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Your data matches 39 different statistics following compositions of up to 3 maps.
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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: 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
{{1},{2,3,4,5,6,7,8,9,10,11}}
=> [1,10] => [2,1,1,1,1,1,1,1,1,1] => ? = 2
Description
The first part of an integer composition.
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
{{1},{2},{3},{4},{5},{6,7,8}}
=> [1,1,1,1,1,3] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 + 1
{{1},{2},{3},{4},{5,6,7,8}}
=> [1,1,1,1,4] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 5 + 1
{{1},{2},{3},{4},{5},{6},{7,8,9}}
=> [1,1,1,1,1,1,3] => [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 7 + 1
{{1},{2,3,4,5,6,7,8,9,10,11}}
=> [1,10] => [2,1,1,1,1,1,1,1,1,1] => ?
=> ? = 2 + 1
{{1,2,3,4,5,6,7,8,9}}
=> [9] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
{{1,2,3,4,5,6,7,8,9,10}}
=> [10] => [1,1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1 + 1
Description
The position of the first down step of a Dyck path.
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: 99% ●values known / values provided: 99%●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: 99% ●values known / values provided: 99%●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
{{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => 0111111110 => ? = 1 + 1
{{1},{2,3,4,5,6,7,8,9,10}}
=> [1,9] => 1100000000 => 0011111111 => ? = 2 + 1
{{1,2,3,4,5,6,7,8,9,10},{11}}
=> [10,1] => 10000000001 => 01111111110 => ? = 1 + 1
{{1},{2,3,4,5,6,7,8,9,10,11}}
=> [1,10] => 11000000000 => 00111111111 => ? = 2 + 1
{{1,3,4,5,6,7,8,9,10},{2}}
=> [9,1] => 1000000001 => 0111111110 => ? = 1 + 1
{{1,2,3,4,5,6,7,8,10},{9}}
=> [9,1] => 1000000001 => 0111111110 => ? = 1 + 1
{{1,2,3,4,5,6,7,8,9,11},{10}}
=> [10,1] => 10000000001 => 01111111110 => ? = 1 + 1
{{1,2,3,5,6,7,8,9,10},{4}}
=> [9,1] => 1000000001 => 0111111110 => ? = 1 + 1
{{1,2,3,4,6,7,8,9,10},{5}}
=> [9,1] => 1000000001 => 0111111110 => ? = 1 + 1
{{1,3,4,5,6,7,8,9,10,11},{2}}
=> [10,1] => 10000000001 => 01111111110 => ? = 1 + 1
{{1,2,3,4,5,6,7,9,10},{8}}
=> [9,1] => 1000000001 => 0111111110 => ? = 1 + 1
{{1,2,3,4,5,6,8,9,10},{7}}
=> [9,1] => 1000000001 => 0111111110 => ? = 1 + 1
{{1,2,3,4,5,7,8,9,10},{6}}
=> [9,1] => 1000000001 => 0111111110 => ? = 1 + 1
{{1,2,4,5,6,7,8,9,10},{3}}
=> [9,1] => 1000000001 => 0111111110 => ? = 1 + 1
{{1,2,3,4,5,6,7,8,10,11},{9}}
=> [10,1] => 10000000001 => 01111111110 => ? = 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,…,n,n+1} that contains n+1, this is the minimal element of the set.
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: 90% ●values known / values provided: 99%●distinct values known / distinct values provided: 90%
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 90% ●values known / values provided: 99%●distinct values known / distinct values provided: 90%
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
{{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [1,1,1,1,1,1,1,1,1,1] => 1111111111 => ? = 10
{{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => 1000000001 => ? = 1
{{1},{2,3,4,5,6,7,8,9,10}}
=> [1,9] => 1100000000 => ? = 2
{{1,2,3,4,5,6,7,8,9,10},{11}}
=> [10,1] => 10000000001 => ? = 1
{{1},{2,3,4,5,6,7,8,9,10,11}}
=> [1,10] => 11000000000 => ? = 2
{{1,3,4,5,6,7,8,9,10},{2}}
=> [9,1] => 1000000001 => ? = 1
{{1,2,3,4,5,6,7,8,10},{9}}
=> [9,1] => 1000000001 => ? = 1
{{1,2,3,4,5,6,7,8,9,11},{10}}
=> [10,1] => 10000000001 => ? = 1
{{1,2,3,5,6,7,8,9,10},{4}}
=> [9,1] => 1000000001 => ? = 1
{{1,2,3,4,6,7,8,9,10},{5}}
=> [9,1] => 1000000001 => ? = 1
{{1,3,4,5,6,7,8,9,10,11},{2}}
=> [10,1] => 10000000001 => ? = 1
{{1,2,3,4,5,6,7,9,10},{8}}
=> [9,1] => 1000000001 => ? = 1
{{1,2,3,4,5,6,8,9,10},{7}}
=> [9,1] => 1000000001 => ? = 1
{{1,2,3,4,5,7,8,9,10},{6}}
=> [9,1] => 1000000001 => ? = 1
{{1,2,4,5,6,7,8,9,10},{3}}
=> [9,1] => 1000000001 => ? = 1
{{1,2,3,4,5,6,7,8,10,11},{9}}
=> [10,1] => 10000000001 => ? = 1
Description
The number of leading ones in a binary word.
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: 90% ●values known / values provided: 97%●distinct values known / distinct values provided: 90%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 90% ●values known / values provided: 97%●distinct values known / distinct values provided: 90%
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
{{1},{2},{3},{4},{5},{6},{7},{8}}
=> [1,1,1,1,1,1,1,1] => [8] => ? = 8
{{1},{2},{3},{4},{5},{6,7,8}}
=> [1,1,1,1,1,3] => [1,1,6] => ? = 6
{{1},{2},{3},{4},{5,6,7,8}}
=> [1,1,1,1,4] => [1,1,1,5] => ? = 5
{{1},{2,3,4,5,6,7,8}}
=> [1,7] => [1,1,1,1,1,1,2] => ? = 2
{{1,3,4,5,6,7,8},{2}}
=> [7,1] => [2,1,1,1,1,1,1] => ? = 1
{{1,2,4,5,6,7,8},{3}}
=> [7,1] => [2,1,1,1,1,1,1] => ? = 1
{{1,2,3,5,6,7,8},{4}}
=> [7,1] => [2,1,1,1,1,1,1] => ? = 1
{{1,2,3,4,6,7,8},{5}}
=> [7,1] => [2,1,1,1,1,1,1] => ? = 1
{{1,2,3,4,5,7,8},{6}}
=> [7,1] => [2,1,1,1,1,1,1] => ? = 1
{{1,2,3,4,5,6,7},{8}}
=> [7,1] => [2,1,1,1,1,1,1] => ? = 1
{{1,2,3,4,5,6,8},{7}}
=> [7,1] => [2,1,1,1,1,1,1] => ? = 1
{{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> [1,1,1,1,1,1,1,1,1] => [9] => ? = 9
{{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [1,1,1,1,1,1,1,1,1,1] => [10] => ? = 10
{{1},{2},{3},{4},{5},{6},{7,8,9}}
=> [1,1,1,1,1,1,3] => [1,1,7] => ? = 7
{{1,2,3,4,5,6,7,8},{9}}
=> [8,1] => [2,1,1,1,1,1,1,1] => ? = 1
{{1},{2,3,4,5,6,7,8,9}}
=> [1,8] => [1,1,1,1,1,1,1,2] => ? = 2
{{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => [2,1,1,1,1,1,1,1,1] => ? = 1
{{1},{2,3,4,5,6,7,8,9,10}}
=> [1,9] => [1,1,1,1,1,1,1,1,2] => ? = 2
{{1,2,3,4,5,6,7,8,9,10},{11}}
=> [10,1] => [2,1,1,1,1,1,1,1,1,1] => ? = 1
{{1},{2,3,4,5,6,7,8,9,10,11}}
=> [1,10] => [1,1,1,1,1,1,1,1,1,2] => ? = 2
{{1,2,3,4,5,6,7,8,9}}
=> [9] => [1,1,1,1,1,1,1,1,1] => ? = 1
{{1,3,4,5,6,7,8,9},{2}}
=> [8,1] => [2,1,1,1,1,1,1,1] => ? = 1
{{1,3,4,5,6,7,8,9,10},{2}}
=> [9,1] => [2,1,1,1,1,1,1,1,1] => ? = 1
{{1,2,3,4,5,6,7,9},{8}}
=> [8,1] => [2,1,1,1,1,1,1,1] => ? = 1
{{1,2,3,4,5,6,7,8,10},{9}}
=> [9,1] => [2,1,1,1,1,1,1,1,1] => ? = 1
{{1,2,3,4,5,6,7,8,9,11},{10}}
=> [10,1] => [2,1,1,1,1,1,1,1,1,1] => ? = 1
{{1,2,3,5,6,7,8,9,10},{4}}
=> [9,1] => [2,1,1,1,1,1,1,1,1] => ? = 1
{{1,2,3,4,6,7,8,9,10},{5}}
=> [9,1] => [2,1,1,1,1,1,1,1,1] => ? = 1
{{1,3,4,5,6,7,8,9,10,11},{2}}
=> [10,1] => [2,1,1,1,1,1,1,1,1,1] => ? = 1
{{1,2,3,4,5,6,8,9},{7}}
=> [8,1] => [2,1,1,1,1,1,1,1] => ? = 1
{{1,2,3,4,5,7,8,9},{6}}
=> [8,1] => [2,1,1,1,1,1,1,1] => ? = 1
{{1,2,3,4,6,7,8,9},{5}}
=> [8,1] => [2,1,1,1,1,1,1,1] => ? = 1
{{1,2,3,5,6,7,8,9},{4}}
=> [8,1] => [2,1,1,1,1,1,1,1] => ? = 1
{{1,2,4,5,6,7,8,9},{3}}
=> [8,1] => [2,1,1,1,1,1,1,1] => ? = 1
{{1,2,3,4,5,6,7,9,10},{8}}
=> [9,1] => [2,1,1,1,1,1,1,1,1] => ? = 1
{{1,2,3,4,5,6,8,9,10},{7}}
=> [9,1] => [2,1,1,1,1,1,1,1,1] => ? = 1
{{1,2,3,4,5,7,8,9,10},{6}}
=> [9,1] => [2,1,1,1,1,1,1,1,1] => ? = 1
{{1,2,4,5,6,7,8,9,10},{3}}
=> [9,1] => [2,1,1,1,1,1,1,1,1] => ? = 1
{{1,2,3,4,5,6,7,8,10,11},{9}}
=> [10,1] => [2,1,1,1,1,1,1,1,1,1] => ? = 1
Description
The last part of an integer composition.
Matching statistic: St000678
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 90%●distinct values known / distinct values provided: 80%
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 90%●distinct values known / distinct values provided: 80%
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
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
{{1},{2},{3},{4},{5},{6,7,8}}
=> [1,1,1,1,1,3] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 6
{{1},{2},{3},{4},{5,6,7,8}}
=> [1,1,1,1,4] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 5
{{1},{2,3,4,5,6,7,8}}
=> [1,7] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
{{1,2},{3,4,5,6,7,8}}
=> [2,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
{{1,2,3},{4,5,6,7,8}}
=> [3,5] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
{{1,2,3,4},{5,6,7,8}}
=> [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1,5,6,7,8},{2,3,4}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,2,3,4,5},{6,7,8}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,6,7,8},{2,3,4,5}}
=> [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1,2,6,7,8},{3,4,5}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,7,8},{2,3,4,5,6}}
=> [3,5] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
{{1,2,7,8},{3,4,5,6}}
=> [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1,2,3,7,8},{4,5,6}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,8},{2,3,4,5,6,7}}
=> [2,6] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1
{{1,2,8},{3,4,5,6,7}}
=> [3,5] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 1
{{1,2,3,8},{4,5,6,7}}
=> [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1,2,3,4,8},{5,6,7}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,2,3,4,5,6,7,8}}
=> [8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
{{1,3,5,7,8},{2,4,6}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,2,5,7,8},{3,4,6}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,3,4,7,8},{2,5,6}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,2,4,7,8},{3,5,6}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,3,5,6,8},{2,4,7}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,2,5,6,8},{3,4,7}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,3,4,6,8},{2,5,7}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,2,4,6,8},{3,5,7}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,2,3,6,8},{4,5,7}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,2,4,5,8},{3,6,7}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,2,3,5,8},{4,6,7}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,3,5,6,7},{2,4,8}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,2,5,6,7},{3,4,8}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,3,4,6,7},{2,5,8}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,2,4,6,7},{3,5,8}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,2,3,6,7},{4,5,8}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,3,4,5,7},{2,6,8}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,2,4,5,7},{3,6,8}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,2,3,5,7},{4,6,8}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,2,3,4,7},{5,6,8}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,2,4,5,6},{3,7,8}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,2,3,5,6},{4,7,8}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,2,3,4,6},{5,7,8}}
=> [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
{{1,3,5,7},{2,4,6,8}}
=> [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1,2,5,7},{3,4,6,8}}
=> [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1,3,4,7},{2,5,6,8}}
=> [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1,2,4,7},{3,5,6,8}}
=> [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1,2,3,7},{4,5,6,8}}
=> [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1,3,5,6},{2,4,7,8}}
=> [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1,2,5,6},{3,4,7,8}}
=> [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
{{1,3,4,6},{2,5,7,8}}
=> [4,4] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
Description
The number of up steps after the last double rise of a 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: 80% ●values known / values provided: 88%●distinct values known / distinct values provided: 80%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001733: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 88%●distinct values known / distinct values provided: 80%
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
{{1},{2},{3},{4},{5},{6},{7,8}}
=> [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
{{1},{2},{3},{4},{5},{6,7,8}}
=> [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 6
{{1},{2},{3},{4},{5,6,7,8}}
=> [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 5
{{1},{2,3,4,5,6,7,8}}
=> [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
{{1,2},{3,4,5,6,7,8}}
=> [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 1
{{1,3,4,5,6,7,8},{2}}
=> [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,2,3},{4,5,6,7,8}}
=> [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 1
{{1,4,5,6,7,8},{2,3}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
{{1,2,4,5,6,7,8},{3}}
=> [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,2,3,4},{5,6,7,8}}
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 1
{{1,5,6,7,8},{2,3,4}}
=> [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 1
{{1,2,5,6,7,8},{3,4}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
{{1,2,3,5,6,7,8},{4}}
=> [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,2,3,4,5},{6,7,8}}
=> [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 1
{{1,6,7,8},{2,3,4,5}}
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 1
{{1,2,6,7,8},{3,4,5}}
=> [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 1
{{1,2,3,6,7,8},{4,5}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
{{1,2,3,4,6,7,8},{5}}
=> [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,2,3,4,5,6},{7,8}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
{{1,7,8},{2,3,4,5,6}}
=> [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 1
{{1,2,7,8},{3,4,5,6}}
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 1
{{1,2,3,7,8},{4,5,6}}
=> [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 1
{{1,2,3,4,7,8},{5,6}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
{{1,2,3,4,5,7,8},{6}}
=> [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,2,3,4,5,6,7},{8}}
=> [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,8},{2,3,4,5,6,7}}
=> [2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 1
{{1,2,8},{3,4,5,6,7}}
=> [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 1
{{1,2,3,8},{4,5,6,7}}
=> [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 1
{{1,2,3,4,8},{5,6,7}}
=> [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 1
{{1,2,3,4,5,8},{6,7}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
{{1,2,3,4,5,6,8},{7}}
=> [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,2,3,4,5,6,7,8}}
=> [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
{{1,3,5,6,7,8},{2,4}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
{{1,3,4,6,7,8},{2,5}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
{{1,2,4,6,7,8},{3,5}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
{{1,3,4,5,7,8},{2,6}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
{{1,2,4,5,7,8},{3,6}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
{{1,2,3,5,7,8},{4,6}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
{{1,3,4,5,6,8},{2,7}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
{{1,2,4,5,6,8},{3,7}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
{{1,2,3,5,6,8},{4,7}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
{{1,2,3,4,6,8},{5,7}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
{{1,3,4,5,6,7},{2,8}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
{{1,2,4,5,6,7},{3,8}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
{{1,2,3,5,6,7},{4,8}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
{{1,2,3,4,6,7},{5,8}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
{{1,2,3,4,5,7},{6,8}}
=> [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1
{{1,3,5,7,8},{2,4,6}}
=> [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 1
{{1,2,5,7,8},{3,4,6}}
=> [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 1
{{1,3,4,7,8},{2,5,6}}
=> [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ? = 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: St000729
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00249: Set partitions —Callan switch⟶ Set partitions
Mp00217: Set partitions —Wachs-White-rho ⟶ Set partitions
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
St000729: Set partitions ⟶ ℤResult quality: 70% ●values known / values provided: 88%●distinct values known / distinct values provided: 70%
Mp00217: Set partitions —Wachs-White-rho ⟶ Set partitions
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
St000729: Set partitions ⟶ ℤResult quality: 70% ●values known / values provided: 88%●distinct values known / distinct values provided: 70%
Values
{{1}}
=> {{1}}
=> {{1}}
=> {{1}}
=> ? = 1
{{1,2}}
=> {{1,2}}
=> {{1,2}}
=> {{1,2}}
=> 1
{{1},{2}}
=> {{1},{2}}
=> {{1},{2}}
=> {{1},{2}}
=> 2
{{1,2,3}}
=> {{1,3},{2}}
=> {{1,3},{2}}
=> {{1},{2,3}}
=> 1
{{1,2},{3}}
=> {{1,2},{3}}
=> {{1,2},{3}}
=> {{1,2},{3}}
=> 1
{{1,3},{2}}
=> {{1,2,3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> 1
{{1},{2,3}}
=> {{1},{2,3}}
=> {{1},{2,3}}
=> {{1,3},{2}}
=> 2
{{1},{2},{3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 3
{{1,2,3,4}}
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> 1
{{1,2,3},{4}}
=> {{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> 1
{{1,2,4},{3}}
=> {{1,3,4},{2}}
=> {{1,3,4},{2}}
=> {{1},{2,3,4}}
=> 1
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> 1
{{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 1
{{1,3,4},{2}}
=> {{1,2,4},{3}}
=> {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> 1
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> {{1,3,4},{2}}
=> 1
{{1,3},{2},{4}}
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 1
{{1,4},{2,3}}
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> 1
{{1},{2,3,4}}
=> {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> {{1,3},{2,4}}
=> 2
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> 2
{{1,4},{2},{3}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 1
{{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 2
{{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> 3
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 4
{{1,2,3,4,5}}
=> {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> {{1},{2},{3},{4,5}}
=> 1
{{1,2,3,4},{5}}
=> {{1,4},{2},{3},{5}}
=> {{1,4},{2},{3},{5}}
=> {{1},{2},{3,4},{5}}
=> 1
{{1,2,3,5},{4}}
=> {{1,4,5},{2},{3}}
=> {{1,4,5},{2},{3}}
=> {{1},{2},{3,4,5}}
=> 1
{{1,2,3},{4,5}}
=> {{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> {{1,5},{2,3},{4}}
=> 1
{{1,2,3},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> 1
{{1,2,4,5},{3}}
=> {{1,3,4,5},{2}}
=> {{1,3,4,5},{2}}
=> {{1},{2,3,4,5}}
=> 1
{{1,2,4},{3,5}}
=> {{1,4},{2},{3,5}}
=> {{1,5},{2},{3,4}}
=> {{1,4,5},{2},{3}}
=> 1
{{1,2,4},{3},{5}}
=> {{1,3,4},{2},{5}}
=> {{1,3,4},{2},{5}}
=> {{1},{2,3,4},{5}}
=> 1
{{1,2,5},{3,4}}
=> {{1,5},{2},{3,4}}
=> {{1,4},{2},{3,5}}
=> {{1,5},{2},{3,4}}
=> 1
{{1,2},{3,4,5}}
=> {{1,2},{3,4,5}}
=> {{1,2},{3,4,5}}
=> {{1,2,4},{3,5}}
=> 1
{{1,2},{3,4},{5}}
=> {{1,2},{3,4},{5}}
=> {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> 1
{{1,2,5},{3},{4}}
=> {{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> {{1},{2,3},{4,5}}
=> 1
{{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> 1
{{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> {{1,2,5},{3},{4}}
=> 1
{{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> 1
{{1,3,4,5},{2}}
=> {{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> {{1,2},{3,4,5}}
=> 1
{{1,3,4},{2,5}}
=> {{1,4},{2,5},{3}}
=> {{1,5},{2,4},{3}}
=> {{1},{2,4,5},{3}}
=> 1
{{1,3,4},{2},{5}}
=> {{1,2,4},{3},{5}}
=> {{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> 1
{{1,3,5},{2,4}}
=> {{1,5},{2,4},{3}}
=> {{1,4},{2,5},{3}}
=> {{1},{2,5},{3,4}}
=> 1
{{1,3},{2,4,5}}
=> {{1,3},{2,4,5}}
=> {{1,4,5},{2,3}}
=> {{1,3,4,5},{2}}
=> 1
{{1,3},{2,4},{5}}
=> {{1,3},{2,4},{5}}
=> {{1,4},{2,3},{5}}
=> {{1,3,4},{2},{5}}
=> 1
{{1,3,5},{2},{4}}
=> {{1,2,5},{3},{4}}
=> {{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> 1
{{1,3},{2,5},{4}}
=> {{1,3},{2,5},{4}}
=> {{1,5},{2,3},{4}}
=> {{1,3},{2},{4,5}}
=> 1
{{1,3},{2},{4,5}}
=> {{1,2,3},{4,5}}
=> {{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> 1
{{1,3},{2},{4},{5}}
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 1
{{1,4,5},{2,3}}
=> {{1,4,5},{2,3}}
=> {{1,3},{2,4,5}}
=> {{1,4},{2,3,5}}
=> 1
{{1,4},{2,3,5}}
=> {{1,4},{2,3,5}}
=> {{1,3,5},{2,4}}
=> {{1,4,5},{2,3}}
=> 1
{{1,4},{2,3},{5}}
=> {{1,4},{2,3},{5}}
=> {{1,3},{2,4},{5}}
=> {{1,4},{2,3},{5}}
=> 1
{{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 8
{{1},{2},{3},{4},{5},{6},{7,8}}
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 7
{{1},{2},{3},{4},{5},{6,7,8}}
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> {{1},{2},{3},{4},{5},{6,7,8}}
=> {{1,7},{2,8},{3},{4},{5},{6}}
=> ? = 6
{{1},{2},{3},{4},{5,6,7,8}}
=> {{1},{2},{3},{4},{5,6,7,8}}
=> {{1},{2},{3},{4},{5,6,7,8}}
=> {{1,6},{2,7},{3,8},{4},{5}}
=> ? = 5
{{1},{2,3,4,5,6,7,8}}
=> {{1},{2,3,4,5,6,7,8}}
=> {{1},{2,3,4,5,6,7,8}}
=> {{1,3,5,7},{2,4,6,8}}
=> ? = 2
{{1,2},{3},{4},{5},{6},{7},{8}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,2},{3,4,5,6,7,8}}
=> {{1,2},{3,4,5,6,7,8}}
=> {{1,2},{3,4,5,6,7,8}}
=> {{1,2,4,6,8},{3,5,7}}
=> ? = 1
{{1,3},{2},{4},{5},{6},{7},{8}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,4},{2},{3},{5},{6},{7},{8}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,5},{2},{3},{4},{6},{7},{8}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,6},{2},{3},{4},{5},{7},{8}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,7},{2},{3},{4},{5},{6},{8}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,3,4,5,6,7,8},{2}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,2,3},{4,5,6,7,8}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,4,5,6,7,8},{2,3}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,2,4,5,6,7,8},{3}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,2,3,4},{5,6,7,8}}
=> {{1,4},{2},{3},{5,6,7,8}}
=> {{1,4},{2},{3},{5,6,7,8}}
=> {{1,6},{2,7},{3,4,8},{5}}
=> ? = 1
{{1,5,6,7,8},{2,3,4}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,2,5,6,7,8},{3,4}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,2,3,5,6,7,8},{4}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,2,3,4,5},{6,7,8}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,6,7,8},{2,3,4,5}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,2,6,7,8},{3,4,5}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,2,3,6,7,8},{4,5}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,2,3,4,6,7,8},{5}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,2,3,4,5,6},{7,8}}
=> {{1,6},{2},{3},{4},{5},{7,8}}
=> {{1,6},{2},{3},{4},{5},{7,8}}
=> {{1,8},{2},{3},{4},{5,6},{7}}
=> ? = 1
{{1,7,8},{2,3,4,5,6}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,2,7,8},{3,4,5,6}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,2,3,7,8},{4,5,6}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,2,3,4,7,8},{5,6}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,2,3,4,5,7,8},{6}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,2,3,4,5,6,7},{8}}
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> {{1},{2},{3},{4},{5},{6,7},{8}}
=> ? = 1
{{1,8},{2,3,4,5,6,7}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,2,8},{3,4,5,6,7}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,2,3,8},{4,5,6,7}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,2,3,4,8},{5,6,7}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,2,3,4,5,8},{6,7}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,2,3,4,5,6,8},{7}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,2,3,4,5,6,7,8}}
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 1
{{1,3,5,6,7,8},{2,4}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,3,4,6,7,8},{2,5}}
=> {{1,6,7,8},{2,5},{3},{4}}
=> {{1,5},{2,6,7,8},{3},{4}}
=> {{1},{2},{3,6,8},{4,5,7}}
=> ? = 1
{{1,2,4,6,7,8},{3,5}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,3,4,5,7,8},{2,6}}
=> {{1,7,8},{2,6},{3},{4},{5}}
=> {{1,6},{2,7,8},{3},{4},{5}}
=> ?
=> ? = 1
{{1,2,4,5,7,8},{3,6}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,2,3,5,7,8},{4,6}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,3,4,5,6,8},{2,7}}
=> ?
=> ?
=> ?
=> ? = 1
{{1,2,4,5,6,8},{3,7}}
=> {{1,8},{2},{3,7},{4},{5},{6}}
=> {{1,7},{2},{3,8},{4},{5},{6}}
=> ?
=> ? = 1
{{1,2,3,5,6,8},{4,7}}
=> {{1,8},{2},{3},{4,7},{5},{6}}
=> {{1,7},{2},{3},{4,8},{5},{6}}
=> ?
=> ? = 1
{{1,2,3,4,6,8},{5,7}}
=> ?
=> ?
=> ?
=> ? = 1
Description
The minimal arc length of a set partition.
The arcs of a set partition are those i<j that are consecutive elements in the blocks. If there are no arcs, the minimal arc length is the size of the ground set (as the minimum of the empty set in the universe of arcs of length less than the size of the ground set).
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: 70% ●values known / values provided: 88%●distinct values known / distinct values provided: 70%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 88%●distinct values known / distinct values provided: 70%
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
{{1},{2},{3},{4},{5},{6},{7},{8}}
=> [1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8
{{1},{2},{3},{4},{5},{6},{7,8}}
=> [1,1,1,1,1,1,2] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
{{1},{2},{3},{4},{5},{6,7,8}}
=> [1,1,1,1,1,3] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 6
{{1},{2},{3},{4},{5,6,7,8}}
=> [1,1,1,1,4] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 5
{{1},{2,3,4,5,6,7,8}}
=> [1,7] => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
{{1,2},{3},{4},{5},{6},{7},{8}}
=> [2,1,1,1,1,1,1] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1,2},{3,4,5,6,7,8}}
=> [2,6] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,3},{2},{4},{5},{6},{7},{8}}
=> [2,1,1,1,1,1,1] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1,4},{2},{3},{5},{6},{7},{8}}
=> [2,1,1,1,1,1,1] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1,5},{2},{3},{4},{6},{7},{8}}
=> [2,1,1,1,1,1,1] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1,6},{2},{3},{4},{5},{7},{8}}
=> [2,1,1,1,1,1,1] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1,7},{2},{3},{4},{5},{6},{8}}
=> [2,1,1,1,1,1,1] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1,8},{2},{3},{4},{5},{6},{7}}
=> [2,1,1,1,1,1,1] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
{{1,3,4,5,6,7,8},{2}}
=> [7,1] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,2,3},{4,5,6,7,8}}
=> [3,5] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,4,5,6,7,8},{2,3}}
=> [6,2] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
{{1,2,4,5,6,7,8},{3}}
=> [7,1] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,2,3,4},{5,6,7,8}}
=> [4,4] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
{{1,5,6,7,8},{2,3,4}}
=> [5,3] => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
{{1,2,5,6,7,8},{3,4}}
=> [6,2] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
{{1,2,3,5,6,7,8},{4}}
=> [7,1] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,2,3,4,5},{6,7,8}}
=> [5,3] => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
{{1,6,7,8},{2,3,4,5}}
=> [4,4] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
{{1,2,6,7,8},{3,4,5}}
=> [5,3] => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
{{1,2,3,6,7,8},{4,5}}
=> [6,2] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
{{1,2,3,4,6,7,8},{5}}
=> [7,1] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,2,3,4,5,6},{7,8}}
=> [6,2] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
{{1,7,8},{2,3,4,5,6}}
=> [3,5] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,2,7,8},{3,4,5,6}}
=> [4,4] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
{{1,2,3,7,8},{4,5,6}}
=> [5,3] => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
{{1,2,3,4,7,8},{5,6}}
=> [6,2] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
{{1,2,3,4,5,7,8},{6}}
=> [7,1] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,2,3,4,5,6,7},{8}}
=> [7,1] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,8},{2,3,4,5,6,7}}
=> [2,6] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,2,8},{3,4,5,6,7}}
=> [3,5] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,2,3,8},{4,5,6,7}}
=> [4,4] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
{{1,2,3,4,8},{5,6,7}}
=> [5,3] => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
{{1,2,3,4,5,8},{6,7}}
=> [6,2] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
{{1,2,3,4,5,6,8},{7}}
=> [7,1] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1,2,3,4,5,6,7,8}}
=> [8] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,3,5,6,7,8},{2,4}}
=> [6,2] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
{{1,3,4,6,7,8},{2,5}}
=> [6,2] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
{{1,2,4,6,7,8},{3,5}}
=> [6,2] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
{{1,3,4,5,7,8},{2,6}}
=> [6,2] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
{{1,2,4,5,7,8},{3,6}}
=> [6,2] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
{{1,2,3,5,7,8},{4,6}}
=> [6,2] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
{{1,3,4,5,6,8},{2,7}}
=> [6,2] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
{{1,2,4,5,6,8},{3,7}}
=> [6,2] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
{{1,2,3,5,6,8},{4,7}}
=> [6,2] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
{{1,2,3,4,6,8},{5,7}}
=> [6,2] => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of D.
The following 29 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000026The position of the first return of a Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the 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. St000617The number of global maxima of a Dyck path. St000501The size of the first part in the decomposition of a permutation. 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 S0 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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