- FindStat

Your data matches 167 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000010
(load all 10 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the partition.

Matching statistic: St000329
(load all 3 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.

Matching statistic: St000147
(load all 10 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest part of an integer partition.

Matching statistic: St000288
(load all 15 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0,1,0,1,0]
 => [2,1]
 => 1010 => 2 = 1 + 1
[1,0,1,1,0,0]
 => [1,1]
 => 110 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [2]
 => 100 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 101010 => 3 = 2 + 1
[1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 11010 => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 100110 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 10110 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 1110 => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
 => [3,2]
 => 10100 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => [2,2]
 => 1100 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [3,1]
 => 10010 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
 => [2,1]
 => 1010 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [1,1]
 => 110 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [3]
 => 1000 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [2]
 => 100 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => 1101010 => 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => 1011010 => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => 111010 => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => 1100110 => 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [3,2,1,1]
 => 1010110 => 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => 110110 => 4 = 3 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => 1001110 => 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => 101110 => 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 11110 => 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [4,3,2]
 => 1010100 => 3 = 2 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [3,3,2]
 => 110100 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [4,2,2]
 => 1001100 => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [3,2,2]
 => 101100 => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,2,2]
 => 11100 => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [4,3,1]
 => 1010010 => 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,3,1]
 => 110010 => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,2,1]
 => 1001010 => 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [3,2,1]
 => 101010 => 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,2,1]
 => 11010 => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1,1]
 => 1000110 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [3,1,1]
 => 100110 => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,1,1]
 => 10110 => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1]
 => 1110 => 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [4,3]
 => 101000 => 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,3]
 => 11000 => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,2]
 => 100100 => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,2]
 => 10100 => 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [2,2]
 => 1100 => 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [4,1]
 => 100010 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,1]
 => 10010 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [2,1]
 => 1010 => 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [1,1]
 => 110 => 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [4]
 => 10000 => 1 = 0 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [3]
 => 1000 => 1 = 0 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [2]
 => 100 => 1 = 0 + 1

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

Matching statistic: St000378
(load all 2 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The diagonal inversion number of an integer partition. The dinv of a partition is the number of cells

$c$ in the diagram of an integer partition

$\lambda$ for which

$\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$ . See also exercise 3.19 of [2]. This statistic is equidistributed with the length of the partition, see [3].

Matching statistic: St000053

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of valleys of the Dyck path.

Matching statistic: St000155
(load all 3 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of exceedances (also excedences) of a permutation. This is defined as

$exc(\sigma) = \#\{ i : \sigma(i) > i \}$ . It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic

$(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here,

$den$ is the Denert index of a permutation, see [[St000156]].

Matching statistic: St000216
(load all 9 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000216: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The absolute length of a permutation. The absolute length of a permutation

$\sigma$ of length

$n$ is the shortest

$k$ such that

$\sigma = t_1 \dots t_k$ for transpositions

$t_i$ . Also, this is equal to

$n$ minus the number of cycles of

$\sigma$ .

Matching statistic: St000331

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000331: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of upper interactions of a Dyck path. An ''upper interaction'' in a Dyck path is defined as the occurrence of a factor '''

$A^{k}$

$B^{k}$ ''' for any '''

${k ≥ 1}$ ''', where '''

${A}$ ''' is a down-step and '''

${B}$ ''' is a up-step.

Matching statistic: St000809
(load all 6 compositions to match this statistic)

Mapping

Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000809: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The reduced reflection length of the permutation. Let

$T$ be the set of reflections in a Coxeter group and let

$\ell(w)$ be the usual length function. Then the reduced reflection length of

$w$ is

$\min\{r\in\mathbb N \mid w = t_1\cdots t_r,\quad t_1,\dots,t_r \in T,\quad \ell(w)=\sum \ell(t_i)\}.$ In the case of the symmetric group, this is twice the depth [[St000029]] minus the usual length [[St000018]].

The following 157 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000015The number of peaks of a Dyck path. St000676The number of odd rises of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000734The last entry in the first row of a standard tableau. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000157The number of descents of a standard tableau. St000733The row containing the largest entry of a standard tableau. St000546The number of global descents of a permutation. St001777The number of weak descents in an integer composition. St000007The number of saliances of the permutation. St000507The number of ascents of a standard tableau. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000653The last descent of a permutation. St001480The number of simple summands of the module J^2/J^3. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St001462The number of factors of a standard tableaux under concatenation. St000738The first entry in the last row of a standard tableau. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000006The dinv of a Dyck path. St000013The height of a Dyck path. St000009The charge of a standard tableau. St001280The number of parts of an integer partition that are at least two. St000105The number of blocks in the set partition. St000925The number of topologically connected components of a set partition. St000439The position of the first down step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000141The maximum drop size of a permutation. St000054The first entry of the permutation. St000839The largest opener of a set partition. St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001152The number of pairs with even minimum in a perfect matching. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000931The number of occurrences of the pattern UUU in a Dyck path. St000444The length of the maximal rise of a Dyck path. St000021The number of descents of a permutation. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000451The length of the longest pattern of the form k 1 2. St000161The sum of the sizes of the right subtrees of a binary tree. St000730The maximal arc length of a set partition. St000874The position of the last double rise in a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St000204The number of internal nodes of a binary tree. St000304The load of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St001723The differential of a graph. St001724The 2-packing differential of a graph. St000024The number of double up and double down steps of a Dyck path. St000171The degree of the graph. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000209Maximum difference of elements in cycles. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001725The harmonious chromatic number of a graph. St000675The number of centered multitunnels of a Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000845The maximal number of elements covered by an element in a poset. St000446The disorder of a permutation. St000019The cardinality of the support of a permutation. St000673The number of non-fixed points of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000028The number of stack-sorts needed to sort a permutation. St000651The maximal size of a rise in a permutation. St000740The last entry of a permutation. St001497The position of the largest weak excedence of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000356The number of occurrences of the pattern 13-2. St000067The inversion number of the alternating sign matrix. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000652The maximal difference between successive positions of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000051The size of the left subtree of a binary tree. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St000235The number of indices that are not cyclical small weak excedances. St000240The number of indices that are not small excedances. St000692Babson and Steingrímsson's statistic of a permutation. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St000030The sum of the descent differences of a permutations. St000133The "bounce" of a permutation. St000305The inverse major index of a permutation. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000082The number of elements smaller than a binary tree in Tamari order. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000443The number of long tunnels of a Dyck path. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St001645The pebbling number of a connected graph. St000840The number of closers smaller than the largest opener in a perfect matching. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000989The number of final rises of a permutation. St000461The rix statistic of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000338The number of pixed points of a permutation. St001557The number of inversions of the second entry of a permutation. St000327The number of cover relations in a poset. St000654The first descent of a permutation. St000873The aix statistic of a permutation. St000990The first ascent of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001434The number of negative sum pairs of a signed permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001861The number of Bruhat lower covers of a permutation. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation.