)

Mapping

Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of parts equal to 1 in a partition.

Matching statistic: St000895
(load all 8 compositions to match this statistic)

Mapping

Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of ones on the main diagonal of an alternating sign matrix.

Matching statistic: St001008
(load all 11 compositions to match this statistic)

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001008: 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
[1,0,1,0]
 => [2,1] => [2,1] => [1,1,0,0]
 => 2
[1,1,0,0]
 => [1,2] => [1,2] => [1,0,1,0]
 => 0
[1,0,1,0,1,0]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 1
[1,0,1,1,0,0]
 => [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
 => 3
[1,1,0,0,1,0]
 => [3,1,2] => [2,3,1] => [1,1,0,1,0,0]
 => 0
[1,1,0,1,0,0]
 => [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 2
[1,0,1,0,1,1,0,0]
 => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 1
[1,0,1,1,0,0,1,0]
 => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 1
[1,0,1,1,0,1,0,0]
 => [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 4
[1,1,0,0,1,0,1,0]
 => [3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,0]
 => [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 0
[1,1,0,1,1,0,0,0]
 => [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 0
[1,1,1,0,1,0,0,0]
 => [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
 => 3
[1,0,1,0,1,1,0,1,0,0]
 => [2,4,1,5,3] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
 => 2
[1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,1,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 3
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,5,2,4] => [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3

Description

Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.

Matching statistic: St000674
(load all 11 compositions to match this statistic)

Mapping

Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of hills of a Dyck path. A hill is a peak with up step starting and down step ending at height zero.

The following 3 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000247The number of singleton blocks of a set partition. St000894The trace of an alternating sign matrix. St001903The number of fixed points of a parking function.