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Your data matches 39 different statistics following compositions of up to 3 maps.
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Matching statistic: St000297
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: 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
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => 1 => 1
{{1,2}}
=> [2] => 10 => 1
{{1},{2}}
=> [1,1] => 11 => 2
{{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 => 2
{{1},{2},{3}}
=> [1,1,1] => 111 => 3
{{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 => 1
{{1,2},{3},{4}}
=> [2,1,1] => 1011 => 1
{{1,3,4},{2}}
=> [3,1] => 1001 => 1
{{1,3},{2,4}}
=> [2,2] => 1010 => 1
{{1,3},{2},{4}}
=> [2,1,1] => 1011 => 1
{{1,4},{2,3}}
=> [2,2] => 1010 => 1
{{1},{2,3,4}}
=> [1,3] => 1100 => 2
{{1},{2,3},{4}}
=> [1,2,1] => 1101 => 2
{{1,4},{2},{3}}
=> [2,1,1] => 1011 => 1
{{1},{2,4},{3}}
=> [1,2,1] => 1101 => 2
{{1},{2},{3,4}}
=> [1,1,2] => 1110 => 3
{{1},{2},{3},{4}}
=> [1,1,1,1] => 1111 => 4
{{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 => 1
{{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 => 1
{{1,2,4},{3},{5}}
=> [3,1,1] => 10011 => 1
{{1,2,5},{3,4}}
=> [3,2] => 10010 => 1
{{1,2},{3,4,5}}
=> [2,3] => 10100 => 1
{{1,2},{3,4},{5}}
=> [2,2,1] => 10101 => 1
{{1,2,5},{3},{4}}
=> [3,1,1] => 10011 => 1
{{1,2},{3,5},{4}}
=> [2,2,1] => 10101 => 1
{{1,2},{3},{4,5}}
=> [2,1,2] => 10110 => 1
{{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 => 1
{{1,3,4},{2},{5}}
=> [3,1,1] => 10011 => 1
{{1,3,5},{2,4}}
=> [3,2] => 10010 => 1
{{1,3},{2,4,5}}
=> [2,3] => 10100 => 1
{{1,3},{2,4},{5}}
=> [2,2,1] => 10101 => 1
{{1,3,5},{2},{4}}
=> [3,1,1] => 10011 => 1
{{1,3},{2,5},{4}}
=> [2,2,1] => 10101 => 1
{{1,3},{2},{4,5}}
=> [2,1,2] => 10110 => 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => 10111 => 1
{{1,4,5},{2,3}}
=> [3,2] => 10010 => 1
{{1,4},{2,3,5}}
=> [2,3] => 10100 => 1
Description
The number of leading ones in a binary word.
Matching statistic: St000382
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00039: Integer compositions —complement⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 1
{{1,2}}
=> [2] => [1,1] => 1
{{1},{2}}
=> [1,1] => [2] => 2
{{1,2,3}}
=> [3] => [1,1,1] => 1
{{1,2},{3}}
=> [2,1] => [1,2] => 1
{{1,3},{2}}
=> [2,1] => [1,2] => 1
{{1},{2,3}}
=> [1,2] => [2,1] => 2
{{1},{2},{3}}
=> [1,1,1] => [3] => 3
{{1,2,3,4}}
=> [4] => [1,1,1,1] => 1
{{1,2,3},{4}}
=> [3,1] => [1,1,2] => 1
{{1,2,4},{3}}
=> [3,1] => [1,1,2] => 1
{{1,2},{3,4}}
=> [2,2] => [1,2,1] => 1
{{1,2},{3},{4}}
=> [2,1,1] => [1,3] => 1
{{1,3,4},{2}}
=> [3,1] => [1,1,2] => 1
{{1,3},{2,4}}
=> [2,2] => [1,2,1] => 1
{{1,3},{2},{4}}
=> [2,1,1] => [1,3] => 1
{{1,4},{2,3}}
=> [2,2] => [1,2,1] => 1
{{1},{2,3,4}}
=> [1,3] => [2,1,1] => 2
{{1},{2,3},{4}}
=> [1,2,1] => [2,2] => 2
{{1,4},{2},{3}}
=> [2,1,1] => [1,3] => 1
{{1},{2,4},{3}}
=> [1,2,1] => [2,2] => 2
{{1},{2},{3,4}}
=> [1,1,2] => [3,1] => 3
{{1},{2},{3},{4}}
=> [1,1,1,1] => [4] => 4
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1] => 1
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,2] => 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,2] => 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,2,1] => 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,3] => 1
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,2] => 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,2,1] => 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,3] => 1
{{1,2,5},{3,4}}
=> [3,2] => [1,1,2,1] => 1
{{1,2},{3,4,5}}
=> [2,3] => [1,2,1,1] => 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,2,2] => 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,3] => 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,2,2] => 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,3,1] => 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,4] => 1
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,2] => 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,2,1] => 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,3] => 1
{{1,3,5},{2,4}}
=> [3,2] => [1,1,2,1] => 1
{{1,3},{2,4,5}}
=> [2,3] => [1,2,1,1] => 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,2,2] => 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,3] => 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,2,2] => 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,3,1] => 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,4] => 1
{{1,4,5},{2,3}}
=> [3,2] => [1,1,2,1] => 1
{{1,4},{2,3,5}}
=> [2,3] => [1,2,1,1] => 1
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 1
{{1,2}}
=> [2] => [1,1] => 1
{{1},{2}}
=> [1,1] => [2] => 2
{{1,2,3}}
=> [3] => [1,1,1] => 1
{{1,2},{3}}
=> [2,1] => [2,1] => 1
{{1,3},{2}}
=> [2,1] => [2,1] => 1
{{1},{2,3}}
=> [1,2] => [1,2] => 2
{{1},{2},{3}}
=> [1,1,1] => [3] => 3
{{1,2,3,4}}
=> [4] => [1,1,1,1] => 1
{{1,2,3},{4}}
=> [3,1] => [2,1,1] => 1
{{1,2,4},{3}}
=> [3,1] => [2,1,1] => 1
{{1,2},{3,4}}
=> [2,2] => [1,2,1] => 1
{{1,2},{3},{4}}
=> [2,1,1] => [3,1] => 1
{{1,3,4},{2}}
=> [3,1] => [2,1,1] => 1
{{1,3},{2,4}}
=> [2,2] => [1,2,1] => 1
{{1,3},{2},{4}}
=> [2,1,1] => [3,1] => 1
{{1,4},{2,3}}
=> [2,2] => [1,2,1] => 1
{{1},{2,3,4}}
=> [1,3] => [1,1,2] => 2
{{1},{2,3},{4}}
=> [1,2,1] => [2,2] => 2
{{1,4},{2},{3}}
=> [2,1,1] => [3,1] => 1
{{1},{2,4},{3}}
=> [1,2,1] => [2,2] => 2
{{1},{2},{3,4}}
=> [1,1,2] => [1,3] => 3
{{1},{2},{3},{4}}
=> [1,1,1,1] => [4] => 4
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1] => 1
{{1,2,3,4},{5}}
=> [4,1] => [2,1,1,1] => 1
{{1,2,3,5},{4}}
=> [4,1] => [2,1,1,1] => 1
{{1,2,3},{4,5}}
=> [3,2] => [1,2,1,1] => 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [3,1,1] => 1
{{1,2,4,5},{3}}
=> [4,1] => [2,1,1,1] => 1
{{1,2,4},{3,5}}
=> [3,2] => [1,2,1,1] => 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [3,1,1] => 1
{{1,2,5},{3,4}}
=> [3,2] => [1,2,1,1] => 1
{{1,2},{3,4,5}}
=> [2,3] => [1,1,2,1] => 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [2,2,1] => 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [3,1,1] => 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [2,2,1] => 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,3,1] => 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [4,1] => 1
{{1,3,4,5},{2}}
=> [4,1] => [2,1,1,1] => 1
{{1,3,4},{2,5}}
=> [3,2] => [1,2,1,1] => 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [3,1,1] => 1
{{1,3,5},{2,4}}
=> [3,2] => [1,2,1,1] => 1
{{1,3},{2,4,5}}
=> [2,3] => [1,1,2,1] => 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [2,2,1] => 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [3,1,1] => 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [2,2,1] => 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,3,1] => 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [4,1] => 1
{{1,4,5},{2,3}}
=> [3,2] => [1,2,1,1] => 1
{{1,4},{2,3,5}}
=> [2,3] => [1,1,2,1] => 1
Description
The last part of an integer composition.
Matching statistic: St000025
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1,0]
=> 1
{{1,2}}
=> [2] => [1,1] => [1,0,1,0]
=> 1
{{1},{2}}
=> [1,1] => [2] => [1,1,0,0]
=> 2
{{1,2,3}}
=> [3] => [1,1,1] => [1,0,1,0,1,0]
=> 1
{{1,2},{3}}
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
{{1,3},{2}}
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
{{1},{2,3}}
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
{{1},{2},{3}}
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 3
{{1,2,3,4}}
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
{{1,2,3},{4}}
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
{{1,2,4},{3}}
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
{{1,2},{3,4}}
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
{{1,2},{3},{4}}
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,3,4},{2}}
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
{{1,3},{2,4}}
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
{{1,3},{2},{4}}
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,4},{2,3}}
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
{{1},{2,3,4}}
=> [1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
{{1},{2,3},{4}}
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1,4},{2},{3}}
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1},{2,4},{3}}
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1},{2},{3,4}}
=> [1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
{{1},{2},{3},{4}}
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 4
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
{{1,2,5},{3,4}}
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
{{1,2},{3,4,5}}
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
{{1,3,5},{2,4}}
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
{{1,3},{2,4,5}}
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,4,5},{2,3}}
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
{{1,4},{2,3,5}}
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000026
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1,0]
=> 1
{{1,2}}
=> [2] => [1,1] => [1,0,1,0]
=> 1
{{1},{2}}
=> [1,1] => [2] => [1,1,0,0]
=> 2
{{1,2,3}}
=> [3] => [1,1,1] => [1,0,1,0,1,0]
=> 1
{{1,2},{3}}
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
{{1,3},{2}}
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
{{1},{2,3}}
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
{{1},{2},{3}}
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 3
{{1,2,3,4}}
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
{{1,2,3},{4}}
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
{{1,2,4},{3}}
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
{{1,2},{3,4}}
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
{{1,2},{3},{4}}
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,3,4},{2}}
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
{{1,3},{2,4}}
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
{{1,3},{2},{4}}
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,4},{2,3}}
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
{{1},{2,3,4}}
=> [1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
{{1},{2,3},{4}}
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1,4},{2},{3}}
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1},{2,4},{3}}
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
{{1},{2},{3,4}}
=> [1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 3
{{1},{2},{3},{4}}
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 4
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
{{1,2,5},{3,4}}
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
{{1,2},{3,4,5}}
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
{{1,3,5},{2,4}}
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
{{1,3},{2,4,5}}
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,4,5},{2,3}}
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
{{1,4},{2,3,5}}
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
Description
The position of the first return of a Dyck path.
Matching statistic: St000759
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000759: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000759: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> []
=> 1
{{1,2}}
=> [2] => [1,1,0,0]
=> []
=> 1
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [1]
=> 2
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> []
=> 1
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [2]
=> 1
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [2]
=> 1
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [1,1]
=> 2
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [2,1]
=> 3
{{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]
=> [3]
=> 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 2
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 2
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 3
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 4
{{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]
=> [4]
=> 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 1
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 1
Description
The smallest missing part in an integer partition.
In [3], this is referred to as the mex, the minimal excluded part of the partition.
For compositions, this is studied in [sec.3.2., 1].
Matching statistic: St001135
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001135: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001135: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1,0]
=> 1
{{1,2}}
=> [2] => [2] => [1,1,0,0]
=> 1
{{1},{2}}
=> [1,1] => [1,1] => [1,0,1,0]
=> 2
{{1,2,3}}
=> [3] => [3] => [1,1,1,0,0,0]
=> 1
{{1,2},{3}}
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
{{1,3},{2}}
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
{{1},{2,3}}
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 2
{{1},{2},{3}}
=> [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
{{1,2,3,4}}
=> [4] => [4] => [1,1,1,1,0,0,0,0]
=> 1
{{1,2,3},{4}}
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,2,4},{3}}
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,2},{3,4}}
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
{{1,3,4},{2}}
=> [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1,3},{2,4}}
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
{{1,4},{2,3}}
=> [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
{{1},{2,3,4}}
=> [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
{{1},{2,3},{4}}
=> [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
{{1},{2,4},{3}}
=> [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
{{1},{2},{3,4}}
=> [1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
{{1,2,3,4,5}}
=> [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
{{1,2,3,4},{5}}
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,3,5},{4}}
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,3},{4,5}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
{{1,2,4,5},{3}}
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,2,4},{3,5}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
{{1,2,5},{3,4}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
{{1,2},{3,4,5}}
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,3,4,5},{2}}
=> [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
{{1,3,4},{2,5}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
{{1,3,5},{2,4}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
{{1,3},{2,4,5}}
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
{{1,4,5},{2,3}}
=> [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
{{1,4},{2,3,5}}
=> [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
Description
The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001733
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001733: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001733: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> [1,0]
=> 1
{{1,2}}
=> [2] => [1,1,0,0]
=> [1,1,0,0]
=> 1
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [1,0,1,0]
=> 2
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 2
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [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]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [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]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [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]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [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]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [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]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [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]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [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]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [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]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
Description
The number of weak left to right maxima of a Dyck path.
A weak left to right maximum is a peak whose height is larger than or equal to the height of all peaks to its
left.
Matching statistic: St000326
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000326: 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
Mp00105: Binary words —complement⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => 1 => 0 => 2 = 1 + 1
{{1,2}}
=> [2] => 10 => 01 => 2 = 1 + 1
{{1},{2}}
=> [1,1] => 11 => 00 => 3 = 2 + 1
{{1,2,3}}
=> [3] => 100 => 011 => 2 = 1 + 1
{{1,2},{3}}
=> [2,1] => 101 => 010 => 2 = 1 + 1
{{1,3},{2}}
=> [2,1] => 101 => 010 => 2 = 1 + 1
{{1},{2,3}}
=> [1,2] => 110 => 001 => 3 = 2 + 1
{{1},{2},{3}}
=> [1,1,1] => 111 => 000 => 4 = 3 + 1
{{1,2,3,4}}
=> [4] => 1000 => 0111 => 2 = 1 + 1
{{1,2,3},{4}}
=> [3,1] => 1001 => 0110 => 2 = 1 + 1
{{1,2,4},{3}}
=> [3,1] => 1001 => 0110 => 2 = 1 + 1
{{1,2},{3,4}}
=> [2,2] => 1010 => 0101 => 2 = 1 + 1
{{1,2},{3},{4}}
=> [2,1,1] => 1011 => 0100 => 2 = 1 + 1
{{1,3,4},{2}}
=> [3,1] => 1001 => 0110 => 2 = 1 + 1
{{1,3},{2,4}}
=> [2,2] => 1010 => 0101 => 2 = 1 + 1
{{1,3},{2},{4}}
=> [2,1,1] => 1011 => 0100 => 2 = 1 + 1
{{1,4},{2,3}}
=> [2,2] => 1010 => 0101 => 2 = 1 + 1
{{1},{2,3,4}}
=> [1,3] => 1100 => 0011 => 3 = 2 + 1
{{1},{2,3},{4}}
=> [1,2,1] => 1101 => 0010 => 3 = 2 + 1
{{1,4},{2},{3}}
=> [2,1,1] => 1011 => 0100 => 2 = 1 + 1
{{1},{2,4},{3}}
=> [1,2,1] => 1101 => 0010 => 3 = 2 + 1
{{1},{2},{3,4}}
=> [1,1,2] => 1110 => 0001 => 4 = 3 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => 1111 => 0000 => 5 = 4 + 1
{{1,2,3,4,5}}
=> [5] => 10000 => 01111 => 2 = 1 + 1
{{1,2,3,4},{5}}
=> [4,1] => 10001 => 01110 => 2 = 1 + 1
{{1,2,3,5},{4}}
=> [4,1] => 10001 => 01110 => 2 = 1 + 1
{{1,2,3},{4,5}}
=> [3,2] => 10010 => 01101 => 2 = 1 + 1
{{1,2,3},{4},{5}}
=> [3,1,1] => 10011 => 01100 => 2 = 1 + 1
{{1,2,4,5},{3}}
=> [4,1] => 10001 => 01110 => 2 = 1 + 1
{{1,2,4},{3,5}}
=> [3,2] => 10010 => 01101 => 2 = 1 + 1
{{1,2,4},{3},{5}}
=> [3,1,1] => 10011 => 01100 => 2 = 1 + 1
{{1,2,5},{3,4}}
=> [3,2] => 10010 => 01101 => 2 = 1 + 1
{{1,2},{3,4,5}}
=> [2,3] => 10100 => 01011 => 2 = 1 + 1
{{1,2},{3,4},{5}}
=> [2,2,1] => 10101 => 01010 => 2 = 1 + 1
{{1,2,5},{3},{4}}
=> [3,1,1] => 10011 => 01100 => 2 = 1 + 1
{{1,2},{3,5},{4}}
=> [2,2,1] => 10101 => 01010 => 2 = 1 + 1
{{1,2},{3},{4,5}}
=> [2,1,2] => 10110 => 01001 => 2 = 1 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => 10111 => 01000 => 2 = 1 + 1
{{1,3,4,5},{2}}
=> [4,1] => 10001 => 01110 => 2 = 1 + 1
{{1,3,4},{2,5}}
=> [3,2] => 10010 => 01101 => 2 = 1 + 1
{{1,3,4},{2},{5}}
=> [3,1,1] => 10011 => 01100 => 2 = 1 + 1
{{1,3,5},{2,4}}
=> [3,2] => 10010 => 01101 => 2 = 1 + 1
{{1,3},{2,4,5}}
=> [2,3] => 10100 => 01011 => 2 = 1 + 1
{{1,3},{2,4},{5}}
=> [2,2,1] => 10101 => 01010 => 2 = 1 + 1
{{1,3,5},{2},{4}}
=> [3,1,1] => 10011 => 01100 => 2 = 1 + 1
{{1,3},{2,5},{4}}
=> [2,2,1] => 10101 => 01010 => 2 = 1 + 1
{{1,3},{2},{4,5}}
=> [2,1,2] => 10110 => 01001 => 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => 10111 => 01000 => 2 = 1 + 1
{{1,4,5},{2,3}}
=> [3,2] => 10010 => 01101 => 2 = 1 + 1
{{1,4},{2,3,5}}
=> [2,3] => 10100 => 01011 => 2 = 1 + 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000439
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [1,0]
=> 2 = 1 + 1
{{1,2}}
=> [2] => [1,1] => [1,0,1,0]
=> 2 = 1 + 1
{{1},{2}}
=> [1,1] => [2] => [1,1,0,0]
=> 3 = 2 + 1
{{1,2,3}}
=> [3] => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 1 + 1
{{1,2},{3}}
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3},{2}}
=> [2,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 1 + 1
{{1},{2,3}}
=> [1,2] => [2,1] => [1,1,0,0,1,0]
=> 3 = 2 + 1
{{1},{2},{3}}
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> 4 = 3 + 1
{{1,2,3,4}}
=> [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
{{1,2,3},{4}}
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,4},{3}}
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2},{3,4}}
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
{{1,2},{3},{4}}
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
{{1,3,4},{2}}
=> [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3},{2,4}}
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
{{1,3},{2},{4}}
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
{{1,4},{2,3}}
=> [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
{{1},{2,3,4}}
=> [1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
{{1},{2,3},{4}}
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
{{1,4},{2},{3}}
=> [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
{{1},{2,4},{3}}
=> [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
{{1},{2},{3,4}}
=> [1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,3},{4,5}}
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,4},{3,5}}
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
{{1,2,5},{3,4}}
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
{{1,2},{3,4,5}}
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3,4},{2,5}}
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
{{1,3,5},{2,4}}
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
{{1,3},{2,4,5}}
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
{{1,4,5},{2,3}}
=> [3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
{{1,4},{2,3,5}}
=> [2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
Description
The position of the first down step of a Dyck path.
The following 29 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000729The minimal arc length of a set partition. St000678The number of up steps after the last double rise of a Dyck path. St000363The number of minimal vertex covers of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001829The common independence number of a graph. St001316The domatic number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St000501The size of the first part in the decomposition of a permutation. St000617The number of global maxima of a Dyck path. St000546The number of global descents of a permutation. St000654The first descent of a permutation. St000989The number of final rises of a permutation. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000335The difference of lower and upper interactions. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001481The minimal height of a peak of a Dyck path. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex 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. St000054The first entry of the permutation. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function.
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