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Your data matches 141 different statistics following compositions of up to 3 maps.
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Matching statistic: St000251
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
St000251: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> 1
{{1},{2}}
=> 0
{{1,2,3}}
=> 1
{{1,2},{3}}
=> 1
{{1,3},{2}}
=> 1
{{1},{2,3}}
=> 1
{{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> 1
{{1,2,3},{4}}
=> 1
{{1,2,4},{3}}
=> 1
{{1,2},{3,4}}
=> 2
{{1,2},{3},{4}}
=> 1
{{1,3,4},{2}}
=> 1
{{1,3},{2,4}}
=> 2
{{1,3},{2},{4}}
=> 1
{{1,4},{2,3}}
=> 2
{{1},{2,3,4}}
=> 1
{{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> 1
{{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> 1
{{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> 1
{{1,2,3,4},{5}}
=> 1
{{1,2,3,5},{4}}
=> 1
{{1,2,3},{4,5}}
=> 2
{{1,2,3},{4},{5}}
=> 1
{{1,2,4,5},{3}}
=> 1
{{1,2,4},{3,5}}
=> 2
{{1,2,4},{3},{5}}
=> 1
{{1,2,5},{3,4}}
=> 2
{{1,2},{3,4,5}}
=> 2
{{1,2},{3,4},{5}}
=> 2
{{1,2,5},{3},{4}}
=> 1
{{1,2},{3,5},{4}}
=> 2
{{1,2},{3},{4,5}}
=> 2
{{1,2},{3},{4},{5}}
=> 1
{{1,3,4,5},{2}}
=> 1
{{1,3,4},{2,5}}
=> 2
{{1,3,4},{2},{5}}
=> 1
{{1,3,5},{2,4}}
=> 2
{{1,3},{2,4,5}}
=> 2
{{1,3},{2,4},{5}}
=> 2
{{1,3,5},{2},{4}}
=> 1
{{1,3},{2,5},{4}}
=> 2
{{1,3},{2},{4,5}}
=> 2
{{1,3},{2},{4},{5}}
=> 1
{{1,4,5},{2,3}}
=> 2
{{1,4},{2,3,5}}
=> 2
{{1,4},{2,3},{5}}
=> 2
Description
The number of nonsingleton blocks of a set partition.
Matching statistic: St001280
Mp00079: Set partitions —shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> 1
{{1},{2}}
=> [1,1]
=> 0
{{1,2,3}}
=> [3]
=> 1
{{1,2},{3}}
=> [2,1]
=> 1
{{1,3},{2}}
=> [2,1]
=> 1
{{1},{2,3}}
=> [2,1]
=> 1
{{1},{2},{3}}
=> [1,1,1]
=> 0
{{1,2,3,4}}
=> [4]
=> 1
{{1,2,3},{4}}
=> [3,1]
=> 1
{{1,2,4},{3}}
=> [3,1]
=> 1
{{1,2},{3,4}}
=> [2,2]
=> 2
{{1,2},{3},{4}}
=> [2,1,1]
=> 1
{{1,3,4},{2}}
=> [3,1]
=> 1
{{1,3},{2,4}}
=> [2,2]
=> 2
{{1,3},{2},{4}}
=> [2,1,1]
=> 1
{{1,4},{2,3}}
=> [2,2]
=> 2
{{1},{2,3,4}}
=> [3,1]
=> 1
{{1},{2,3},{4}}
=> [2,1,1]
=> 1
{{1,4},{2},{3}}
=> [2,1,1]
=> 1
{{1},{2,4},{3}}
=> [2,1,1]
=> 1
{{1},{2},{3,4}}
=> [2,1,1]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> 0
{{1,2,3,4,5}}
=> [5]
=> 1
{{1,2,3,4},{5}}
=> [4,1]
=> 1
{{1,2,3,5},{4}}
=> [4,1]
=> 1
{{1,2,3},{4,5}}
=> [3,2]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> 1
{{1,2,4,5},{3}}
=> [4,1]
=> 1
{{1,2,4},{3,5}}
=> [3,2]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> 1
{{1,2,5},{3,4}}
=> [3,2]
=> 2
{{1,2},{3,4,5}}
=> [3,2]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> 2
{{1,2,5},{3},{4}}
=> [3,1,1]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> 2
{{1,2},{3},{4,5}}
=> [2,2,1]
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> 1
{{1,3,4},{2,5}}
=> [3,2]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> 1
{{1,3,5},{2,4}}
=> [3,2]
=> 2
{{1,3},{2,4,5}}
=> [3,2]
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> 2
{{1,3,5},{2},{4}}
=> [3,1,1]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> 2
{{1,3},{2},{4,5}}
=> [2,2,1]
=> 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> 2
{{1,4},{2,3,5}}
=> [3,2]
=> 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> 2
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000024
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ 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
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [1,0,1,0]
=> 0
{{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> 0
{{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
{{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
{{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
{{1,2,3,4,5}}
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
{{1,2,3,4},{5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,3},{4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,4},{3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,2,5},{3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,3,4},{2,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,3,5},{2,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,3},{2,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,4},{2,3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
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: St000291
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2] => 10 => 1
{{1},{2}}
=> [1,1] => 11 => 0
{{1,2,3}}
=> [3] => 100 => 1
{{1,2},{3}}
=> [2,1] => 101 => 1
{{1,3},{2}}
=> [2,1] => 101 => 1
{{1},{2,3}}
=> [1,2] => 110 => 1
{{1},{2},{3}}
=> [1,1,1] => 111 => 0
{{1,2,3,4}}
=> [4] => 1000 => 1
{{1,2,3},{4}}
=> [3,1] => 1001 => 1
{{1,2,4},{3}}
=> [3,1] => 1001 => 1
{{1,2},{3,4}}
=> [2,2] => 1010 => 2
{{1,2},{3},{4}}
=> [2,1,1] => 1011 => 1
{{1,3,4},{2}}
=> [3,1] => 1001 => 1
{{1,3},{2,4}}
=> [2,2] => 1010 => 2
{{1,3},{2},{4}}
=> [2,1,1] => 1011 => 1
{{1,4},{2,3}}
=> [2,2] => 1010 => 2
{{1},{2,3,4}}
=> [1,3] => 1100 => 1
{{1},{2,3},{4}}
=> [1,2,1] => 1101 => 1
{{1,4},{2},{3}}
=> [2,1,1] => 1011 => 1
{{1},{2,4},{3}}
=> [1,2,1] => 1101 => 1
{{1},{2},{3,4}}
=> [1,1,2] => 1110 => 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => 1111 => 0
{{1,2,3,4,5}}
=> [5] => 10000 => 1
{{1,2,3,4},{5}}
=> [4,1] => 10001 => 1
{{1,2,3,5},{4}}
=> [4,1] => 10001 => 1
{{1,2,3},{4,5}}
=> [3,2] => 10010 => 2
{{1,2,3},{4},{5}}
=> [3,1,1] => 10011 => 1
{{1,2,4,5},{3}}
=> [4,1] => 10001 => 1
{{1,2,4},{3,5}}
=> [3,2] => 10010 => 2
{{1,2,4},{3},{5}}
=> [3,1,1] => 10011 => 1
{{1,2,5},{3,4}}
=> [3,2] => 10010 => 2
{{1,2},{3,4,5}}
=> [2,3] => 10100 => 2
{{1,2},{3,4},{5}}
=> [2,2,1] => 10101 => 2
{{1,2,5},{3},{4}}
=> [3,1,1] => 10011 => 1
{{1,2},{3,5},{4}}
=> [2,2,1] => 10101 => 2
{{1,2},{3},{4,5}}
=> [2,1,2] => 10110 => 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => 10111 => 1
{{1,3,4,5},{2}}
=> [4,1] => 10001 => 1
{{1,3,4},{2,5}}
=> [3,2] => 10010 => 2
{{1,3,4},{2},{5}}
=> [3,1,1] => 10011 => 1
{{1,3,5},{2,4}}
=> [3,2] => 10010 => 2
{{1,3},{2,4,5}}
=> [2,3] => 10100 => 2
{{1,3},{2,4},{5}}
=> [2,2,1] => 10101 => 2
{{1,3,5},{2},{4}}
=> [3,1,1] => 10011 => 1
{{1,3},{2,5},{4}}
=> [2,2,1] => 10101 => 2
{{1,3},{2},{4,5}}
=> [2,1,2] => 10110 => 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => 10111 => 1
{{1,4,5},{2,3}}
=> [3,2] => 10010 => 2
{{1,4},{2,3,5}}
=> [2,3] => 10100 => 2
{{1,4},{2,3},{5}}
=> [2,2,1] => 10101 => 2
Description
The number of descents of a binary word.
Matching statistic: St000442
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ 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
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [1,0,1,0]
=> 0
{{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> 0
{{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
{{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
{{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
{{1,2,3,4,5}}
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
{{1,2,3,4},{5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,3},{4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,4},{3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,2,5},{3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,3,4},{2,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,3,5},{2,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,3},{2,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,4},{2,3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St000659
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2] => [1,1,0,0]
=> 1
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> 0
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> 1
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> 1
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> 1
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> 1
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> 0
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 1
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
Description
The number of rises of length at least 2 of a Dyck path.
Matching statistic: St000010
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [1,1]
=> 2 = 1 + 1
{{1},{2}}
=> [1,1]
=> [2]
=> 1 = 0 + 1
{{1,2,3}}
=> [3]
=> [2,1]
=> 2 = 1 + 1
{{1,2},{3}}
=> [2,1]
=> [2,1]
=> 2 = 1 + 1
{{1,3},{2}}
=> [2,1]
=> [2,1]
=> 2 = 1 + 1
{{1},{2,3}}
=> [2,1]
=> [2,1]
=> 2 = 1 + 1
{{1},{2},{3}}
=> [1,1,1]
=> [3]
=> 1 = 0 + 1
{{1,2,3,4}}
=> [4]
=> [3,1]
=> 2 = 1 + 1
{{1,2,3},{4}}
=> [3,1]
=> [2,2]
=> 2 = 1 + 1
{{1,2,4},{3}}
=> [3,1]
=> [2,2]
=> 2 = 1 + 1
{{1,2},{3,4}}
=> [2,2]
=> [2,1,1]
=> 3 = 2 + 1
{{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
{{1,3,4},{2}}
=> [3,1]
=> [2,2]
=> 2 = 1 + 1
{{1,3},{2,4}}
=> [2,2]
=> [2,1,1]
=> 3 = 2 + 1
{{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
{{1,4},{2,3}}
=> [2,2]
=> [2,1,1]
=> 3 = 2 + 1
{{1},{2,3,4}}
=> [3,1]
=> [2,2]
=> 2 = 1 + 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> 2 = 1 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [4]
=> 1 = 0 + 1
{{1,2,3,4,5}}
=> [5]
=> [4,1]
=> 2 = 1 + 1
{{1,2,3,4},{5}}
=> [4,1]
=> [3,2]
=> 2 = 1 + 1
{{1,2,3,5},{4}}
=> [4,1]
=> [3,2]
=> 2 = 1 + 1
{{1,2,3},{4,5}}
=> [3,2]
=> [2,2,1]
=> 3 = 2 + 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [3,2]
=> 2 = 1 + 1
{{1,2,4,5},{3}}
=> [4,1]
=> [3,2]
=> 2 = 1 + 1
{{1,2,4},{3,5}}
=> [3,2]
=> [2,2,1]
=> 3 = 2 + 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [3,2]
=> 2 = 1 + 1
{{1,2,5},{3,4}}
=> [3,2]
=> [2,2,1]
=> 3 = 2 + 1
{{1,2},{3,4,5}}
=> [3,2]
=> [2,2,1]
=> 3 = 2 + 1
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [3,1,1]
=> 3 = 2 + 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [3,2]
=> 2 = 1 + 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [3,1,1]
=> 3 = 2 + 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [3,1,1]
=> 3 = 2 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
{{1,3,4,5},{2}}
=> [4,1]
=> [3,2]
=> 2 = 1 + 1
{{1,3,4},{2,5}}
=> [3,2]
=> [2,2,1]
=> 3 = 2 + 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [3,2]
=> 2 = 1 + 1
{{1,3,5},{2,4}}
=> [3,2]
=> [2,2,1]
=> 3 = 2 + 1
{{1,3},{2,4,5}}
=> [3,2]
=> [2,2,1]
=> 3 = 2 + 1
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [3,1,1]
=> 3 = 2 + 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [3,2]
=> 2 = 1 + 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [3,1,1]
=> 3 = 2 + 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [3,1,1]
=> 3 = 2 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 2 = 1 + 1
{{1,4,5},{2,3}}
=> [3,2]
=> [2,2,1]
=> 3 = 2 + 1
{{1,4},{2,3,5}}
=> [3,2]
=> [2,2,1]
=> 3 = 2 + 1
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [3,1,1]
=> 3 = 2 + 1
Description
The length of the partition.
Matching statistic: St000013
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [1,0,1,0]
=> 1 = 0 + 1
{{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> 2 = 1 + 1
{{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
{{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
{{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
{{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
{{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2,3,4,5}}
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,3},{4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,4},{3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2,5},{3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3,4},{2,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,3,5},{2,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3},{2,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
{{1,4,5},{2,3}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,4},{2,3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000444
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [1,0,1,0]
=> 1 = 0 + 1
{{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> 2 = 1 + 1
{{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
{{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
{{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
{{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
{{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2,3,4,5}}
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,3},{4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,4},{3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2,5},{3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3,4},{2,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,3,5},{2,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3},{2,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
{{1,4,5},{2,3}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,4},{2,3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St001007
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [1,0,1,0]
=> 1 = 0 + 1
{{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> 2 = 1 + 1
{{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
{{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
{{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
{{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> 3 = 2 + 1
{{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2,3,4,5}}
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
{{1,2,3,4},{5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,3},{4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,4},{3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2,5},{3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3,4},{2,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,3,5},{2,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3},{2,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2 = 1 + 1
{{1,4,5},{2,3}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,4},{2,3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
The following 131 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001471The magnitude of a Dyck path. St000053The number of valleys of the Dyck path. St000147The largest part of an integer partition. St000157The number of descents of a standard tableau. St000211The rank of the set partition. St000288The number of ones in a binary word. St000292The number of ascents of a binary word. St000306The bounce count of a Dyck path. St000374The number of exclusive right-to-left minima of a permutation. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000662The staircase size of the code of a permutation. St000665The number of rafts of a permutation. St000703The number of deficiencies of a permutation. St000834The number of right outer peaks of a permutation. St000919The number of maximal left branches of a binary tree. St001011Number of simple modules of projective dimension 2 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 $eA$. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St000381The largest part of an integer composition. St000451The length of the longest pattern of the form k 1 2. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000808The number of up steps of the associated bargraph. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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:
St000521The number of distinct subtrees of an ordered tree. 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. St000354The number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001489The maximum of the number of descents and the number of inverse descents. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000470The number of runs in a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001354The number of series nodes in the modular decomposition of a graph. St000668The least common multiple of the parts of the partition. St000141The maximum drop size of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000021The number of descents of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. 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. St000316The number of non-left-to-right-maxima of a permutation. St000331The number of upper interactions of a Dyck path. St000332The positive inversions of an alternating sign matrix. St001046The maximal number of arcs nesting a given arc of a perfect matching. 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. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000308The height of the tree associated to a permutation. St000325The width of the tree associated to a permutation. St000392The length of the longest run of ones in a binary word. St000702The number of weak deficiencies of a permutation. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000982The length of the longest constant subword. St000991The number of right-to-left minima of a permutation. St001372The length of a longest cyclic run of ones of a binary word. St000094The depth of an ordered tree. St000539The number of odd inversions of a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St000259The diameter of a connected graph. St000884The number of isolated descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000260The radius of a connected graph. St000670The reversal length of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St001589The nesting number of a perfect matching. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001665The number of pure excedances of a permutation. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St000035The number of left outer peaks of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001427The number of descents of a signed permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. 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. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001874Lusztig's a-function for the symmetric group. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001394The genus of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000023The number of inner peaks of a permutation. St000710The number of big deficiencies of a permutation. St000779The tier of a permutation. St000099The number of valleys of a permutation, including the boundary. St000455The second largest eigenvalue of a graph if it is integral. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001905The number of preferred parking spots in a parking function less than the index of the car. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001946The number of descents in a parking function.
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