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Your data matches 262 different statistics following compositions of up to 3 maps.
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Matching statistic: St000147
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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 + 1
[1,0,1,1,0,0]
=> [1,1]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [2]
=> 2 = 1 + 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]
=> 2 = 1 + 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]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 4 = 3 + 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]
=> 4 = 3 + 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]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 4 = 3 + 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]
=> 4 = 3 + 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]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 4 = 3 + 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]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 4 = 3 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 4 = 3 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 3 = 2 + 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]
=> 4 = 3 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 3 = 2 + 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]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 4 = 3 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 3 = 2 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2 = 1 + 1
Description
The largest part of an integer partition.
Matching statistic: St000010
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ 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]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1]
=> [2]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [2]
=> [1,1]
=> 2 = 1 + 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]
=> 2 = 1 + 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]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [3]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [2,2,1]
=> 3 = 2 + 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]
=> 3 = 2 + 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]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [4,3,2]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [4,3,1]
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [4,3]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [4,2,2]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [4,2,1]
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [4,2]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [4,1,1]
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,3,2,1]
=> 4 = 3 + 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]
=> 4 = 3 + 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]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,2,2,1]
=> 4 = 3 + 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]
=> 4 = 3 + 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]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [3,1,1,1]
=> 4 = 3 + 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]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [3]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [2,2,2,1]
=> 4 = 3 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [2,2,2]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2,2,1,1]
=> 4 = 3 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2,2,1]
=> 3 = 2 + 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]
=> 4 = 3 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [2,1,1]
=> 3 = 2 + 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]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1]
=> 4 = 3 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1]
=> 3 = 2 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1]
=> 2 = 1 + 1
Description
The length of the partition.
Matching statistic: St000676
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: 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]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 2 = 1 + 1
[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 = 2 + 1
[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 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[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 = 2 + 1
[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 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[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 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[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
[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]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3 = 2 + 1
[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]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[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]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[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]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[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]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> 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]
=> 4 = 3 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,0,1,0]
=> 2 = 1 + 1
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St000734
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [2]
=> [[1,2]]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [[1,2,3]]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [[1,2]]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> 4 = 3 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 4 = 3 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4 = 3 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,2,3],[4]]
=> 3 = 2 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [[1,2,3,4]]
=> 4 = 3 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [[1,2,3]]
=> 3 = 2 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [[1,2]]
=> 2 = 1 + 1
Description
The last entry in the first row of a standard tableau.
Matching statistic: St001039
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: 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]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> 2 = 1 + 1
[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 = 2 + 1
[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 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[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 = 2 + 1
[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 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[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 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
[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
[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]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3 = 2 + 1
[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]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[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]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[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]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[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]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4 = 3 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> 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]
=> 4 = 3 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,0,1,0]
=> 2 = 1 + 1
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
Matching statistic: St000024
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000024: 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]
=> 0
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[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]
=> 1
[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]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[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]
=> 2
[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]
=> 2
[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]
=> 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[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]
=> 2
[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]
=> 2
[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]
=> 1
[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]
=> 2
[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]
=> 2
[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]
=> 1
[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]
=> 2
[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]
=> 1
[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]
=> 0
[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]
=> 3
[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]
=> 3
[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]
=> 1
[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]
=> 3
[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]
=> 3
[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]
=> 1
[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]
=> 3
[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]
=> 1
[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]
=> 0
[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]
=> 3
[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]
=> 2
[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]
=> 3
[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]
=> 2
[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]
=> 3
[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]
=> 2
[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]
=> 0
[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]
=> 3
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000083
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
St000083: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
St000083: Binary trees ⟶ 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]
=> [[.,.],.]
=> 0
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> [.,[.,.]]
=> 1
[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]
=> [[.,.],[[.,.],.]]
=> 1
[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]
=> [.,[[[.,.],.],.]]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[[.,.],.],.]
=> 0
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> 2
[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]
=> [.,[.,[[.,.],.]]]
=> 2
[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]
=> [[.,.],.]
=> 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 2
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> [.,[.,.]]
=> 1
[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]
=> [[.,[[.,.],.]],[[.,.],.]]
=> 2
[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]
=> [.,[[[.,.],.],[[.,.],.]]]
=> 2
[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
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[.,[.,.]],[[[.,.],.],.]]
=> 2
[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]
=> [.,[[.,.],[[[.,.],.],.]]]
=> 2
[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
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [.,[.,[[[[.,.],.],.],.]]]
=> 2
[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
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[[[.,.],.],.],.]
=> 0
[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]
=> [.,[[.,[[.,.],.]],[.,.]]]
=> 3
[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]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> 3
[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]
=> [[[.,.],.],[.,.]]
=> 1
[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]
=> [.,[[.,[.,.]],[[.,.],.]]]
=> 3
[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]
=> [.,[.,[[.,.],[[.,.],.]]]]
=> 3
[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]
=> [[.,.],[[.,.],.]]
=> 1
[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]
=> [.,[.,[.,[[[.,.],.],.]]]]
=> 3
[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]
=> [.,[[[.,.],.],.]]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [[[.,.],.],.]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [.,[[.,[.,.]],[.,.]]]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [[.,[.,.]],[.,.]]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> 2
[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]
=> [.,[.,[.,[[.,.],.]]]]
=> 3
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 2
[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]
=> [[.,.],.]
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 3
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,0,1,0]
=> [.,[.,.]]
=> 1
Description
The number of left oriented leafs of a binary tree except the first one.
In other other words, this is the sum of canopee vector of the tree.
The canopee of a non empty binary tree T with n internal nodes is the list l of 0 and 1 of length n-1 obtained by going along the leaves of T from left to right except the two extremal ones, writing 0 if the leaf is a right leaf and 1 if the leaf is a left leaf.
This is also the number of nodes having a right child. Indeed each of said right children will give exactly one left oriented leaf.
Matching statistic: St000329
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ 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]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 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]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[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]
=> 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[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]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 2
[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]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[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]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 2
[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]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[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]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 2
[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]
=> 1
[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]
=> 0
[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]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 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]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[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]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 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]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[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]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 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]
=> 1
[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]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[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]
=> 0
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
Description
The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.
Matching statistic: St000442
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000442: 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]
=> 0
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[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]
=> 1
[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]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[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]
=> 2
[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]
=> 2
[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]
=> 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[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]
=> 2
[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]
=> 2
[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]
=> 1
[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]
=> 2
[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]
=> 2
[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]
=> 1
[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]
=> 2
[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]
=> 1
[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]
=> 0
[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]
=> 3
[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]
=> 3
[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]
=> 1
[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]
=> 3
[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]
=> 3
[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]
=> 1
[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]
=> 3
[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]
=> 1
[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]
=> 0
[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]
=> 3
[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]
=> 2
[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]
=> 3
[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]
=> 2
[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]
=> 3
[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]
=> 2
[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]
=> 0
[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]
=> 3
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St000874
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000874: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000874: 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]
=> 0
[1,1,0,0,1,0]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[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]
=> 1
[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]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[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]
=> 2
[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]
=> 2
[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]
=> 0
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[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]
=> 2
[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]
=> 2
[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]
=> 1
[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]
=> 2
[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]
=> 2
[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]
=> 1
[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]
=> 2
[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]
=> 1
[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]
=> 0
[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]
=> 3
[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]
=> 3
[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]
=> 1
[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]
=> 3
[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]
=> 3
[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]
=> 1
[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]
=> 3
[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]
=> 1
[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]
=> 0
[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]
=> 3
[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]
=> 2
[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]
=> 3
[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]
=> 2
[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]
=> 3
[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]
=> 2
[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]
=> 0
[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]
=> 3
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
Description
The position of the last double rise in a Dyck path.
If the Dyck path has no double rises, this statistic is $0$.
The following 252 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000288The number of ones in a binary word. St000378The diagonal inversion number of an integer partition. St000443The number of long tunnels of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000691The number of changes of a binary word. St000702The number of weak deficiencies of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. 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. St001461The number of topologically connected components of the chord diagram of a permutation. St001809The index of the step at the first peak of maximal height in a Dyck path. St000439The position of the first down step of a 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. St000141The maximum drop size of a permutation. St000054The first entry of the permutation. St000384The maximal part of the shifted composition of an integer partition. St000653The last descent of a permutation. St000784The maximum of the length and the largest part of the integer partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000157The number of descents of a standard tableau. St000326The position of the first one in a binary word after appending a 1 at the end. St000382The first part of an integer composition. St000733The row containing the largest entry of a standard tableau. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000019The cardinality of the support of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000738The first entry in the last row of a standard tableau. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. 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. St001480The number of simple summands of the module J^2/J^3. St001497The position of the largest weak excedence of a permutation. St000381The largest part of an integer composition. St000808The number of up steps of the associated bargraph. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St001721The degree of a binary word. St000383The last part of an integer composition. St000746The number of pairs with odd minimum in a perfect matching. St000840The number of closers smaller than the largest opener in a perfect matching. St000011The number of touch points (or returns) of a Dyck path. St000505The biggest entry in the block containing the 1. St000971The smallest closer 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. St000839The largest opener of a set partition. 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. St000740The last entry of a permutation. St000503The maximal difference between two elements in a common block. St000225Difference between largest and smallest parts in a partition. St000504The cardinality of the first block of a set partition. St000823The number of unsplittable factors of the set partition. St001062The maximal size of a block of a set partition. St000651The maximal size of a rise in a permutation. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000932The number of occurrences of the pattern UDU in a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000245The number of ascents of a permutation. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000989The number of final rises of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000062The length of the longest increasing subsequence of the permutation. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000991The number of right-to-left minima of a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000143The largest repeated part of a partition. St000306The bounce count of a Dyck path. St000730The maximal arc length of a set partition. St000521The number of distinct subtrees of an ordered tree. St000026The position of the first return of a Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001723The differential of a graph. St001724The 2-packing differential of a graph. St000053The number of valleys of the Dyck path. St000171The degree of the graph. St000211The rank of the set partition. St000271The chromatic index of a graph. St000728The dimension of a set partition. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001118The acyclic chromatic index of a graph. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001498The normalised height of a Nakayama algebra with magnitude 1. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000058The order of a permutation. St000686The finitistic dominant dimension of a Dyck path. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001725The harmonious chromatic number of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000209Maximum difference of elements in cycles. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000673The number of non-fixed points of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. 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. St000652The maximal difference between successive positions of a permutation. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000809The reduced reflection length of the permutation. St000485The length of the longest cycle of a permutation. St000668The least common multiple of the parts of the partition. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000497The lcb statistic of a set partition. St000572The dimension exponent of a set partition. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000332The positive inversions of an alternating sign matrix. St001046The maximal number of arcs nesting a given arc of a perfect matching. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000297The number of leading ones in a binary word. St000392The length of the longest run of ones in a binary word. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000982The length of the longest constant subword. St001372The length of a longest cyclic run of ones of a binary word. St000067The inversion number of the alternating sign matrix. St001268The size of the largest ordinal summand in the poset. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St001298The number of repeated entries in the Lehmer code 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. St000051The size of the left subtree of a binary tree. St000204The number of internal nodes of a binary tree. St000216The absolute length of a permutation. 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. St000133The "bounce" of a permutation. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000021The number of descents of a permutation. St000030The sum of the descent differences of a permutations. St000120The number of left tunnels of a Dyck path. St000168The number of internal nodes of an ordered tree. St000238The number of indices that are not small weak excedances. St000304The load of a permutation. St000331The number of upper interactions of a Dyck path. St000354The number of recoils of a permutation. St000356The number of occurrences of the pattern 13-2. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000795The mad of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St001077The prefix exchange distance of a permutation. St001117The game chromatic index of a graph. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001489The maximum of the number of descents and the number of inverse descents. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001841The number of inversions of a set partition. St000015The number of peaks of a Dyck path. St000061The number of nodes on the left branch of a binary tree. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000144The pyramid weight of the Dyck path. St000166The depth minus 1 of an ordered tree. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000692Babson and Steingrímsson's statistic of a permutation. 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. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St001778The largest greatest common divisor of an element and its image in a permutation. St000094The depth of an ordered tree. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001468The smallest fixpoint of a permutation. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000984The number of boxes below precisely one peak. St000012The area of a Dyck path. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001726The number of visible inversions of a permutation. St000947The major index east count of a Dyck path. St001161The major index north count of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001136The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001557The number of inversions of the second entry of a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001960The number of descents of a permutation minus one if its first entry is not one. St000005The bounce statistic of a Dyck path. St000029The depth of a permutation. St000224The sorting index of a permutation. St000327The number of cover relations in a poset. St000461The rix statistic of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000833The comajor index of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001082The number of boxed occurrences of 123 in a permutation. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001584The area statistic between a Dyck path and its bounce path. St000060The greater neighbor of the maximum. St000155The number of exceedances (also excedences) of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001589The nesting number of a perfect matching. St000617The number of global maxima of a Dyck path. St001861The number of Bruhat lower covers of a permutation. St000896The number of zeros on the main diagonal of an alternating sign matrix.
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