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Your data matches 25 different statistics following compositions of up to 3 maps.
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Matching statistic: St000993
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [1,1]
=> 2
{{1},{2}}
=> [1,1]
=> [2]
=> 1
{{1,2,3}}
=> [3]
=> [1,1,1]
=> 3
{{1,2},{3}}
=> [2,1]
=> [2,1]
=> 1
{{1,3},{2}}
=> [2,1]
=> [2,1]
=> 1
{{1},{2,3}}
=> [2,1]
=> [2,1]
=> 1
{{1},{2},{3}}
=> [1,1,1]
=> [3]
=> 1
{{1,2,3,4}}
=> [4]
=> [1,1,1,1]
=> 4
{{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]
=> [2,2]
=> 2
{{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]
=> [2,2]
=> 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> 1
{{1,4},{2,3}}
=> [2,2]
=> [2,2]
=> 2
{{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [4]
=> 1
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1]
=> 5
{{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]
=> [2,2,1]
=> 2
{{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]
=> [2,2,1]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [3,1,1]
=> 1
{{1,2,5},{3,4}}
=> [3,2]
=> [2,2,1]
=> 2
{{1,2},{3,4,5}}
=> [3,2]
=> [2,2,1]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [3,2]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [3,1,1]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [3,2]
=> 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [3,2]
=> 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]
=> [2,2,1]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [3,1,1]
=> 1
{{1,3,5},{2,4}}
=> [3,2]
=> [2,2,1]
=> 2
{{1,3},{2,4,5}}
=> [3,2]
=> [2,2,1]
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [3,2]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [3,1,1]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [3,2]
=> 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [3,2]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [2,2,1]
=> 2
{{1,4},{2,3,5}}
=> [3,2]
=> [2,2,1]
=> 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [3,2]
=> 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St001038
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(load all 4 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 98%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 98%●distinct values known / distinct values provided: 50%
Values
{{1,2}}
=> [2]
=> []
=> []
=> ? = 2
{{1},{2}}
=> [1,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3}}
=> [3]
=> []
=> []
=> ? = 3
{{1,2},{3}}
=> [2,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,3},{2}}
=> [2,1]
=> [1]
=> [1,0]
=> ? = 1
{{1},{2,3}}
=> [2,1]
=> [1]
=> [1,0]
=> ? = 1
{{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,2,3,4}}
=> [4]
=> []
=> []
=> ? = 4
{{1,2,3},{4}}
=> [3,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,4},{3}}
=> [3,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2},{3,4}}
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,3,4},{2}}
=> [3,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,3},{2,4}}
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,4},{2,3}}
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 2
{{1},{2,3,4}}
=> [3,1]
=> [1]
=> [1,0]
=> ? = 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1,2,3,4,5}}
=> [5]
=> []
=> []
=> ? = 5
{{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,2,5},{3,4}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
{{1,2},{3,4,5}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,3,4},{2,5}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,3,5},{2,4}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
{{1,3},{2,4,5}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
{{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,5},{2,3,4}}
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
{{1},{2,3,4,5}}
=> [4,1]
=> [1]
=> [1,0]
=> ? = 1
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
{{1,2,3,4,5,6}}
=> [6]
=> []
=> []
=> ? = 6
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1]
=> [1,0]
=> ? = 1
{{1},{2,3,4,5,6}}
=> [5,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,4,5,6,7}}
=> [7]
=> []
=> []
=> ? = 7
{{1,2,3,4,5,6},{7}}
=> [6,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,4,5,7},{6}}
=> [6,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,4,6,7},{5}}
=> [6,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,5,6,7},{4}}
=> [6,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,4,5,6,7},{3}}
=> [6,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,3,4,5,6,7},{2}}
=> [6,1]
=> [1]
=> [1,0]
=> ? = 1
{{1},{2,3,4,5,6,7}}
=> [6,1]
=> [1]
=> [1,0]
=> ? = 1
{{1},{2,3,4,5,6,7,8}}
=> [7,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,3,4,5,6,7,8},{2}}
=> [7,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,4,5,6,7,8},{3}}
=> [7,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,5,6,7,8},{4}}
=> [7,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,4,6,7,8},{5}}
=> [7,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,4,5,7,8},{6}}
=> [7,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,4,5,6,7},{8}}
=> [7,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,4,5,6,8},{7}}
=> [7,1]
=> [1]
=> [1,0]
=> ? = 1
{{1,2,3,4,5,6,7,8}}
=> [8]
=> []
=> []
=> ? = 8
{{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3},{4},{5},{6},{7},{8},{9,10}}
=> [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3},{4},{5},{6},{7},{8,10},{9}}
=> [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3},{4},{5},{6},{7,10},{8},{9}}
=> [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3},{4},{5},{6,10},{7},{8},{9}}
=> [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3},{4},{5,10},{6},{7},{8},{9}}
=> [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3},{4,10},{5},{6},{7},{8},{9}}
=> [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
{{1},{2},{3,10},{4},{5},{6},{7},{8},{9}}
=> [2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000667
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 94%●distinct values known / distinct values provided: 20%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 20% ●values known / values provided: 94%●distinct values known / distinct values provided: 20%
Values
{{1,2}}
=> [2]
=> []
=> ?
=> ? = 2
{{1},{2}}
=> [1,1]
=> [1]
=> []
=> ? = 1
{{1,2,3}}
=> [3]
=> []
=> ?
=> ? = 3
{{1,2},{3}}
=> [2,1]
=> [1]
=> []
=> ? = 1
{{1,3},{2}}
=> [2,1]
=> [1]
=> []
=> ? = 1
{{1},{2,3}}
=> [2,1]
=> [1]
=> []
=> ? = 1
{{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
{{1,2,3,4}}
=> [4]
=> []
=> ?
=> ? = 4
{{1,2,3},{4}}
=> [3,1]
=> [1]
=> []
=> ? = 1
{{1,2,4},{3}}
=> [3,1]
=> [1]
=> []
=> ? = 1
{{1,2},{3,4}}
=> [2,2]
=> [2]
=> []
=> ? = 2
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
{{1,3,4},{2}}
=> [3,1]
=> [1]
=> []
=> ? = 1
{{1,3},{2,4}}
=> [2,2]
=> [2]
=> []
=> ? = 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
{{1,4},{2,3}}
=> [2,2]
=> [2]
=> []
=> ? = 2
{{1},{2,3,4}}
=> [3,1]
=> [1]
=> []
=> ? = 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,2,3,4,5}}
=> [5]
=> []
=> ?
=> ? = 5
{{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> []
=> ? = 1
{{1,2,3,5},{4}}
=> [4,1]
=> [1]
=> []
=> ? = 1
{{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> []
=> ? = 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
{{1,2,4,5},{3}}
=> [4,1]
=> [1]
=> []
=> ? = 1
{{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> []
=> ? = 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
{{1,2,5},{3,4}}
=> [3,2]
=> [2]
=> []
=> ? = 2
{{1,2},{3,4,5}}
=> [3,2]
=> [2]
=> []
=> ? = 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [1]
=> []
=> ? = 1
{{1,3,4},{2,5}}
=> [3,2]
=> [2]
=> []
=> ? = 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
{{1,3,5},{2,4}}
=> [3,2]
=> [2]
=> []
=> ? = 2
{{1,3},{2,4,5}}
=> [3,2]
=> [2]
=> []
=> ? = 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [2]
=> []
=> ? = 2
{{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> []
=> ? = 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
{{1,5},{2,3,4}}
=> [3,2]
=> [2]
=> []
=> ? = 2
{{1},{2,3,4,5}}
=> [4,1]
=> [1]
=> []
=> ? = 1
{{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
{{1},{2,3,5},{4}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,4,5},{2},{3}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
{{1},{2,4,5},{3}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
{{1},{2},{3,4,5}}
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
{{1,2,3,4,5,6}}
=> [6]
=> []
=> ?
=> ? = 6
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1]
=> []
=> ? = 1
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1]
=> []
=> ? = 1
{{1,2,3,4},{5,6}}
=> [4,2]
=> [2]
=> []
=> ? = 2
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1]
=> []
=> ? = 1
{{1,2,3,5},{4,6}}
=> [4,2]
=> [2]
=> []
=> ? = 2
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
{{1,2,3,6},{4,5}}
=> [4,2]
=> [2]
=> []
=> ? = 2
{{1,2,3},{4,5,6}}
=> [3,3]
=> [3]
=> []
=> ? = 3
{{1,2,3},{4,5},{6}}
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,1]
=> [1]
=> 1
{{1,2,3},{4,6},{5}}
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
{{1,2,3},{4},{5,6}}
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1]
=> []
=> ? = 1
{{1,2,4,5},{3,6}}
=> [4,2]
=> [2]
=> []
=> ? = 2
{{1,2,4,6},{3,5}}
=> [4,2]
=> [2]
=> []
=> ? = 2
{{1,2,4},{3,5,6}}
=> [3,3]
=> [3]
=> []
=> ? = 3
{{1,2,5,6},{3,4}}
=> [4,2]
=> [2]
=> []
=> ? = 2
{{1,2,5},{3,4,6}}
=> [3,3]
=> [3]
=> []
=> ? = 3
{{1,2,6},{3,4,5}}
=> [3,3]
=> [3]
=> []
=> ? = 3
{{1,2},{3,4,5,6}}
=> [4,2]
=> [2]
=> []
=> ? = 2
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1]
=> []
=> ? = 1
{{1,3,4,5},{2,6}}
=> [4,2]
=> [2]
=> []
=> ? = 2
{{1,3,4,6},{2,5}}
=> [4,2]
=> [2]
=> []
=> ? = 2
{{1,3,4},{2,5,6}}
=> [3,3]
=> [3]
=> []
=> ? = 3
Description
The greatest common divisor of the parts of the partition.
Matching statistic: St000326
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> 100 => 001 => 3 = 2 + 1
{{1},{2}}
=> [1,1]
=> 110 => 011 => 2 = 1 + 1
{{1,2,3}}
=> [3]
=> 1000 => 0001 => 4 = 3 + 1
{{1,2},{3}}
=> [2,1]
=> 1010 => 0101 => 2 = 1 + 1
{{1,3},{2}}
=> [2,1]
=> 1010 => 0101 => 2 = 1 + 1
{{1},{2,3}}
=> [2,1]
=> 1010 => 0101 => 2 = 1 + 1
{{1},{2},{3}}
=> [1,1,1]
=> 1110 => 0111 => 2 = 1 + 1
{{1,2,3,4}}
=> [4]
=> 10000 => 00001 => 5 = 4 + 1
{{1,2,3},{4}}
=> [3,1]
=> 10010 => 01001 => 2 = 1 + 1
{{1,2,4},{3}}
=> [3,1]
=> 10010 => 01001 => 2 = 1 + 1
{{1,2},{3,4}}
=> [2,2]
=> 1100 => 0011 => 3 = 2 + 1
{{1,2},{3},{4}}
=> [2,1,1]
=> 10110 => 01101 => 2 = 1 + 1
{{1,3,4},{2}}
=> [3,1]
=> 10010 => 01001 => 2 = 1 + 1
{{1,3},{2,4}}
=> [2,2]
=> 1100 => 0011 => 3 = 2 + 1
{{1,3},{2},{4}}
=> [2,1,1]
=> 10110 => 01101 => 2 = 1 + 1
{{1,4},{2,3}}
=> [2,2]
=> 1100 => 0011 => 3 = 2 + 1
{{1},{2,3,4}}
=> [3,1]
=> 10010 => 01001 => 2 = 1 + 1
{{1},{2,3},{4}}
=> [2,1,1]
=> 10110 => 01101 => 2 = 1 + 1
{{1,4},{2},{3}}
=> [2,1,1]
=> 10110 => 01101 => 2 = 1 + 1
{{1},{2,4},{3}}
=> [2,1,1]
=> 10110 => 01101 => 2 = 1 + 1
{{1},{2},{3,4}}
=> [2,1,1]
=> 10110 => 01101 => 2 = 1 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> 11110 => 01111 => 2 = 1 + 1
{{1,2,3,4,5}}
=> [5]
=> 100000 => 000001 => 6 = 5 + 1
{{1,2,3,4},{5}}
=> [4,1]
=> 100010 => 010001 => 2 = 1 + 1
{{1,2,3,5},{4}}
=> [4,1]
=> 100010 => 010001 => 2 = 1 + 1
{{1,2,3},{4,5}}
=> [3,2]
=> 10100 => 00101 => 3 = 2 + 1
{{1,2,3},{4},{5}}
=> [3,1,1]
=> 100110 => 011001 => 2 = 1 + 1
{{1,2,4,5},{3}}
=> [4,1]
=> 100010 => 010001 => 2 = 1 + 1
{{1,2,4},{3,5}}
=> [3,2]
=> 10100 => 00101 => 3 = 2 + 1
{{1,2,4},{3},{5}}
=> [3,1,1]
=> 100110 => 011001 => 2 = 1 + 1
{{1,2,5},{3,4}}
=> [3,2]
=> 10100 => 00101 => 3 = 2 + 1
{{1,2},{3,4,5}}
=> [3,2]
=> 10100 => 00101 => 3 = 2 + 1
{{1,2},{3,4},{5}}
=> [2,2,1]
=> 11010 => 01011 => 2 = 1 + 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> 100110 => 011001 => 2 = 1 + 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> 11010 => 01011 => 2 = 1 + 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> 11010 => 01011 => 2 = 1 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> 101110 => 011101 => 2 = 1 + 1
{{1,3,4,5},{2}}
=> [4,1]
=> 100010 => 010001 => 2 = 1 + 1
{{1,3,4},{2,5}}
=> [3,2]
=> 10100 => 00101 => 3 = 2 + 1
{{1,3,4},{2},{5}}
=> [3,1,1]
=> 100110 => 011001 => 2 = 1 + 1
{{1,3,5},{2,4}}
=> [3,2]
=> 10100 => 00101 => 3 = 2 + 1
{{1,3},{2,4,5}}
=> [3,2]
=> 10100 => 00101 => 3 = 2 + 1
{{1,3},{2,4},{5}}
=> [2,2,1]
=> 11010 => 01011 => 2 = 1 + 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> 100110 => 011001 => 2 = 1 + 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> 11010 => 01011 => 2 = 1 + 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> 11010 => 01011 => 2 = 1 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> 101110 => 011101 => 2 = 1 + 1
{{1,4,5},{2,3}}
=> [3,2]
=> 10100 => 00101 => 3 = 2 + 1
{{1,4},{2,3,5}}
=> [3,2]
=> 10100 => 00101 => 3 = 2 + 1
{{1,4},{2,3},{5}}
=> [2,2,1]
=> 11010 => 01011 => 2 = 1 + 1
{{1,2,3,4,5,6,7,8,9,10},{11}}
=> [10,1]
=> 100000000010 => 010000000001 => ? = 1 + 1
{{1},{2,3,4,5,6,7,8,9,10,11}}
=> [10,1]
=> 100000000010 => 010000000001 => ? = 1 + 1
{{1,2,4,8},{3,6,12},{5,10},{7},{9},{11}}
=> [4,3,2,1,1,1]
=> 1010101110 => 0111010101 => ? = 1 + 1
{{1,2,3,4,11},{5,6},{7},{8},{9},{10}}
=> [5,2,1,1,1,1]
=> 10001011110 => 01111010001 => ? = 1 + 1
{{1,11},{2,3,4,5,6,7,8,9,10}}
=> [9,2]
=> 10000000100 => 00100000001 => ? = 2 + 1
{{1},{2},{3},{4},{5},{6},{7},{8},{9},{10,11}}
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => 011111111101 => ? = 1 + 1
{{1,2,3,4,5,6,7,8,9,11},{10}}
=> [10,1]
=> 100000000010 => 010000000001 => ? = 1 + 1
{{1,3,4,5,6,7,8,9,10,11},{2}}
=> [10,1]
=> 100000000010 => 010000000001 => ? = 1 + 1
{{1,2},{3},{4},{5},{6},{7},{8},{9},{10},{11}}
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => 011111111101 => ? = 1 + 1
{{1,11},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => 011111111101 => ? = 1 + 1
{{1,2,4,8},{3,6,12},{5,11},{7},{9},{10}}
=> [4,3,2,1,1,1]
=> 1010101110 => 0111010101 => ? = 1 + 1
{{1,2,4,9},{3,6,12},{5,10},{7},{8},{11}}
=> [4,3,2,1,1,1]
=> 1010101110 => 0111010101 => ? = 1 + 1
{{1,2,4,9},{3,6,12},{5,11},{7},{8},{10}}
=> [4,3,2,1,1,1]
=> 1010101110 => 0111010101 => ? = 1 + 1
{{1,2,4,10},{3,6,12},{5,11},{7},{8},{9}}
=> [4,3,2,1,1,1]
=> 1010101110 => 0111010101 => ? = 1 + 1
{{1,2,4,8},{3,7},{5,10},{6,12},{9},{11}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,2,4,8},{3,7},{5,11},{6,12},{9},{10}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,2,4,9},{3,7},{5,10},{6,12},{8},{11}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,2,4,9},{3,7},{5,11},{6,12},{8},{10}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,2,4,10},{3,7},{5,11},{6,12},{8},{9}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,2,4,9},{3,8},{5,10},{6,12},{7},{11}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,2,4,9},{3,8},{5,11},{6,12},{7},{10}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,2,4,10},{3,8},{5,11},{6,12},{7},{9}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,2,4,10},{3,9},{5,11},{6,12},{7},{8}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,2,5,10},{3,6,12},{4,8},{7},{9},{11}}
=> [4,3,2,1,1,1]
=> 1010101110 => 0111010101 => ? = 1 + 1
{{1,2,5,11},{3,6,12},{4,8},{7},{9},{10}}
=> [4,3,2,1,1,1]
=> 1010101110 => 0111010101 => ? = 1 + 1
{{1,2,5,10},{3,6,12},{4,9},{7},{8},{11}}
=> [4,3,2,1,1,1]
=> 1010101110 => 0111010101 => ? = 1 + 1
{{1,2,5,11},{3,6,12},{4,9},{7},{8},{10}}
=> [4,3,2,1,1,1]
=> 1010101110 => 0111010101 => ? = 1 + 1
{{1,2,5,11},{3,6,12},{4,10},{7},{8},{9}}
=> [4,3,2,1,1,1]
=> 1010101110 => 0111010101 => ? = 1 + 1
{{1,2,5,10},{3,7},{4,8},{6,12},{9},{11}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,2,5,11},{3,7},{4,8},{6,12},{9},{10}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,2,5,10},{3,7},{4,9},{6,12},{8},{11}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,2,5,11},{3,7},{4,9},{6,12},{8},{10}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,2,5,11},{3,7},{4,10},{6,12},{8},{9}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,2,5,10},{3,8},{4,9},{6,12},{7},{11}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,2,5,11},{3,8},{4,9},{6,12},{7},{10}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,2,5,11},{3,8},{4,10},{6,12},{7},{9}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,2,5,11},{3,9},{4,10},{6,12},{7},{8}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,2,6,12},{3,7},{4,8},{5,10},{9},{11}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,2,6,12},{3,7},{4,8},{5,11},{9},{10}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,2,6,12},{3,7},{4,9},{5,10},{8},{11}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,2,6,12},{3,7},{4,9},{5,11},{8},{10}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,2,6,12},{3,7},{4,10},{5,11},{8},{9}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,2,6,12},{3,8},{4,9},{5,10},{7},{11}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,2,6,12},{3,8},{4,9},{5,11},{7},{10}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,2,6,12},{3,8},{4,10},{5,11},{7},{9}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,2,6,12},{3,9},{4,10},{5,11},{7},{8}}
=> [4,2,2,2,1,1]
=> 1001110110 => 0110111001 => ? = 1 + 1
{{1,3,6,12},{2,4,8},{5,10},{7},{9},{11}}
=> [4,3,2,1,1,1]
=> 1010101110 => 0111010101 => ? = 1 + 1
{{1,3,6,12},{2,4,8},{5,11},{7},{9},{10}}
=> [4,3,2,1,1,1]
=> 1010101110 => 0111010101 => ? = 1 + 1
{{1,3,6,12},{2,4,9},{5,10},{7},{8},{11}}
=> [4,3,2,1,1,1]
=> 1010101110 => 0111010101 => ? = 1 + 1
{{1,3,6,12},{2,4,9},{5,11},{7},{8},{10}}
=> [4,3,2,1,1,1]
=> 1010101110 => 0111010101 => ? = 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
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [1,1]
=> 110 => 2
{{1},{2}}
=> [1,1]
=> [2]
=> 100 => 1
{{1,2,3}}
=> [3]
=> [1,1,1]
=> 1110 => 3
{{1,2},{3}}
=> [2,1]
=> [2,1]
=> 1010 => 1
{{1,3},{2}}
=> [2,1]
=> [2,1]
=> 1010 => 1
{{1},{2,3}}
=> [2,1]
=> [2,1]
=> 1010 => 1
{{1},{2},{3}}
=> [1,1,1]
=> [3]
=> 1000 => 1
{{1,2,3,4}}
=> [4]
=> [1,1,1,1]
=> 11110 => 4
{{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> 10110 => 1
{{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> 10110 => 1
{{1,2},{3,4}}
=> [2,2]
=> [2,2]
=> 1100 => 2
{{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> 10010 => 1
{{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> 10110 => 1
{{1,3},{2,4}}
=> [2,2]
=> [2,2]
=> 1100 => 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> 10010 => 1
{{1,4},{2,3}}
=> [2,2]
=> [2,2]
=> 1100 => 2
{{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> 10110 => 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> 10010 => 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> 10010 => 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> 10010 => 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> 10010 => 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [4]
=> 10000 => 1
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1]
=> 111110 => 5
{{1,2,3,4},{5}}
=> [4,1]
=> [2,1,1,1]
=> 101110 => 1
{{1,2,3,5},{4}}
=> [4,1]
=> [2,1,1,1]
=> 101110 => 1
{{1,2,3},{4,5}}
=> [3,2]
=> [2,2,1]
=> 11010 => 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [3,1,1]
=> 100110 => 1
{{1,2,4,5},{3}}
=> [4,1]
=> [2,1,1,1]
=> 101110 => 1
{{1,2,4},{3,5}}
=> [3,2]
=> [2,2,1]
=> 11010 => 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [3,1,1]
=> 100110 => 1
{{1,2,5},{3,4}}
=> [3,2]
=> [2,2,1]
=> 11010 => 2
{{1,2},{3,4,5}}
=> [3,2]
=> [2,2,1]
=> 11010 => 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [3,2]
=> 10100 => 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [3,1,1]
=> 100110 => 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [3,2]
=> 10100 => 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [3,2]
=> 10100 => 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 100010 => 1
{{1,3,4,5},{2}}
=> [4,1]
=> [2,1,1,1]
=> 101110 => 1
{{1,3,4},{2,5}}
=> [3,2]
=> [2,2,1]
=> 11010 => 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [3,1,1]
=> 100110 => 1
{{1,3,5},{2,4}}
=> [3,2]
=> [2,2,1]
=> 11010 => 2
{{1,3},{2,4,5}}
=> [3,2]
=> [2,2,1]
=> 11010 => 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [3,2]
=> 10100 => 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [3,1,1]
=> 100110 => 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [3,2]
=> 10100 => 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [3,2]
=> 10100 => 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> 100010 => 1
{{1,4,5},{2,3}}
=> [3,2]
=> [2,2,1]
=> 11010 => 2
{{1,4},{2,3,5}}
=> [3,2]
=> [2,2,1]
=> 11010 => 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [3,2]
=> 10100 => 1
{{1,2,3,4,5,6,7,8,9,10},{11}}
=> [10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 1
{{1},{2,3,4,5,6,7,8,9,10,11}}
=> [10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 1
{{1,2,4,8},{3,6,12},{5,10},{7},{9},{11}}
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> 1000101010 => ? = 1
{{1,2,3,4,10,11},{5},{6},{7},{8},{9}}
=> [6,1,1,1,1,1]
=> [6,1,1,1,1,1]
=> 100000111110 => ? = 1
{{1},{2},{3},{4},{5},{6},{7},{8},{9},{10,11}}
=> [2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> 100000000010 => ? = 1
{{1,2,3,4,5,6,7,8,9,11},{10}}
=> [10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 1
{{1,3,4,5,6,7,8,9,10,11},{2}}
=> [10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? = 1
{{1,2},{3},{4},{5},{6},{7},{8},{9},{10},{11}}
=> [2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> 100000000010 => ? = 1
{{1,11},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> 100000000010 => ? = 1
{{1,2,4,8},{3,6,12},{5,11},{7},{9},{10}}
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> 1000101010 => ? = 1
{{1,2,4,9},{3,6,12},{5,10},{7},{8},{11}}
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> 1000101010 => ? = 1
{{1,2,4,9},{3,6,12},{5,11},{7},{8},{10}}
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> 1000101010 => ? = 1
{{1,2,4,10},{3,6,12},{5,11},{7},{8},{9}}
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> 1000101010 => ? = 1
{{1,2,4,8},{3,7},{5,10},{6,12},{9},{11}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,2,4,8},{3,7},{5,11},{6,12},{9},{10}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,2,4,9},{3,7},{5,10},{6,12},{8},{11}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,2,4,9},{3,7},{5,11},{6,12},{8},{10}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,2,4,10},{3,7},{5,11},{6,12},{8},{9}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,2,4,9},{3,8},{5,10},{6,12},{7},{11}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,2,4,9},{3,8},{5,11},{6,12},{7},{10}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,2,4,10},{3,8},{5,11},{6,12},{7},{9}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,2,4,10},{3,9},{5,11},{6,12},{7},{8}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,2,5,10},{3,6,12},{4,8},{7},{9},{11}}
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> 1000101010 => ? = 1
{{1,2,5,11},{3,6,12},{4,8},{7},{9},{10}}
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> 1000101010 => ? = 1
{{1,2,5,10},{3,6,12},{4,9},{7},{8},{11}}
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> 1000101010 => ? = 1
{{1,2,5,11},{3,6,12},{4,9},{7},{8},{10}}
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> 1000101010 => ? = 1
{{1,2,5,11},{3,6,12},{4,10},{7},{8},{9}}
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> 1000101010 => ? = 1
{{1,2,5,10},{3,7},{4,8},{6,12},{9},{11}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,2,5,11},{3,7},{4,8},{6,12},{9},{10}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,2,5,10},{3,7},{4,9},{6,12},{8},{11}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,2,5,11},{3,7},{4,9},{6,12},{8},{10}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,2,5,11},{3,7},{4,10},{6,12},{8},{9}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,2,5,10},{3,8},{4,9},{6,12},{7},{11}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,2,5,11},{3,8},{4,9},{6,12},{7},{10}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,2,5,11},{3,8},{4,10},{6,12},{7},{9}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,2,5,11},{3,9},{4,10},{6,12},{7},{8}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,2,6,12},{3,7},{4,8},{5,10},{9},{11}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,2,6,12},{3,7},{4,8},{5,11},{9},{10}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,2,6,12},{3,7},{4,9},{5,10},{8},{11}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,2,6,12},{3,7},{4,9},{5,11},{8},{10}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,2,6,12},{3,7},{4,10},{5,11},{8},{9}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,2,6,12},{3,8},{4,9},{5,10},{7},{11}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,2,6,12},{3,8},{4,9},{5,11},{7},{10}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,2,6,12},{3,8},{4,10},{5,11},{7},{9}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,2,6,12},{3,9},{4,10},{5,11},{7},{8}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> 1001000110 => ? = 1
{{1,3,6,12},{2,4,8},{5,10},{7},{9},{11}}
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> 1000101010 => ? = 1
{{1,3,6,12},{2,4,8},{5,11},{7},{9},{10}}
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> 1000101010 => ? = 1
{{1,3,6,12},{2,4,9},{5,10},{7},{8},{11}}
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> 1000101010 => ? = 1
{{1,3,6,12},{2,4,9},{5,11},{7},{8},{10}}
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> 1000101010 => ? = 1
{{1,3,6,12},{2,4,10},{5,11},{7},{8},{9}}
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> 1000101010 => ? = 1
Description
The number of leading ones in a binary word.
Matching statistic: St000382
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [[1,2]]
=> [2] => 2
{{1},{2}}
=> [1,1]
=> [[1],[2]]
=> [1,1] => 1
{{1,2,3}}
=> [3]
=> [[1,2,3]]
=> [3] => 3
{{1,2},{3}}
=> [2,1]
=> [[1,3],[2]]
=> [1,2] => 1
{{1,3},{2}}
=> [2,1]
=> [[1,3],[2]]
=> [1,2] => 1
{{1},{2,3}}
=> [2,1]
=> [[1,3],[2]]
=> [1,2] => 1
{{1},{2},{3}}
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1
{{1,2,3,4}}
=> [4]
=> [[1,2,3,4]]
=> [4] => 4
{{1,2,3},{4}}
=> [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
{{1,2,4},{3}}
=> [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
{{1,2},{3,4}}
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
{{1,2},{3},{4}}
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
{{1,3,4},{2}}
=> [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
{{1,3},{2,4}}
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
{{1,4},{2,3}}
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
{{1},{2,3,4}}
=> [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1
{{1,2,3,4,5}}
=> [5]
=> [[1,2,3,4,5]]
=> [5] => 5
{{1,2,3,4},{5}}
=> [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
{{1,2,3,5},{4}}
=> [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
{{1,2,3},{4,5}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
{{1,2,4,5},{3}}
=> [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
{{1,2,4},{3,5}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
{{1,2,5},{3,4}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 2
{{1,2},{3,4,5}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
{{1,3,4,5},{2}}
=> [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
{{1,3,4},{2,5}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
{{1,3,5},{2,4}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 2
{{1,3},{2,4,5}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
{{1,4,5},{2,3}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 2
{{1,4},{2,3,5}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
{{1,2,3,4,5,6,7,8,9,10},{11}}
=> [10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ? = 1
{{1},{2,3,4,5,6,7,8,9,10,11}}
=> [10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ? = 1
{{1,2,4,8},{3,6,12},{5,10},{7},{9},{11}}
=> [4,3,2,1,1,1]
=> [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ? = 1
{{1,2,3,4,10,11},{5},{6},{7},{8},{9}}
=> [6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ? => ? = 1
{{1,2,3,4,11},{5,6},{7},{8},{9},{10}}
=> [5,2,1,1,1,1]
=> [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
=> ? => ? = 1
{{1,11},{2,3,4,5,6,7,8,9,10}}
=> [9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> ? => ? = 2
{{1},{2},{3},{4},{5},{6},{7},{8},{9},{10,11}}
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? => ? = 1
{{1,2,3,4,5,6,7,8,9,11},{10}}
=> [10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ? = 1
{{1,3,4,5,6,7,8,9,10,11},{2}}
=> [10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ? = 1
{{1,2},{3},{4},{5},{6},{7},{8},{9},{10},{11}}
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? => ? = 1
{{1,11},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? => ? = 1
{{1,2,4,8},{3,6,12},{5,11},{7},{9},{10}}
=> [4,3,2,1,1,1]
=> [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ? = 1
{{1,2,4,9},{3,6,12},{5,10},{7},{8},{11}}
=> [4,3,2,1,1,1]
=> [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ? = 1
{{1,2,4,9},{3,6,12},{5,11},{7},{8},{10}}
=> [4,3,2,1,1,1]
=> [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ? = 1
{{1,2,4,10},{3,6,12},{5,11},{7},{8},{9}}
=> [4,3,2,1,1,1]
=> [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ? = 1
{{1,2,4,8},{3,7},{5,10},{6,12},{9},{11}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,4,8},{3,7},{5,11},{6,12},{9},{10}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,4,9},{3,7},{5,10},{6,12},{8},{11}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,4,9},{3,7},{5,11},{6,12},{8},{10}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,4,10},{3,7},{5,11},{6,12},{8},{9}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,4,9},{3,8},{5,10},{6,12},{7},{11}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,4,9},{3,8},{5,11},{6,12},{7},{10}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,4,10},{3,8},{5,11},{6,12},{7},{9}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,4,10},{3,9},{5,11},{6,12},{7},{8}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,5,10},{3,6,12},{4,8},{7},{9},{11}}
=> [4,3,2,1,1,1]
=> [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ? = 1
{{1,2,5,11},{3,6,12},{4,8},{7},{9},{10}}
=> [4,3,2,1,1,1]
=> [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ? = 1
{{1,2,5,10},{3,6,12},{4,9},{7},{8},{11}}
=> [4,3,2,1,1,1]
=> [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ? = 1
{{1,2,5,11},{3,6,12},{4,9},{7},{8},{10}}
=> [4,3,2,1,1,1]
=> [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ? = 1
{{1,2,5,11},{3,6,12},{4,10},{7},{8},{9}}
=> [4,3,2,1,1,1]
=> [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ? = 1
{{1,2,5,10},{3,7},{4,8},{6,12},{9},{11}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,5,11},{3,7},{4,8},{6,12},{9},{10}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,5,10},{3,7},{4,9},{6,12},{8},{11}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,5,11},{3,7},{4,9},{6,12},{8},{10}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,5,11},{3,7},{4,10},{6,12},{8},{9}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,5,10},{3,8},{4,9},{6,12},{7},{11}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,5,11},{3,8},{4,9},{6,12},{7},{10}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,5,11},{3,8},{4,10},{6,12},{7},{9}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,5,11},{3,9},{4,10},{6,12},{7},{8}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,6,12},{3,7},{4,8},{5,10},{9},{11}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,6,12},{3,7},{4,8},{5,11},{9},{10}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,6,12},{3,7},{4,9},{5,10},{8},{11}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,6,12},{3,7},{4,9},{5,11},{8},{10}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,6,12},{3,7},{4,10},{5,11},{8},{9}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,6,12},{3,8},{4,9},{5,10},{7},{11}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,6,12},{3,8},{4,9},{5,11},{7},{10}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,6,12},{3,8},{4,10},{5,11},{7},{9}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,6,12},{3,9},{4,10},{5,11},{7},{8}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
{{1,2,7},{3,8},{4,9},{5,10},{6,12},{11}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,2,7},{3,8},{4,9},{5,11},{6,12},{10}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
{{1,2,7},{3,8},{4,10},{5,11},{6,12},{9}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? => ? = 1
Description
The first part of an integer composition.
Matching statistic: St000733
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [1,1]
=> [[1],[2]]
=> 2
{{1},{2}}
=> [1,1]
=> [2]
=> [[1,2]]
=> 1
{{1,2,3}}
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
{{1,2},{3}}
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 1
{{1,3},{2}}
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 1
{{1},{2,3}}
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 1
{{1},{2},{3}}
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1
{{1,2,3,4}}
=> [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
{{1,2,3},{4}}
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 1
{{1,2,4},{3}}
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 1
{{1,2},{3,4}}
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
{{1,2},{3},{4}}
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
{{1,3,4},{2}}
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 1
{{1,3},{2,4}}
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
{{1,4},{2,3}}
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
{{1},{2,3,4}}
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 1
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 5
{{1,2,3,4},{5}}
=> [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1
{{1,2,3,5},{4}}
=> [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1
{{1,2,3},{4,5}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1
{{1,2,4,5},{3}}
=> [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1
{{1,2,4},{3,5}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1
{{1,2,5},{3,4}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
{{1,2},{3,4,5}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 1
{{1,3,4},{2,5}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1
{{1,3,5},{2,4}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
{{1,3},{2,4,5}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
{{1,4},{2,3,5}}
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1
{{1,2,3,4,5,6,7,8,9,10},{11}}
=> [10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 1
{{1},{2,3,4,5,6,7,8,9,10,11}}
=> [10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 1
{{1,2,4,8},{3,6,12},{5,10},{7},{9},{11}}
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> [[1,3,6,10,11,12],[2,5,9],[4,8],[7]]
=> ? = 1
{{1,2,3,4,10,11},{5},{6},{7},{8},{9}}
=> [6,1,1,1,1,1]
=> [6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ? = 1
{{1,2,3,4,11},{5,6},{7},{8},{9},{10}}
=> [5,2,1,1,1,1]
=> [6,2,1,1,1]
=> [[1,5,8,9,10,11],[2,7],[3],[4],[6]]
=> ? = 1
{{1,11},{2,3,4,5,6,7,8,9,10}}
=> [9,2]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,9],[2,11],[3],[4],[5],[6],[7],[8],[10]]
=> ? = 2
{{1},{2},{3},{4},{5},{6},{7},{8},{9},{10,11}}
=> [2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 1
{{1,2,3,4,5,6,7,8,9,11},{10}}
=> [10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 1
{{1,3,4,5,6,7,8,9,10,11},{2}}
=> [10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 1
{{1,2},{3},{4},{5},{6},{7},{8},{9},{10},{11}}
=> [2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 1
{{1,11},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 1
{{1,2,4,8},{3,6,12},{5,11},{7},{9},{10}}
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> [[1,3,6,10,11,12],[2,5,9],[4,8],[7]]
=> ? = 1
{{1,2,4,9},{3,6,12},{5,10},{7},{8},{11}}
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> [[1,3,6,10,11,12],[2,5,9],[4,8],[7]]
=> ? = 1
{{1,2,4,9},{3,6,12},{5,11},{7},{8},{10}}
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> [[1,3,6,10,11,12],[2,5,9],[4,8],[7]]
=> ? = 1
{{1,2,4,10},{3,6,12},{5,11},{7},{8},{9}}
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> [[1,3,6,10,11,12],[2,5,9],[4,8],[7]]
=> ? = 1
{{1,2,4,8},{3,7},{5,10},{6,12},{9},{11}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,4,8},{3,7},{5,11},{6,12},{9},{10}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,4,9},{3,7},{5,10},{6,12},{8},{11}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,4,9},{3,7},{5,11},{6,12},{8},{10}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,4,10},{3,7},{5,11},{6,12},{8},{9}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,4,9},{3,8},{5,10},{6,12},{7},{11}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,4,9},{3,8},{5,11},{6,12},{7},{10}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,4,10},{3,8},{5,11},{6,12},{7},{9}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,4,10},{3,9},{5,11},{6,12},{7},{8}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,5,10},{3,6,12},{4,8},{7},{9},{11}}
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> [[1,3,6,10,11,12],[2,5,9],[4,8],[7]]
=> ? = 1
{{1,2,5,11},{3,6,12},{4,8},{7},{9},{10}}
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> [[1,3,6,10,11,12],[2,5,9],[4,8],[7]]
=> ? = 1
{{1,2,5,10},{3,6,12},{4,9},{7},{8},{11}}
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> [[1,3,6,10,11,12],[2,5,9],[4,8],[7]]
=> ? = 1
{{1,2,5,11},{3,6,12},{4,9},{7},{8},{10}}
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> [[1,3,6,10,11,12],[2,5,9],[4,8],[7]]
=> ? = 1
{{1,2,5,11},{3,6,12},{4,10},{7},{8},{9}}
=> [4,3,2,1,1,1]
=> [6,3,2,1]
=> [[1,3,6,10,11,12],[2,5,9],[4,8],[7]]
=> ? = 1
{{1,2,5,10},{3,7},{4,8},{6,12},{9},{11}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,5,11},{3,7},{4,8},{6,12},{9},{10}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,5,10},{3,7},{4,9},{6,12},{8},{11}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,5,11},{3,7},{4,9},{6,12},{8},{10}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,5,11},{3,7},{4,10},{6,12},{8},{9}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,5,10},{3,8},{4,9},{6,12},{7},{11}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,5,11},{3,8},{4,9},{6,12},{7},{10}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,5,11},{3,8},{4,10},{6,12},{7},{9}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,5,11},{3,9},{4,10},{6,12},{7},{8}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,6,12},{3,7},{4,8},{5,10},{9},{11}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,6,12},{3,7},{4,8},{5,11},{9},{10}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,6,12},{3,7},{4,9},{5,10},{8},{11}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,6,12},{3,7},{4,9},{5,11},{8},{10}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,6,12},{3,7},{4,10},{5,11},{8},{9}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,6,12},{3,8},{4,9},{5,10},{7},{11}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,6,12},{3,8},{4,9},{5,11},{7},{10}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,6,12},{3,8},{4,10},{5,11},{7},{9}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,6,12},{3,9},{4,10},{5,11},{7},{8}}
=> [4,2,2,2,1,1]
=> [6,4,1,1]
=> [[1,4,5,6,11,12],[2,8,9,10],[3],[7]]
=> ? = 1
{{1,2,7},{3,8},{4,9},{5,10},{6,12},{11}}
=> [3,2,2,2,2,1]
=> [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? = 1
{{1,2,7},{3,8},{4,9},{5,11},{6,12},{10}}
=> [3,2,2,2,2,1]
=> [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? = 1
{{1,2,7},{3,8},{4,10},{5,11},{6,12},{9}}
=> [3,2,2,2,2,1]
=> [6,5,1]
=> [[1,3,4,5,6,12],[2,8,9,10,11],[7]]
=> ? = 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000745
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 2
{{1},{2}}
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 1
{{1,2,3}}
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 3
{{1,2},{3}}
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
{{1,3},{2}}
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
{{1},{2,3}}
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
{{1},{2},{3}}
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 1
{{1,2,3,4}}
=> [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 4
{{1,2,3},{4}}
=> [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 1
{{1,2,4},{3}}
=> [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 1
{{1,2},{3,4}}
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
{{1,2},{3},{4}}
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
{{1,3,4},{2}}
=> [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 1
{{1,3},{2,4}}
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
{{1,3},{2},{4}}
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
{{1,4},{2,3}}
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
{{1},{2,3,4}}
=> [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 1
{{1},{2,3},{4}}
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
{{1,4},{2},{3}}
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
{{1},{2,4},{3}}
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
{{1},{2},{3,4}}
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 1
{{1,2,3,4,5}}
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 5
{{1,2,3,4},{5}}
=> [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 1
{{1,2,3,5},{4}}
=> [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 1
{{1,2,3},{4,5}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
{{1,2,4,5},{3}}
=> [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 1
{{1,2,4},{3,5}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
{{1,2,5},{3,4}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
{{1,2},{3,4,5}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
{{1,3,4,5},{2}}
=> [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 1
{{1,3,4},{2,5}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
{{1,3,5},{2,4}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
{{1,3},{2,4,5}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
{{1,4,5},{2,3}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
{{1,4},{2,3,5}}
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
{{1,2,3,4,5,6,7,8,9,10},{11}}
=> [10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 1
{{1},{2,3,4,5,6,7,8,9,10,11}}
=> [10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 1
{{1,2,4,8},{3,6,12},{5,10},{7},{9},{11}}
=> [4,3,2,1,1,1]
=> [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
=> ?
=> ? = 1
{{1,2,3,4,10,11},{5},{6},{7},{8},{9}}
=> [6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ?
=> ? = 1
{{1,2,3,4,11},{5,6},{7},{8},{9},{10}}
=> [5,2,1,1,1,1]
=> [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
=> [[1,2,3,4,5,7],[6,8],[9],[10],[11]]
=> ? = 1
{{1,11},{2,3,4,5,6,7,8,9,10}}
=> [9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 2
{{1},{2},{3},{4},{5},{6},{7},{8},{9},{10,11}}
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 1
{{1,2,3,4,5,6,7,8,9,11},{10}}
=> [10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 1
{{1,3,4,5,6,7,8,9,10,11},{2}}
=> [10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 1
{{1,2},{3},{4},{5},{6},{7},{8},{9},{10},{11}}
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 1
{{1,11},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 1
{{1,2,4,8},{3,6,12},{5,11},{7},{9},{10}}
=> [4,3,2,1,1,1]
=> [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
=> ?
=> ? = 1
{{1,2,4,9},{3,6,12},{5,10},{7},{8},{11}}
=> [4,3,2,1,1,1]
=> [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
=> ?
=> ? = 1
{{1,2,4,9},{3,6,12},{5,11},{7},{8},{10}}
=> [4,3,2,1,1,1]
=> [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
=> ?
=> ? = 1
{{1,2,4,10},{3,6,12},{5,11},{7},{8},{9}}
=> [4,3,2,1,1,1]
=> [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
=> ?
=> ? = 1
{{1,2,4,8},{3,7},{5,10},{6,12},{9},{11}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,4,8},{3,7},{5,11},{6,12},{9},{10}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,4,9},{3,7},{5,10},{6,12},{8},{11}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,4,9},{3,7},{5,11},{6,12},{8},{10}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,4,10},{3,7},{5,11},{6,12},{8},{9}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,4,9},{3,8},{5,10},{6,12},{7},{11}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,4,9},{3,8},{5,11},{6,12},{7},{10}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,4,10},{3,8},{5,11},{6,12},{7},{9}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,4,10},{3,9},{5,11},{6,12},{7},{8}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,5,10},{3,6,12},{4,8},{7},{9},{11}}
=> [4,3,2,1,1,1]
=> [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
=> ?
=> ? = 1
{{1,2,5,11},{3,6,12},{4,8},{7},{9},{10}}
=> [4,3,2,1,1,1]
=> [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
=> ?
=> ? = 1
{{1,2,5,10},{3,6,12},{4,9},{7},{8},{11}}
=> [4,3,2,1,1,1]
=> [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
=> ?
=> ? = 1
{{1,2,5,11},{3,6,12},{4,9},{7},{8},{10}}
=> [4,3,2,1,1,1]
=> [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
=> ?
=> ? = 1
{{1,2,5,11},{3,6,12},{4,10},{7},{8},{9}}
=> [4,3,2,1,1,1]
=> [[1,5,8,12],[2,7,11],[3,10],[4],[6],[9]]
=> ?
=> ? = 1
{{1,2,5,10},{3,7},{4,8},{6,12},{9},{11}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,5,11},{3,7},{4,8},{6,12},{9},{10}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,5,10},{3,7},{4,9},{6,12},{8},{11}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,5,11},{3,7},{4,9},{6,12},{8},{10}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,5,11},{3,7},{4,10},{6,12},{8},{9}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,5,10},{3,8},{4,9},{6,12},{7},{11}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,5,11},{3,8},{4,9},{6,12},{7},{10}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,5,11},{3,8},{4,10},{6,12},{7},{9}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,5,11},{3,9},{4,10},{6,12},{7},{8}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,6,12},{3,7},{4,8},{5,10},{9},{11}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,6,12},{3,7},{4,8},{5,11},{9},{10}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,6,12},{3,7},{4,9},{5,10},{8},{11}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,6,12},{3,7},{4,9},{5,11},{8},{10}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,6,12},{3,7},{4,10},{5,11},{8},{9}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,6,12},{3,8},{4,9},{5,10},{7},{11}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,6,12},{3,8},{4,9},{5,11},{7},{10}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,6,12},{3,8},{4,10},{5,11},{7},{9}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,6,12},{3,9},{4,10},{5,11},{7},{8}}
=> [4,2,2,2,1,1]
=> [[1,4,11,12],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
{{1,2,7},{3,8},{4,9},{5,10},{6,12},{11}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,2,7},{3,8},{4,9},{5,11},{6,12},{10}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
{{1,2,7},{3,8},{4,10},{5,11},{6,12},{9}}
=> [3,2,2,2,2,1]
=> [[1,3,12],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11],[12]]
=> ? = 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000383
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> 100 => [1,2] => 2
{{1},{2}}
=> [1,1]
=> 110 => [2,1] => 1
{{1,2,3}}
=> [3]
=> 1000 => [1,3] => 3
{{1,2},{3}}
=> [2,1]
=> 1010 => [1,1,1,1] => 1
{{1,3},{2}}
=> [2,1]
=> 1010 => [1,1,1,1] => 1
{{1},{2,3}}
=> [2,1]
=> 1010 => [1,1,1,1] => 1
{{1},{2},{3}}
=> [1,1,1]
=> 1110 => [3,1] => 1
{{1,2,3,4}}
=> [4]
=> 10000 => [1,4] => 4
{{1,2,3},{4}}
=> [3,1]
=> 10010 => [1,2,1,1] => 1
{{1,2,4},{3}}
=> [3,1]
=> 10010 => [1,2,1,1] => 1
{{1,2},{3,4}}
=> [2,2]
=> 1100 => [2,2] => 2
{{1,2},{3},{4}}
=> [2,1,1]
=> 10110 => [1,1,2,1] => 1
{{1,3,4},{2}}
=> [3,1]
=> 10010 => [1,2,1,1] => 1
{{1,3},{2,4}}
=> [2,2]
=> 1100 => [2,2] => 2
{{1,3},{2},{4}}
=> [2,1,1]
=> 10110 => [1,1,2,1] => 1
{{1,4},{2,3}}
=> [2,2]
=> 1100 => [2,2] => 2
{{1},{2,3,4}}
=> [3,1]
=> 10010 => [1,2,1,1] => 1
{{1},{2,3},{4}}
=> [2,1,1]
=> 10110 => [1,1,2,1] => 1
{{1,4},{2},{3}}
=> [2,1,1]
=> 10110 => [1,1,2,1] => 1
{{1},{2,4},{3}}
=> [2,1,1]
=> 10110 => [1,1,2,1] => 1
{{1},{2},{3,4}}
=> [2,1,1]
=> 10110 => [1,1,2,1] => 1
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> 11110 => [4,1] => 1
{{1,2,3,4,5}}
=> [5]
=> 100000 => [1,5] => 5
{{1,2,3,4},{5}}
=> [4,1]
=> 100010 => [1,3,1,1] => 1
{{1,2,3,5},{4}}
=> [4,1]
=> 100010 => [1,3,1,1] => 1
{{1,2,3},{4,5}}
=> [3,2]
=> 10100 => [1,1,1,2] => 2
{{1,2,3},{4},{5}}
=> [3,1,1]
=> 100110 => [1,2,2,1] => 1
{{1,2,4,5},{3}}
=> [4,1]
=> 100010 => [1,3,1,1] => 1
{{1,2,4},{3,5}}
=> [3,2]
=> 10100 => [1,1,1,2] => 2
{{1,2,4},{3},{5}}
=> [3,1,1]
=> 100110 => [1,2,2,1] => 1
{{1,2,5},{3,4}}
=> [3,2]
=> 10100 => [1,1,1,2] => 2
{{1,2},{3,4,5}}
=> [3,2]
=> 10100 => [1,1,1,2] => 2
{{1,2},{3,4},{5}}
=> [2,2,1]
=> 11010 => [2,1,1,1] => 1
{{1,2,5},{3},{4}}
=> [3,1,1]
=> 100110 => [1,2,2,1] => 1
{{1,2},{3,5},{4}}
=> [2,2,1]
=> 11010 => [2,1,1,1] => 1
{{1,2},{3},{4,5}}
=> [2,2,1]
=> 11010 => [2,1,1,1] => 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> 101110 => [1,1,3,1] => 1
{{1,3,4,5},{2}}
=> [4,1]
=> 100010 => [1,3,1,1] => 1
{{1,3,4},{2,5}}
=> [3,2]
=> 10100 => [1,1,1,2] => 2
{{1,3,4},{2},{5}}
=> [3,1,1]
=> 100110 => [1,2,2,1] => 1
{{1,3,5},{2,4}}
=> [3,2]
=> 10100 => [1,1,1,2] => 2
{{1,3},{2,4,5}}
=> [3,2]
=> 10100 => [1,1,1,2] => 2
{{1,3},{2,4},{5}}
=> [2,2,1]
=> 11010 => [2,1,1,1] => 1
{{1,3,5},{2},{4}}
=> [3,1,1]
=> 100110 => [1,2,2,1] => 1
{{1,3},{2,5},{4}}
=> [2,2,1]
=> 11010 => [2,1,1,1] => 1
{{1,3},{2},{4,5}}
=> [2,2,1]
=> 11010 => [2,1,1,1] => 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> 101110 => [1,1,3,1] => 1
{{1,4,5},{2,3}}
=> [3,2]
=> 10100 => [1,1,1,2] => 2
{{1,4},{2,3,5}}
=> [3,2]
=> 10100 => [1,1,1,2] => 2
{{1,4},{2,3},{5}}
=> [2,2,1]
=> 11010 => [2,1,1,1] => 1
{{1,2,3,4,5,6,7,8,9,10},{11}}
=> [10,1]
=> 100000000010 => ? => ? = 1
{{1},{2,3,4,5,6,7,8,9,10,11}}
=> [10,1]
=> 100000000010 => ? => ? = 1
{{1,2,4,8},{3,6,12},{5,10},{7},{9},{11}}
=> [4,3,2,1,1,1]
=> 1010101110 => ? => ? = 1
{{1,2,3,4,10,11},{5},{6},{7},{8},{9}}
=> [6,1,1,1,1,1]
=> 100000111110 => [1,5,5,1] => ? = 1
{{1,2,3,4,11},{5,6},{7},{8},{9},{10}}
=> [5,2,1,1,1,1]
=> 10001011110 => [1,3,1,1,4,1] => ? = 1
{{1,11},{2,3,4,5,6,7,8,9,10}}
=> [9,2]
=> 10000000100 => [1,7,1,2] => ? = 2
{{1},{2},{3},{4},{5},{6},{7},{8},{9},{10,11}}
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? = 1
{{1,2,3,4,5,6,7,8,9,11},{10}}
=> [10,1]
=> 100000000010 => ? => ? = 1
{{1,7},{2,8},{3,9},{4,10},{5,11},{6,12}}
=> [2,2,2,2,2,2]
=> 11111100 => [6,2] => ? = 2
{{1,3,4,5,6,7,8,9,10,11},{2}}
=> [10,1]
=> 100000000010 => ? => ? = 1
{{1,2},{3},{4},{5},{6},{7},{8},{9},{10},{11}}
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? = 1
{{1,11},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? = 1
{{1,2,4,8},{3,6,12},{5,11},{7},{9},{10}}
=> [4,3,2,1,1,1]
=> 1010101110 => ? => ? = 1
{{1,2,4,9},{3,6,12},{5,10},{7},{8},{11}}
=> [4,3,2,1,1,1]
=> 1010101110 => ? => ? = 1
{{1,2,4,9},{3,6,12},{5,11},{7},{8},{10}}
=> [4,3,2,1,1,1]
=> 1010101110 => ? => ? = 1
{{1,2,4,10},{3,6,12},{5,11},{7},{8},{9}}
=> [4,3,2,1,1,1]
=> 1010101110 => ? => ? = 1
{{1,2,4,8},{3,7},{5,10},{6,12},{9},{11}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,4,8},{3,7},{5,11},{6,12},{9},{10}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,4,9},{3,7},{5,10},{6,12},{8},{11}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,4,9},{3,7},{5,11},{6,12},{8},{10}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,4,10},{3,7},{5,11},{6,12},{8},{9}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,4,9},{3,8},{5,10},{6,12},{7},{11}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,4,9},{3,8},{5,11},{6,12},{7},{10}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,4,10},{3,8},{5,11},{6,12},{7},{9}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,4,10},{3,9},{5,11},{6,12},{7},{8}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,5,10},{3,6,12},{4,8},{7},{9},{11}}
=> [4,3,2,1,1,1]
=> 1010101110 => ? => ? = 1
{{1,2,5,11},{3,6,12},{4,8},{7},{9},{10}}
=> [4,3,2,1,1,1]
=> 1010101110 => ? => ? = 1
{{1,2,5,10},{3,6,12},{4,9},{7},{8},{11}}
=> [4,3,2,1,1,1]
=> 1010101110 => ? => ? = 1
{{1,2,5,11},{3,6,12},{4,9},{7},{8},{10}}
=> [4,3,2,1,1,1]
=> 1010101110 => ? => ? = 1
{{1,2,5,11},{3,6,12},{4,10},{7},{8},{9}}
=> [4,3,2,1,1,1]
=> 1010101110 => ? => ? = 1
{{1,2,5,10},{3,7},{4,8},{6,12},{9},{11}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,5,11},{3,7},{4,8},{6,12},{9},{10}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,5,10},{3,7},{4,9},{6,12},{8},{11}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,5,11},{3,7},{4,9},{6,12},{8},{10}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,5,11},{3,7},{4,10},{6,12},{8},{9}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,5,10},{3,8},{4,9},{6,12},{7},{11}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,5,11},{3,8},{4,9},{6,12},{7},{10}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,5,11},{3,8},{4,10},{6,12},{7},{9}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,5,11},{3,9},{4,10},{6,12},{7},{8}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,6,12},{3,7},{4,8},{5,10},{9},{11}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,6,12},{3,7},{4,8},{5,11},{9},{10}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,6,12},{3,7},{4,9},{5,10},{8},{11}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,6,12},{3,7},{4,9},{5,11},{8},{10}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,6,12},{3,7},{4,10},{5,11},{8},{9}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,6,12},{3,8},{4,9},{5,10},{7},{11}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,6,12},{3,8},{4,9},{5,11},{7},{10}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,6,12},{3,8},{4,10},{5,11},{7},{9}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,6,12},{3,9},{4,10},{5,11},{7},{8}}
=> [4,2,2,2,1,1]
=> 1001110110 => ? => ? = 1
{{1,2,7},{3,8},{4,9},{5,10},{6,12},{11}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
{{1,2,7},{3,8},{4,9},{5,11},{6,12},{10}}
=> [3,2,2,2,2,1]
=> 101111010 => [1,1,4,1,1,1] => ? = 1
Description
The last part of an integer composition.
Matching statistic: St000657
(load all 50 compositions to match this statistic)
(load all 50 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 90%
St000657: Integer compositions ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 90%
Values
{{1,2}}
=> [2] => 2
{{1},{2}}
=> [1,1] => 1
{{1,2,3}}
=> [3] => 3
{{1,2},{3}}
=> [2,1] => 1
{{1,3},{2}}
=> [2,1] => 1
{{1},{2,3}}
=> [1,2] => 1
{{1},{2},{3}}
=> [1,1,1] => 1
{{1,2,3,4}}
=> [4] => 4
{{1,2,3},{4}}
=> [3,1] => 1
{{1,2,4},{3}}
=> [3,1] => 1
{{1,2},{3,4}}
=> [2,2] => 2
{{1,2},{3},{4}}
=> [2,1,1] => 1
{{1,3,4},{2}}
=> [3,1] => 1
{{1,3},{2,4}}
=> [2,2] => 2
{{1,3},{2},{4}}
=> [2,1,1] => 1
{{1,4},{2,3}}
=> [2,2] => 2
{{1},{2,3,4}}
=> [1,3] => 1
{{1},{2,3},{4}}
=> [1,2,1] => 1
{{1,4},{2},{3}}
=> [2,1,1] => 1
{{1},{2,4},{3}}
=> [1,2,1] => 1
{{1},{2},{3,4}}
=> [1,1,2] => 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => 1
{{1,2,3,4,5}}
=> [5] => 5
{{1,2,3,4},{5}}
=> [4,1] => 1
{{1,2,3,5},{4}}
=> [4,1] => 1
{{1,2,3},{4,5}}
=> [3,2] => 2
{{1,2,3},{4},{5}}
=> [3,1,1] => 1
{{1,2,4,5},{3}}
=> [4,1] => 1
{{1,2,4},{3,5}}
=> [3,2] => 2
{{1,2,4},{3},{5}}
=> [3,1,1] => 1
{{1,2,5},{3,4}}
=> [3,2] => 2
{{1,2},{3,4,5}}
=> [2,3] => 2
{{1,2},{3,4},{5}}
=> [2,2,1] => 1
{{1,2,5},{3},{4}}
=> [3,1,1] => 1
{{1,2},{3,5},{4}}
=> [2,2,1] => 1
{{1,2},{3},{4,5}}
=> [2,1,2] => 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => 1
{{1,3,4,5},{2}}
=> [4,1] => 1
{{1,3,4},{2,5}}
=> [3,2] => 2
{{1,3,4},{2},{5}}
=> [3,1,1] => 1
{{1,3,5},{2,4}}
=> [3,2] => 2
{{1,3},{2,4,5}}
=> [2,3] => 2
{{1,3},{2,4},{5}}
=> [2,2,1] => 1
{{1,3,5},{2},{4}}
=> [3,1,1] => 1
{{1,3},{2,5},{4}}
=> [2,2,1] => 1
{{1,3},{2},{4,5}}
=> [2,1,2] => 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => 1
{{1,4,5},{2,3}}
=> [3,2] => 2
{{1,4},{2,3,5}}
=> [2,3] => 2
{{1,4},{2,3},{5}}
=> [2,2,1] => 1
{{1,6},{2,7},{3,8},{4,9},{5,10}}
=> [2,2,2,2,2] => ? = 2
{{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [1,1,1,1,1,1,1,1,1,1] => ? = 1
{{1},{2},{3},{4},{5},{6},{7},{8},{9,10}}
=> [1,1,1,1,1,1,1,1,2] => ? = 1
{{1},{2},{3},{4},{5},{6},{7},{8,10},{9}}
=> [1,1,1,1,1,1,1,2,1] => ? = 1
{{1},{2},{3},{4},{5},{6},{7,10},{8},{9}}
=> [1,1,1,1,1,1,2,1,1] => ? = 1
{{1},{2},{3},{4},{5},{6,10},{7},{8},{9}}
=> [1,1,1,1,1,2,1,1,1] => ? = 1
{{1},{2},{3},{4},{5,10},{6},{7},{8},{9}}
=> [1,1,1,1,2,1,1,1,1] => ? = 1
{{1},{2},{3},{4,10},{5},{6},{7},{8},{9}}
=> [1,1,1,2,1,1,1,1,1] => ? = 1
{{1},{2},{3,10},{4},{5},{6},{7},{8},{9}}
=> [1,1,2,1,1,1,1,1,1] => ? = 1
{{1},{2,10},{3},{4},{5},{6},{7},{8},{9}}
=> [1,2,1,1,1,1,1,1,1] => ? = 1
{{1,10},{2},{3},{4},{5},{6},{7},{8},{9}}
=> [2,1,1,1,1,1,1,1,1] => ? = 1
{{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1] => ? = 1
{{1},{2,3,4,5,6,7,8,9,10}}
=> [1,9] => ? = 1
{{1,2,3,4,5,6,7,8,9,10},{11}}
=> [10,1] => ? = 1
{{1,2,3,4,5,6,7,9},{8},{10}}
=> [8,1,1] => ? = 1
{{1},{2,3,4,5,6,7,8,10},{9}}
=> [1,8,1] => ? = 1
{{1},{2,3,4,5,6,7,8,9,10,11}}
=> [1,10] => ? = 1
{{1,2,3,4,5},{6,7,8,9,10}}
=> [5,5] => ? = 5
{{1,2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1] => ? = 1
{{1,3},{2},{4},{5},{6},{7},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1] => ? = 1
{{1,4},{2},{3},{5},{6},{7},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1] => ? = 1
{{1,5},{2},{3},{4},{6},{7},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1] => ? = 1
{{1,6},{2},{3},{4},{5},{7},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1] => ? = 1
{{1,7},{2},{3},{4},{5},{6},{8},{9},{10}}
=> [2,1,1,1,1,1,1,1,1] => ? = 1
{{1,8},{2},{3},{4},{5},{6},{7},{9},{10}}
=> [2,1,1,1,1,1,1,1,1] => ? = 1
{{1,9},{2},{3},{4},{5},{6},{7},{8},{10}}
=> [2,1,1,1,1,1,1,1,1] => ? = 1
{{1,2,3,4,5,6,7,8,9,10}}
=> [10] => ? = 10
{{1,2,3,4,5,6,7,8},{9,10}}
=> [8,2] => ? = 2
{{1,2,3,4,5,6,7,8},{9},{10}}
=> [8,1,1] => ? = 1
{{1,2,3,4,5,6,7},{8,9,10}}
=> [7,3] => ? = 3
{{1,2,3,4,5,6,7},{8,9},{10}}
=> [7,2,1] => ? = 1
{{1,2,3,4,5,6,7},{8},{9},{10}}
=> [7,1,1,1] => ? = 1
{{1,2,3,4,5,6},{7,8,9,10}}
=> [6,4] => ? = 4
{{1,2,3,4,5,6},{7},{8},{9},{10}}
=> [6,1,1,1,1] => ? = 1
{{1,2,3,4,5},{6},{7},{8},{9},{10}}
=> [5,1,1,1,1,1] => ? = 1
{{1,2,3,4},{5},{6},{7},{8},{9},{10}}
=> [4,1,1,1,1,1,1] => ? = 1
{{1,2,3},{4},{5},{6},{7},{8},{9},{10}}
=> [3,1,1,1,1,1,1,1] => ? = 1
{{1,3,4,5,6,7,8,9,10},{2}}
=> [9,1] => ? = 1
{{1,8,9,10},{2},{3},{4},{5},{6},{7}}
=> [4,1,1,1,1,1,1] => ? = 1
{{1,9,10},{2},{3},{4},{5},{6},{7},{8}}
=> [3,1,1,1,1,1,1,1] => ? = 1
{{1},{2,4,5,6,7,8,9,10},{3}}
=> [1,8,1] => ? = 1
{{1,3,4,5,6,7,8,9},{2},{10}}
=> [8,1,1] => ? = 1
{{1,2},{3,4,5,6,7,8,9,10}}
=> [2,8] => ? = 2
{{1,2,3,4},{5,6,7,8,9,10}}
=> [4,6] => ? = 4
{{1,2,4,8},{3,6},{5,10},{7},{9}}
=> [4,2,2,1,1] => ? = 1
{{1,2,4,8},{3,6,12},{5,10},{7},{9},{11}}
=> [4,3,2,1,1,1] => ? = 1
{{1,2,3,4,9,10},{5},{6},{7},{8}}
=> [6,1,1,1,1] => ? = 1
{{1,2,3,4,10},{5,6},{7},{8},{9}}
=> [5,2,1,1,1] => ? = 1
{{1,2,3,4,10,11},{5},{6},{7},{8},{9}}
=> [6,1,1,1,1,1] => ? = 1
{{1,2,3,9,10},{4},{5},{6},{7},{8}}
=> [5,1,1,1,1,1] => ? = 1
Description
The smallest part of an integer composition.
The following 15 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000990The first ascent of a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000655The length of the minimal rise of a Dyck path. St000700The protection number of an ordered tree. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St000654The first descent of a permutation. St001075The minimal size of a block of a set partition. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000090The variation of a composition. St000260The radius of a connected graph. St000487The length of the shortest cycle of a permutation. St000210Minimum over maximum difference of elements in cycles. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000314The number of left-to-right-maxima of a permutation.
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