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Your data matches 767 different statistics following compositions of up to 3 maps.
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Matching statistic: St000159
(load all 46 compositions to match this statistic)
(load all 46 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> 1
[1,0,1,0,1,0]
=> [2,1]
=> 2
[1,0,1,1,0,0]
=> [1,1]
=> 1
[1,1,0,0,1,0]
=> [2]
=> 1
[1,1,0,1,0,0]
=> [1]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 1
Description
The number of distinct parts of the integer partition.
This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Matching statistic: St000318
(load all 46 compositions to match this statistic)
(load all 46 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [2,1]
=> 3 = 2 + 1
[1,0,1,1,0,0]
=> [1,1]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [1]
=> 2 = 1 + 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]
=> 2 = 1 + 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]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 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]
=> 2 = 1 + 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]
=> 3 = 2 + 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]
=> 3 = 2 + 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]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 2 = 1 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> 3 = 2 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 2 = 1 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> 3 = 2 + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> 3 = 2 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> 3 = 2 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 2 = 1 + 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> 3 = 2 + 1
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> 3 = 2 + 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> 3 = 2 + 1
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> 3 = 2 + 1
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 2 = 1 + 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 2 = 1 + 1
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St000053
(load all 498 compositions to match this statistic)
(load all 498 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ 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]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 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]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
Description
The number of valleys of the Dyck path.
Matching statistic: St000291
(load all 129 compositions to match this statistic)
(load all 129 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> 10 => 1
[1,0,1,0,1,0]
=> [2,1]
=> 1010 => 2
[1,0,1,1,0,0]
=> [1,1]
=> 110 => 1
[1,1,0,0,1,0]
=> [2]
=> 100 => 1
[1,1,0,1,0,0]
=> [1]
=> 10 => 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 1
[1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 1
[1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 11010 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 100110 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 10110 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1110 => 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 10100 => 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1100 => 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 100010 => 2
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 10010 => 2
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1010 => 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 110 => 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 10000 => 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1000 => 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 100 => 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 10 => 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 111110 => 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> 101110 => 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> 11010 => 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> 100110 => 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> 10110 => 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 1110 => 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> 10100 => 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 1100 => 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> 100010 => 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> 10010 => 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> 1010 => 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 110 => 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 100000 => 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 10000 => 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 1000 => 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 100 => 1
Description
The number of descents of a binary word.
Matching statistic: St000306
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 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]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
Description
The bounce count of a Dyck path.
For a Dyck path $D$ of length $2n$, this is the number of points $(i,i)$ for $1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of $D$.
Matching statistic: St000390
(load all 126 compositions to match this statistic)
(load all 126 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> 10 => 1
[1,0,1,0,1,0]
=> [2,1]
=> 1010 => 2
[1,0,1,1,0,0]
=> [1,1]
=> 110 => 1
[1,1,0,0,1,0]
=> [2]
=> 100 => 1
[1,1,0,1,0,0]
=> [1]
=> 10 => 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 11010 => 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 100110 => 2
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 10110 => 2
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1110 => 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> 10100 => 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> 1100 => 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> 10010 => 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> 1010 => 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> 110 => 1
[1,1,1,0,0,0,1,0]
=> [3]
=> 1000 => 1
[1,1,1,0,0,1,0,0]
=> [2]
=> 100 => 1
[1,1,1,0,1,0,0,0]
=> [1]
=> 10 => 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 101110 => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 11010 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 100110 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 10110 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 1110 => 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 10100 => 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 1100 => 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 100010 => 2
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 10010 => 2
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1010 => 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 110 => 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 10000 => 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1000 => 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 100 => 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 10 => 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 111110 => 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> 101110 => 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 11110 => 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> 11010 => 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> 100110 => 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> 10110 => 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 1110 => 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> 10100 => 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> 1100 => 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> 100010 => 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> 10010 => 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> 1010 => 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 110 => 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 100000 => 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 10000 => 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 1000 => 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 100 => 1
Description
The number of runs of ones in a binary word.
Matching statistic: St001142
(load all 461 compositions to match this statistic)
(load all 461 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001142: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001142: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 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]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
Description
The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001169
(load all 497 compositions to match this statistic)
(load all 497 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001169: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001169: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 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]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
Description
Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St001197
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001197: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001197: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 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]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
Description
The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001199
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 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]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 1
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
The following 757 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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 $eA$. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000015The number of peaks of a Dyck path. St000292The number of ascents of a binary word. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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:
St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000021The number of descents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000069The number of maximal elements of a poset. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000245The number of ascents of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000354The number of recoils of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000522The number of 1-protected nodes of a rooted tree. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000619The number of cyclic descents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000767The number of runs in an integer composition. St000864The number of circled entries of the shifted recording tableau of a permutation. St000903The number of different parts of an integer composition. St000920The logarithmic height of a Dyck path. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000996The number of exclusive left-to-right maxima 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. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001487The number of inner corners of a skew partition. St001489The maximum of the number of descents and the number of inverse descents. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001726The number of visible inversions of a permutation. St001735The number of permutations with the same set of runs. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000105The number of blocks in the set partition. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000386The number of factors DDU in a Dyck path. St000443The number of long tunnels of a Dyck path. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000647The number of big descents of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000761The number of ascents in an integer composition. St000871The number of very big ascents of a permutation. St000925The number of topologically connected components of a set partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001673The degree of asymmetry of an integer composition. St001840The number of descents of a set partition. St001180Number of indecomposable injective modules with projective dimension at most 1. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000345The number of refinements of a partition. St000935The number of ordered refinements of an integer partition. St001389The number of partitions of the same length below the given integer partition. St000806The semiperimeter of the associated bargraph. St000010The length of the partition. St000346The number of coarsenings of a partition. St000519The largest length of a factor maximising the subword complexity. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St000137The Grundy value of an integer partition. St000144The pyramid weight of the Dyck path. St000147The largest part of an integer partition. St000185The weighted size of a partition. St000228The size of a partition. St000374The number of exclusive right-to-left minima of a permutation. St000378The diagonal inversion number of an integer partition. St000395The sum of the heights of the peaks of a Dyck path. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000481The number of upper covers of a partition in dominance order. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000703The number of deficiencies of a permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001176The size of a partition minus its first part. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000317The cycle descent number of a permutation. St000402Half the size of the symmetry class of a permutation. St000568The hook number of a binary tree. St000630The length of the shortest palindromic decomposition of a binary word. St000886The number of permutations with the same antidiagonal sums. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001372The length of a longest cyclic run of ones of a binary word. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000288The number of ones in a binary word. St000486The number of cycles of length at least 3 of a permutation. St000646The number of big ascents of a permutation. St000702The number of weak deficiencies of a permutation. St000991The number of right-to-left minima of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001960The number of descents of a permutation minus one if its first entry is not one. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St000733The row containing the largest entry of a standard tableau. St000392The length of the longest run of ones in a binary word. St000753The Grundy value for the game of Kayles on a binary word. St000982The length of the longest constant subword. St000389The number of runs of ones of odd length in a binary word. St000628The balance of a binary word. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001279The sum of the parts of an integer partition that are at least two. St000377The dinv defect of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St000877The depth of the binary word interpreted as a path. St000326The position of the first one in a binary word after appending a 1 at the end. St001419The length of the longest palindromic factor beginning with a one of a binary word. St000507The number of ascents of a standard tableau. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St000048The multinomial of the parts of a partition. St000160The multiplicity of the smallest part of a partition. St000548The number of different non-empty partial sums of an integer partition. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001933The largest multiplicity of a part in an integer partition. St000143The largest repeated part of a partition. St000150The floored half-sum of the multiplicities of a partition. St000257The number of distinct parts of a partition that occur at least twice. St000734The last entry in the first row of a standard tableau. St001091The number of parts in an integer partition whose next smaller part has the same size. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St001415The length of the longest palindromic prefix of a binary word. St001413Half the length of the longest even length palindromic prefix of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St000617The number of global maxima of a Dyck path. St000473The number of parts of a partition that are strictly bigger than the number of ones. St001280The number of parts of an integer partition that are at least two. St001175The size of a partition minus the hook length of the base cell. St000183The side length of the Durfee square of an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000783The side length of the largest staircase partition fitting into a partition. St001432The order dimension of the partition. St000451The length of the longest pattern of the form k 1 2. St000862The number of parts of the shifted shape of a permutation. St001214The aft of an integer partition. St000356The number of occurrences of the pattern 13-2. St000662The staircase size of the code of a permutation. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000546The number of global descents of a permutation. St000007The number of saliances of the permutation. St000149The number of cells of the partition whose leg is zero and arm is odd. St001128The exponens consonantiae of a partition. St001571The Cartan determinant of the integer partition. St001712The number of natural descents of a standard Young tableau. St001172The number of 1-rises at odd height of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000660The number of rises of length at least 3 of a Dyck path. St000439The position of the first down step of a Dyck path. St000225Difference between largest and smallest parts in a partition. St000336The leg major index of a standard tableau. St000256The number of parts from which one can substract 2 and still get an integer partition. St000035The number of left outer peaks of a permutation. St000141The maximum drop size of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000223The number of nestings in the permutation. St000359The number of occurrences of the pattern 23-1. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000731The number of double exceedences of a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000701The protection number of a binary tree. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000913The number of ways to refine the partition into singletons. St000031The number of cycles in the cycle decomposition of a permutation. St000153The number of adjacent cycles of a permutation. St001092The number of distinct even parts of a partition. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000161The sum of the sizes of the right subtrees of a binary tree. St000097The order of the largest clique of the graph. St001581The achromatic number of a graph. St000012The area of a Dyck path. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000678The number of up steps after the last double rise of a Dyck path. St000744The length of the path to the largest entry in a standard Young tableau. St000984The number of boxes below precisely one peak. St001486The number of corners of the ribbon associated with an integer composition. St000676The number of odd rises of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001462The number of factors of a standard tableaux under concatenation. St001584The area statistic between a Dyck path and its bounce path. St000521The number of distinct subtrees of an ordered tree. St000381The largest part of an integer composition. St000142The number of even parts of a partition. St001058The breadth of the ordered tree. St000071The number of maximal chains in a poset. St000808The number of up steps of the associated bargraph. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000098The chromatic number of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000246The number of non-inversions of a permutation. St000883The number of longest increasing subsequences of a permutation. St000670The reversal length of a permutation. St000001The number of reduced words for a permutation. St000214The number of adjacencies of a permutation. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000843The decomposition number of a perfect matching. St001394The genus of a permutation. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001622The number of join-irreducible elements of a lattice. St001820The size of the image of the pop stack sorting operator. St000068The number of minimal elements in a poset. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001616The number of neutral elements in a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001651The Frankl number of a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001846The number of elements which do not have a complement in the lattice. St001619The number of non-isomorphic sublattices of a lattice. St001625The Möbius invariant of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St000874The position of the last double rise in a Dyck path. St000444The length of the maximal rise of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. 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. St000834The number of right outer peaks of a permutation. St000527The width of the poset. St000028The number of stack-sorts needed to sort a permutation. St000912The number of maximal antichains in a poset. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000528The height of a poset. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St001343The dimension of the reduced incidence algebra of a poset. St000211The rank of the set partition. St000234The number of global ascents of a permutation. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000632The jump number of the poset. St000829The Ulam distance of a permutation to the identity permutation. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001792The arboricity of a graph. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001116The game chromatic number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000537The cutwidth of a graph. St000539The number of odd inversions of a permutation. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000730The maximal arc length of a set partition. St000778The metric dimension of a graph. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000868The aid statistic in the sense of Shareshian-Wachs. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001261The Castelnuovo-Mumford regularity of a graph. St001270The bandwidth of a graph. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001592The maximal number of simple paths between any two different vertices of a graph. St001644The dimension of a graph. St001716The 1-improper chromatic number of a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001777The number of weak descents in an integer composition. St001812The biclique partition number of a graph. St001962The proper pathwidth of a graph. St000017The number of inversions of a standard tableau. St000025The number of initial rises of a Dyck path. St000095The number of triangles of a graph. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000273The domination number of a graph. St000360The number of occurrences of the pattern 32-1. St000382The first part of an integer composition. St000387The matching number of a graph. St000497The lcb statistic of a set partition. St000504The cardinality of the first block of a set partition. St000535The rank-width of a graph. St000544The cop number of a graph. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000624The normalized sum of the minimal distances to a greater element. St000822The Hadwiger number of the graph. St000916The packing number of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001062The maximal size of a block of a set partition. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001322The size of a minimal independent dominating set in a graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001363The Euler characteristic of a graph according to Knill. St001393The induced matching number of a graph. St001469The holeyness of a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001638The book thickness of a graph. St001665The number of pure excedances of a permutation. St001727The number of invisible inversions of a permutation. St001729The number of visible descents of a permutation. St001736The total number of cycles in a graph. St001737The number of descents of type 2 in a permutation. St001743The discrepancy of a graph. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001839The number of excedances of a set partition. St001883The mutual visibility number of a graph. St001928The number of non-overlapping descents in a permutation. 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. St000353The number of inner valleys of a permutation. St000779The tier of a permutation. St001057The Grundy value of the game of creating an independent set in a graph. St001829The common independence number of a graph. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St000643The size of the largest orbit of antichains under Panyushev complementation. St000758The length of the longest staircase fitting into an integer composition. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000201The number of leaf nodes in a binary tree. St000308The height of the tree associated to a permutation. St000454The largest eigenvalue of a graph if it is integral. St000989The number of final rises of a permutation. St000711The number of big exceedences of a permutation. St001330The hat guessing number of a graph. St000396The register function (or Horton-Strahler number) of a binary tree. St000919The number of maximal left branches of a binary tree. St000553The number of blocks of a graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001427The number of descents of a signed permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St000485The length of the longest cycle of a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St000065The number of entries equal to -1 in an alternating sign matrix. St000080The rank of the poset. St000099The number of valleys of a permutation, including the boundary. St000120The number of left tunnels of a Dyck path. St000168The number of internal nodes of an ordered tree. St000216The absolute length of a permutation. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000332The positive inversions of an alternating sign matrix. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000023The number of inner peaks of a permutation. St000056The decomposition (or block) number of a permutation. St000061The number of nodes on the left branch of a binary tree. St000237The number of small exceedances. St000293The number of inversions of a binary word. St000357The number of occurrences of the pattern 12-3. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001731The factorization defect of a permutation. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000092The number of outer peaks of a permutation. St000171The degree of the graph. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000204The number of internal nodes of a binary tree. St000224The sorting index of a permutation. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000450The number of edges minus the number of vertices plus 2 of a graph. St000482The (zero)-forcing number of a graph. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000706The product of the factorials of the multiplicities of an integer partition. St000741The Colin de Verdière graph invariant. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000897The number of different multiplicities of parts of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000948The chromatic discriminant of a graph. St000959The number of strong Bruhat factorizations of a permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000993The multiplicity of the largest part of an integer partition. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001153The number of blocks with even minimum in a set partition. St001220The width of a permutation. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001281The normalized isoperimetric number of a graph. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001345The Hamming dimension of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001480The number of simple summands of the module J^2/J^3. St001568The smallest positive integer that does not appear twice in the partition. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001689The number of celebrities in a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001734The lettericity of a graph. St001869The maximum cut size of a graph. St001949The rigidity index of a graph. St000039The number of crossings of a permutation. St000084The number of subtrees. St000087The number of induced subgraphs. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000166The depth minus 1 of an ordered tree. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000232The number of crossings of a set partition. St000235The number of indices that are not cyclical small weak excedances. St000240The number of indices that are not small excedances. St000242The number of indices that are not cyclical small weak excedances. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000328The maximum number of child nodes in a tree. St000338The number of pixed points of a permutation. St000344The number of strongly connected outdegree sequences of a graph. St000363The number of minimal vertex covers of a graph. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000379The number of Hamiltonian cycles in a graph. St000469The distinguishing number of a graph. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000636The hull number of a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000722The number of different neighbourhoods in a graph. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000740The last entry of a permutation. St000769The major index of a composition regarded as a word. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000807The sum of the heights of the valleys of the associated bargraph. St000926The clique-coclique number of a graph. St000929The constant term of the character polynomial of an integer partition. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001110The 3-dynamic chromatic number of a graph. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. 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. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001316The domatic number of a graph. St001342The number of vertices in the center of a graph. St001346The number of parking functions that give the same permutation. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001471The magnitude of a Dyck path. St001530The depth of a Dyck path. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001725The harmonious chromatic number of a graph. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001746The coalition number of a graph. St001797The number of overfull subgraphs of a graph. St001828The Euler characteristic of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001874Lusztig's a-function for the symmetric group. St000094The depth of an ordered tree. St000203The number of external nodes of a binary tree. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. 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. St001674The number of vertices of the largest induced star graph in the graph. St001706The number of closed sets in a graph. St000477The weight of a partition according to Alladi. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000707The product of the factorials of the parts. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000937The number of positive values of the symmetric group character corresponding to the partition. St000997The even-odd crank of an integer partition. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St000928The sum of the coefficients of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000124The cardinality of the preimage of the Simion-Schmidt map. St000312The number of leaves in a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000846The maximal number of elements covering an element of a poset. St000054The first entry of the permutation. St000352The Elizalde-Pak rank of a permutation. St001323The independence gap of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St000710The number of big deficiencies of a permutation. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St000358The number of occurrences of the pattern 31-2. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000290The major index of a binary word. St000296The length of the symmetric border of a binary word. St000297The number of leading ones in a binary word. St001597The Frobenius rank of a skew partition. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000746The number of pairs with odd minimum in a perfect matching. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St001152The number of pairs with even minimum in a perfect matching. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000347The inversion sum of a binary word. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St000570The Edelman-Greene number of a permutation. St000732The number of double deficiencies of a permutation. St000958The number of Bruhat factorizations of a permutation. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001046The maximal number of arcs nesting a given arc of a perfect matching. 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. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001668The number of points of the poset minus the width of the poset. St000146The Andrews-Garvan crank of a partition. St000177The number of free tiles in the pattern. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001430The number of positive entries in a signed permutation. St001470The cyclic holeyness of a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001856The number of edges in the reduced word graph of a permutation. St001871The number of triconnected components of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St000618The number of self-evacuating tableaux of given shape. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001780The order of promotion on the set of standard tableaux of given shape. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001525The number of symmetric hooks on the diagonal of a partition. St001561The value of the elementary symmetric function evaluated at 1. St001586The number of odd parts smaller than the largest even part in an integer partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000264The girth of a graph, which is not a tree. St000455The second largest eigenvalue of a graph if it is integral. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000845The maximal number of elements covered by an element in a poset. St001896The number of right descents of a signed permutations. St001863The number of weak excedances of a signed permutation. St001889The size of the connectivity set of a signed permutation. St000260The radius of a connected graph. St000259The diameter of a connected graph. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St000691The number of changes of a binary word. St000847The number of standard Young tableaux whose descent set is the binary word. St000295The length of the border of a binary word. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001060The distinguishing index of a graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001864The number of excedances of a signed permutation. St001769The reflection length of a signed permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St000640The rank of the largest boolean interval in a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001866The nesting alignments of a signed permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001488The number of corners of a skew partition. St001964The interval resolution global dimension of a poset. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St000942The number of critical left to right maxima of the parking functions. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000805The number of peaks of the associated bargraph. St000902 The minimal number of repetitions of an integer composition. St001267The length of the Lyndon factorization of the binary word. St001313The number of Dyck paths above the lattice path given by a binary word. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001884The number of borders of a binary word. St001905The number of preferred parking spots in a parking function less than the index of the car. St000348The non-inversion sum of a binary word. St000682The Grundy value of Welter's game on a binary word. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000904The maximal number of repetitions of an integer composition. St001423The number of distinct cubes in a binary word. St001485The modular major index of a binary word. St001589The nesting number of a perfect matching. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001404The number of distinct entries in a Gelfand Tsetlin pattern. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
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