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Your data matches 10 different statistics following compositions of up to 3 maps.
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Matching statistic: St000232
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
St000232: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> 0
{{1,2}}
=> 0
{{1},{2}}
=> 0
{{1,2,3}}
=> 0
{{1,2},{3}}
=> 0
{{1,3},{2}}
=> 0
{{1},{2,3}}
=> 0
{{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> 0
{{1,2,4},{3}}
=> 0
{{1,2},{3,4}}
=> 0
{{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> 0
{{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> 0
{{1},{2,3,4}}
=> 0
{{1},{2,3},{4}}
=> 0
{{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> 0
{{1},{2},{3,4}}
=> 0
{{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> 0
{{1,2,3,5},{4}}
=> 0
{{1,2,3},{4,5}}
=> 0
{{1,2,3},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> 0
{{1,2,4},{3,5}}
=> 1
{{1,2,4},{3},{5}}
=> 0
{{1,2,5},{3,4}}
=> 0
{{1,2},{3,4,5}}
=> 0
{{1,2},{3,4},{5}}
=> 0
{{1,2,5},{3},{4}}
=> 0
{{1,2},{3,5},{4}}
=> 0
{{1,2},{3},{4,5}}
=> 0
{{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> 0
{{1,3,4},{2,5}}
=> 1
{{1,3,4},{2},{5}}
=> 0
{{1,3,5},{2,4}}
=> 2
{{1,3},{2,4,5}}
=> 1
{{1,3},{2,4},{5}}
=> 1
{{1,3,5},{2},{4}}
=> 0
{{1,3},{2,5},{4}}
=> 1
{{1,3},{2},{4,5}}
=> 0
{{1,3},{2},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> 0
{{1,4},{2,3,5}}
=> 1
Description
The number of crossings of a set partition.
This is given by the number of $i < i' < j < j'$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Matching statistic: St000233
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
St000233: Set partitions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
St000233: Set partitions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> {{1}}
=> 0
{{1,2}}
=> {{1,2}}
=> 0
{{1},{2}}
=> {{1},{2}}
=> 0
{{1,2,3}}
=> {{1,2,3}}
=> 0
{{1,2},{3}}
=> {{1,2},{3}}
=> 0
{{1,3},{2}}
=> {{1,3},{2}}
=> 0
{{1},{2,3}}
=> {{1},{2,3}}
=> 0
{{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 0
{{1,2,4},{3}}
=> {{1,2,4},{3}}
=> 0
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 0
{{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> {{1,3,4},{2}}
=> 0
{{1,3},{2,4}}
=> {{1,4},{2,3}}
=> 1
{{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> {{1,3},{2,4}}
=> 0
{{1},{2,3,4}}
=> {{1},{2,3,4}}
=> 0
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 0
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 0
{{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> 0
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> 0
{{1,2,3,5},{4}}
=> {{1,2,3,5},{4}}
=> 0
{{1,2,3},{4,5}}
=> {{1,2,3},{4,5}}
=> 0
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 0
{{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> 0
{{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> 1
{{1,2,4},{3},{5}}
=> {{1,2,4},{3},{5}}
=> 0
{{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> 0
{{1,2},{3,4,5}}
=> {{1,2},{3,4,5}}
=> 0
{{1,2},{3,4},{5}}
=> {{1,2},{3,4},{5}}
=> 0
{{1,2,5},{3},{4}}
=> {{1,2,5},{3},{4}}
=> 0
{{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> 0
{{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> 0
{{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> 0
{{1,3,4,5},{2}}
=> {{1,3,4,5},{2}}
=> 0
{{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> 1
{{1,3,4},{2},{5}}
=> {{1,3,4},{2},{5}}
=> 0
{{1,3,5},{2,4}}
=> {{1,5},{2,3,4}}
=> 2
{{1,3},{2,4,5}}
=> {{1,4,5},{2,3}}
=> 1
{{1,3},{2,4},{5}}
=> {{1,4},{2,3},{5}}
=> 1
{{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> 0
{{1,3},{2,5},{4}}
=> {{1,5},{2,3},{4}}
=> 1
{{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> 0
{{1,3},{2},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> 0
{{1,4,5},{2,3}}
=> {{1,3},{2,4,5}}
=> 0
{{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> 1
{{1,4},{2,3},{5,6},{7,8},{9,10}}
=> {{1,3},{2,4},{5,6},{7,8},{9,10}}
=> ? = 0
{{1,6},{2,3},{4,5},{7,8},{9,10}}
=> {{1,3},{2,5},{4,6},{7,8},{9,10}}
=> ? = 0
{{1,8},{2,3},{4,5},{6,7},{9,10}}
=> {{1,3},{2,5},{4,7},{6,8},{9,10}}
=> ? = 0
{{1,10},{2,3},{4,5},{6,7},{8,9}}
=> {{1,3},{2,5},{4,7},{6,9},{8,10}}
=> ? = 0
{{1,2},{3,6},{4,5},{7,8},{9,10}}
=> {{1,2},{3,5},{4,6},{7,8},{9,10}}
=> ? = 0
{{1,6},{2,5},{3,4},{7,8},{9,10}}
=> {{1,4},{2,5},{3,6},{7,8},{9,10}}
=> ? = 0
{{1,8},{2,5},{3,4},{6,7},{9,10}}
=> {{1,4},{2,5},{3,7},{6,8},{9,10}}
=> ? = 0
{{1,10},{2,5},{3,4},{6,7},{8,9}}
=> {{1,4},{2,5},{3,7},{6,9},{8,10}}
=> ? = 0
{{1,2},{3,8},{4,5},{6,7},{9,10}}
=> {{1,2},{3,5},{4,7},{6,8},{9,10}}
=> ? = 0
{{1,8},{2,7},{3,4},{5,6},{9,10}}
=> {{1,4},{2,6},{3,7},{5,8},{9,10}}
=> ? = 0
{{1,10},{2,7},{3,4},{5,6},{8,9}}
=> {{1,4},{2,6},{3,7},{5,9},{8,10}}
=> ? = 0
{{1,2},{3,10},{4,5},{6,7},{8,9}}
=> {{1,2},{3,5},{4,7},{6,9},{8,10}}
=> ? = 0
{{1,10},{2,9},{3,4},{5,6},{7,8}}
=> {{1,4},{2,6},{3,8},{5,9},{7,10}}
=> ? = 0
{{1,2},{3,4},{5,8},{6,7},{9,10}}
=> {{1,2},{3,4},{5,7},{6,8},{9,10}}
=> ? = 0
{{1,4},{2,3},{5,8},{6,7},{9,10}}
=> {{1,3},{2,4},{5,7},{6,8},{9,10}}
=> ? = 0
{{1,8},{2,3},{4,7},{5,6},{9,10}}
=> {{1,3},{2,6},{4,7},{5,8},{9,10}}
=> ? = 0
{{1,10},{2,3},{4,7},{5,6},{8,9}}
=> {{1,3},{2,6},{4,7},{5,9},{8,10}}
=> ? = 0
{{1,2},{3,8},{4,7},{5,6},{9,10}}
=> {{1,2},{3,6},{4,7},{5,8},{9,10}}
=> ? = 0
{{1,8},{2,7},{3,6},{4,5},{9,10}}
=> {{1,5},{2,6},{3,7},{4,8},{9,10}}
=> ? = 0
{{1,10},{2,7},{3,6},{4,5},{8,9}}
=> {{1,5},{2,6},{3,7},{4,9},{8,10}}
=> ? = 0
{{1,2},{3,10},{4,7},{5,6},{8,9}}
=> {{1,2},{3,6},{4,7},{5,9},{8,10}}
=> ? = 0
{{1,10},{2,9},{3,6},{4,5},{7,8}}
=> {{1,5},{2,6},{3,8},{4,9},{7,10}}
=> ? = 0
{{1,2},{3,4},{5,10},{6,7},{8,9}}
=> {{1,2},{3,4},{5,7},{6,9},{8,10}}
=> ? = 0
{{1,4},{2,3},{5,10},{6,7},{8,9}}
=> {{1,3},{2,4},{5,7},{6,9},{8,10}}
=> ? = 0
{{1,10},{2,3},{4,9},{5,6},{7,8}}
=> {{1,3},{2,6},{4,8},{5,9},{7,10}}
=> ? = 0
{{1,2},{3,10},{4,9},{5,6},{7,8}}
=> {{1,2},{3,6},{4,8},{5,9},{7,10}}
=> ? = 0
{{1,10},{2,9},{3,8},{4,5},{6,7}}
=> {{1,5},{2,7},{3,8},{4,9},{6,10}}
=> ? = 0
{{1,2},{3,4},{5,6},{7,10},{8,9}}
=> {{1,2},{3,4},{5,6},{7,9},{8,10}}
=> ? = 0
{{1,4},{2,3},{5,6},{7,10},{8,9}}
=> {{1,3},{2,4},{5,6},{7,9},{8,10}}
=> ? = 0
{{1,6},{2,3},{4,5},{7,10},{8,9}}
=> {{1,3},{2,5},{4,6},{7,9},{8,10}}
=> ? = 0
{{1,10},{2,3},{4,5},{6,9},{7,8}}
=> {{1,3},{2,5},{4,8},{6,9},{7,10}}
=> ? = 0
{{1,2},{3,6},{4,5},{7,10},{8,9}}
=> {{1,2},{3,5},{4,6},{7,9},{8,10}}
=> ? = 0
{{1,6},{2,5},{3,4},{7,10},{8,9}}
=> {{1,4},{2,5},{3,6},{7,9},{8,10}}
=> ? = 0
{{1,10},{2,5},{3,4},{6,9},{7,8}}
=> {{1,4},{2,5},{3,8},{6,9},{7,10}}
=> ? = 0
{{1,2},{3,10},{4,5},{6,9},{7,8}}
=> {{1,2},{3,5},{4,8},{6,9},{7,10}}
=> ? = 0
{{1,10},{2,9},{3,4},{5,8},{6,7}}
=> {{1,4},{2,7},{3,8},{5,9},{6,10}}
=> ? = 0
{{1,2},{3,4},{5,10},{6,9},{7,8}}
=> {{1,2},{3,4},{5,8},{6,9},{7,10}}
=> ? = 0
{{1,4},{2,3},{5,10},{6,9},{7,8}}
=> {{1,3},{2,4},{5,8},{6,9},{7,10}}
=> ? = 0
{{1,10},{2,3},{4,9},{5,8},{6,7}}
=> {{1,3},{2,7},{4,8},{5,9},{6,10}}
=> ? = 0
{{1,2},{3,10},{4,9},{5,8},{6,7}}
=> {{1,2},{3,7},{4,8},{5,9},{6,10}}
=> ? = 0
{{1,10},{2,9},{3,8},{4,7},{5,6}}
=> {{1,6},{2,7},{3,8},{4,9},{5,10}}
=> ? = 0
{{1},{2,3,4,8},{5,6,7}}
=> {{1},{2,3,4,6,8},{5,7}}
=> ? = 0
{{1},{2,3,4,5,8},{6,7}}
=> {{1},{2,3,4,5,7},{6,8}}
=> ? = 0
{{1,2,5,6,7,8},{3,4}}
=> {{1,2,4},{3,5,6,7,8}}
=> ? = 0
{{1,2,3,6,7,8},{4,5}}
=> {{1,2,3,5},{4,6,7,8}}
=> ? = 0
{{1,2,3,7},{4,5,6},{8}}
=> {{1,2,3,5,7},{4,6},{8}}
=> ? = 0
{{1,2,3,4,7},{5,6},{8}}
=> {{1,2,3,4,6},{5,7},{8}}
=> ? = 0
{{1,2,3,4,7,8},{5,6}}
=> {{1,2,3,4,6},{5,7,8}}
=> ? = 0
{{1,8},{2,3,4,5,6,7}}
=> {{1,3,5,7},{2,4,6,8}}
=> ? = 0
{{1,3,5,6,7,8},{2,4}}
=> {{1,5,6,7,8},{2,3,4}}
=> ? = 2
Description
The number of nestings of a set partition.
This is given by the number of $i < i' < j' < j$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Matching statistic: St001549
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
St001549: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 57%
St001549: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 57%
Values
{{1}}
=> [1] => 0
{{1,2}}
=> [2,1] => 0
{{1},{2}}
=> [1,2] => 0
{{1,2,3}}
=> [2,3,1] => 0
{{1,2},{3}}
=> [2,1,3] => 0
{{1,3},{2}}
=> [3,2,1] => 0
{{1},{2,3}}
=> [1,3,2] => 0
{{1},{2},{3}}
=> [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => 0
{{1,2},{3,4}}
=> [2,1,4,3] => 0
{{1,2},{3},{4}}
=> [2,1,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => 0
{{1,3},{2,4}}
=> [3,4,1,2] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => 0
{{1,4},{2,3}}
=> [4,3,2,1] => 0
{{1},{2,3,4}}
=> [1,3,4,2] => 0
{{1},{2,3},{4}}
=> [1,3,2,4] => 0
{{1,4},{2},{3}}
=> [4,2,3,1] => 0
{{1},{2,4},{3}}
=> [1,4,3,2] => 0
{{1},{2},{3,4}}
=> [1,2,4,3] => 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => 0
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => 0
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => 1
{{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => ? = 0
{{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => ? = 0
{{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => ? = 0
{{1,2,3,4,5},{6,7}}
=> [2,3,4,5,1,7,6] => ? = 0
{{1,2,3,4,5},{6},{7}}
=> [2,3,4,5,1,6,7] => ? = 0
{{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => ? = 0
{{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => ? = 1
{{1,2,3,4,6},{5},{7}}
=> [2,3,4,6,5,1,7] => ? = 0
{{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => ? = 0
{{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => ? = 0
{{1,2,3,4},{5,6},{7}}
=> [2,3,4,1,6,5,7] => ? = 0
{{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => ? = 0
{{1,2,3,4},{5,7},{6}}
=> [2,3,4,1,7,6,5] => ? = 0
{{1,2,3,4},{5},{6,7}}
=> [2,3,4,1,5,7,6] => ? = 0
{{1,2,3,4},{5},{6},{7}}
=> [2,3,4,1,5,6,7] => ? = 0
{{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => ? = 0
{{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => ? = 1
{{1,2,3,5,6},{4},{7}}
=> [2,3,5,4,6,1,7] => ? = 0
{{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => ? = 2
{{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => ? = 1
{{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => ? = 1
{{1,2,3,5,7},{4},{6}}
=> [2,3,5,4,7,6,1] => ? = 0
{{1,2,3,5},{4,7},{6}}
=> [2,3,5,7,1,6,4] => ? = 1
{{1,2,3,5},{4},{6,7}}
=> [2,3,5,4,1,7,6] => ? = 0
{{1,2,3,5},{4},{6},{7}}
=> [2,3,5,4,1,6,7] => ? = 0
{{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => ? = 0
{{1,2,3,6},{4,5,7}}
=> [2,3,6,5,7,1,4] => ? = 1
{{1,2,3,6},{4,5},{7}}
=> [2,3,6,5,4,1,7] => ? = 0
{{1,2,3,7},{4,5,6}}
=> [2,3,7,5,6,4,1] => ? = 0
{{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => ? = 0
{{1,2,3},{4,5,6},{7}}
=> [2,3,1,5,6,4,7] => ? = 0
{{1,2,3,7},{4,5},{6}}
=> [2,3,7,5,4,6,1] => ? = 0
{{1,2,3},{4,5,7},{6}}
=> [2,3,1,5,7,6,4] => ? = 0
{{1,2,3},{4,5},{6,7}}
=> [2,3,1,5,4,7,6] => ? = 0
{{1,2,3},{4,5},{6},{7}}
=> [2,3,1,5,4,6,7] => ? = 0
{{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => ? = 0
{{1,2,3,6},{4,7},{5}}
=> [2,3,6,7,5,1,4] => ? = 1
{{1,2,3,6},{4},{5,7}}
=> [2,3,6,4,7,1,5] => ? = 1
{{1,2,3,6},{4},{5},{7}}
=> [2,3,6,4,5,1,7] => ? = 0
{{1,2,3,7},{4,6},{5}}
=> [2,3,7,6,5,4,1] => ? = 0
{{1,2,3},{4,6,7},{5}}
=> [2,3,1,6,5,7,4] => ? = 0
{{1,2,3},{4,6},{5,7}}
=> [2,3,1,6,7,4,5] => ? = 1
{{1,2,3},{4,6},{5},{7}}
=> [2,3,1,6,5,4,7] => ? = 0
{{1,2,3,7},{4},{5,6}}
=> [2,3,7,4,6,5,1] => ? = 0
{{1,2,3},{4,7},{5,6}}
=> [2,3,1,7,6,5,4] => ? = 0
{{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => ? = 0
{{1,2,3},{4},{5,6},{7}}
=> [2,3,1,4,6,5,7] => ? = 0
{{1,2,3,7},{4},{5},{6}}
=> [2,3,7,4,5,6,1] => ? = 0
{{1,2,3},{4,7},{5},{6}}
=> [2,3,1,7,5,6,4] => ? = 0
{{1,2,3},{4},{5,7},{6}}
=> [2,3,1,4,7,6,5] => ? = 0
Description
The number of restricted non-inversions between exceedances.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{nie}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(i) < \sigma(j) \}.$$
Matching statistic: St001513
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St001513: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 57%
Mp00080: Set partitions —to permutation⟶ Permutations
St001513: Permutations ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 57%
Values
{{1}}
=> {{1}}
=> [1] => 0
{{1,2}}
=> {{1,2}}
=> [2,1] => 0
{{1},{2}}
=> {{1},{2}}
=> [1,2] => 0
{{1,2,3}}
=> {{1,2,3}}
=> [2,3,1] => 0
{{1,2},{3}}
=> {{1,2},{3}}
=> [2,1,3] => 0
{{1,3},{2}}
=> {{1,3},{2}}
=> [3,2,1] => 0
{{1},{2,3}}
=> {{1},{2,3}}
=> [1,3,2] => 0
{{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,2,3] => 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => 0
{{1,2,3},{4}}
=> {{1,2,3},{4}}
=> [2,3,1,4] => 0
{{1,2,4},{3}}
=> {{1,2,4},{3}}
=> [2,4,3,1] => 0
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => 0
{{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,3,4] => 0
{{1,3,4},{2}}
=> {{1,3,4},{2}}
=> [3,2,4,1] => 0
{{1,3},{2,4}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => 1
{{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> [3,2,1,4] => 0
{{1,4},{2,3}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => 0
{{1},{2,3,4}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => 0
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> [1,3,2,4] => 0
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => 0
{{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> [1,4,3,2] => 0
{{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => 0
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => 0
{{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> [2,3,4,5,1] => 0
{{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> [2,3,4,1,5] => 0
{{1,2,3,5},{4}}
=> {{1,2,3,5},{4}}
=> [2,3,5,4,1] => 0
{{1,2,3},{4,5}}
=> {{1,2,3},{4,5}}
=> [2,3,1,5,4] => 0
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => 0
{{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> [2,4,3,5,1] => 0
{{1,2,4},{3,5}}
=> {{1,2,5},{3,4}}
=> [2,5,4,3,1] => 1
{{1,2,4},{3},{5}}
=> {{1,2,4},{3},{5}}
=> [2,4,3,1,5] => 0
{{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> [2,4,5,1,3] => 0
{{1,2},{3,4,5}}
=> {{1,2},{3,4,5}}
=> [2,1,4,5,3] => 0
{{1,2},{3,4},{5}}
=> {{1,2},{3,4},{5}}
=> [2,1,4,3,5] => 0
{{1,2,5},{3},{4}}
=> {{1,2,5},{3},{4}}
=> [2,5,3,4,1] => 0
{{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> [2,1,5,4,3] => 0
{{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> [2,1,3,5,4] => 0
{{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => 0
{{1,3,4,5},{2}}
=> {{1,3,4,5},{2}}
=> [3,2,4,5,1] => 0
{{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> [4,3,5,1,2] => 1
{{1,3,4},{2},{5}}
=> {{1,3,4},{2},{5}}
=> [3,2,4,1,5] => 0
{{1,3,5},{2,4}}
=> {{1,5},{2,3,4}}
=> [5,3,4,2,1] => 2
{{1,3},{2,4,5}}
=> {{1,4,5},{2,3}}
=> [4,3,2,5,1] => 1
{{1,3},{2,4},{5}}
=> {{1,4},{2,3},{5}}
=> [4,3,2,1,5] => 1
{{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> [3,2,5,4,1] => 0
{{1,3},{2,5},{4}}
=> {{1,5},{2,3},{4}}
=> [5,3,2,4,1] => 1
{{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> [3,2,1,5,4] => 0
{{1,3},{2},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => 0
{{1,4,5},{2,3}}
=> {{1,3},{2,4,5}}
=> [3,4,1,5,2] => 0
{{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> [3,5,4,1,2] => 1
{{1,2,3,4,5,6,7}}
=> {{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => ? = 0
{{1,2,3,4,5,6},{7}}
=> {{1,2,3,4,5,6},{7}}
=> [2,3,4,5,6,1,7] => ? = 0
{{1,2,3,4,5,7},{6}}
=> {{1,2,3,4,5,7},{6}}
=> [2,3,4,5,7,6,1] => ? = 0
{{1,2,3,4,5},{6,7}}
=> {{1,2,3,4,5},{6,7}}
=> [2,3,4,5,1,7,6] => ? = 0
{{1,2,3,4,5},{6},{7}}
=> {{1,2,3,4,5},{6},{7}}
=> [2,3,4,5,1,6,7] => ? = 0
{{1,2,3,4,6,7},{5}}
=> {{1,2,3,4,6,7},{5}}
=> [2,3,4,6,5,7,1] => ? = 0
{{1,2,3,4,6},{5,7}}
=> {{1,2,3,4,7},{5,6}}
=> [2,3,4,7,6,5,1] => ? = 1
{{1,2,3,4,6},{5},{7}}
=> {{1,2,3,4,6},{5},{7}}
=> [2,3,4,6,5,1,7] => ? = 0
{{1,2,3,4,7},{5,6}}
=> {{1,2,3,4,6},{5,7}}
=> [2,3,4,6,7,1,5] => ? = 0
{{1,2,3,4},{5,6,7}}
=> {{1,2,3,4},{5,6,7}}
=> [2,3,4,1,6,7,5] => ? = 0
{{1,2,3,4},{5,6},{7}}
=> {{1,2,3,4},{5,6},{7}}
=> [2,3,4,1,6,5,7] => ? = 0
{{1,2,3,4,7},{5},{6}}
=> {{1,2,3,4,7},{5},{6}}
=> [2,3,4,7,5,6,1] => ? = 0
{{1,2,3,4},{5,7},{6}}
=> {{1,2,3,4},{5,7},{6}}
=> [2,3,4,1,7,6,5] => ? = 0
{{1,2,3,4},{5},{6,7}}
=> {{1,2,3,4},{5},{6,7}}
=> [2,3,4,1,5,7,6] => ? = 0
{{1,2,3,4},{5},{6},{7}}
=> {{1,2,3,4},{5},{6},{7}}
=> [2,3,4,1,5,6,7] => ? = 0
{{1,2,3,5,6,7},{4}}
=> {{1,2,3,5,6,7},{4}}
=> [2,3,5,4,6,7,1] => ? = 0
{{1,2,3,5,6},{4,7}}
=> {{1,2,3,6},{4,5,7}}
=> [2,3,6,5,7,1,4] => ? = 1
{{1,2,3,5,6},{4},{7}}
=> {{1,2,3,5,6},{4},{7}}
=> [2,3,5,4,6,1,7] => ? = 0
{{1,2,3,5,7},{4,6}}
=> {{1,2,3,7},{4,5,6}}
=> [2,3,7,5,6,4,1] => ? = 2
{{1,2,3,5},{4,6,7}}
=> {{1,2,3,6,7},{4,5}}
=> [2,3,6,5,4,7,1] => ? = 1
{{1,2,3,5},{4,6},{7}}
=> {{1,2,3,6},{4,5},{7}}
=> [2,3,6,5,4,1,7] => ? = 1
{{1,2,3,5,7},{4},{6}}
=> {{1,2,3,5,7},{4},{6}}
=> [2,3,5,4,7,6,1] => ? = 0
{{1,2,3,5},{4,7},{6}}
=> {{1,2,3,7},{4,5},{6}}
=> [2,3,7,5,4,6,1] => ? = 1
{{1,2,3,5},{4},{6,7}}
=> {{1,2,3,5},{4},{6,7}}
=> [2,3,5,4,1,7,6] => ? = 0
{{1,2,3,5},{4},{6},{7}}
=> {{1,2,3,5},{4},{6},{7}}
=> [2,3,5,4,1,6,7] => ? = 0
{{1,2,3,6,7},{4,5}}
=> {{1,2,3,5},{4,6,7}}
=> [2,3,5,6,1,7,4] => ? = 0
{{1,2,3,6},{4,5,7}}
=> {{1,2,3,5,6},{4,7}}
=> [2,3,5,7,6,1,4] => ? = 1
{{1,2,3,6},{4,5},{7}}
=> {{1,2,3,5},{4,6},{7}}
=> [2,3,5,6,1,4,7] => ? = 0
{{1,2,3,7},{4,5,6}}
=> {{1,2,3,5,7},{4,6}}
=> [2,3,5,6,7,4,1] => ? = 0
{{1,2,3},{4,5,6,7}}
=> {{1,2,3},{4,5,6,7}}
=> [2,3,1,5,6,7,4] => ? = 0
{{1,2,3},{4,5,6},{7}}
=> {{1,2,3},{4,5,6},{7}}
=> [2,3,1,5,6,4,7] => ? = 0
{{1,2,3,7},{4,5},{6}}
=> {{1,2,3,5},{4,7},{6}}
=> [2,3,5,7,1,6,4] => ? = 0
{{1,2,3},{4,5,7},{6}}
=> {{1,2,3},{4,5,7},{6}}
=> [2,3,1,5,7,6,4] => ? = 0
{{1,2,3},{4,5},{6,7}}
=> {{1,2,3},{4,5},{6,7}}
=> [2,3,1,5,4,7,6] => ? = 0
{{1,2,3},{4,5},{6},{7}}
=> {{1,2,3},{4,5},{6},{7}}
=> [2,3,1,5,4,6,7] => ? = 0
{{1,2,3,6,7},{4},{5}}
=> {{1,2,3,6,7},{4},{5}}
=> [2,3,6,4,5,7,1] => ? = 0
{{1,2,3,6},{4,7},{5}}
=> {{1,2,3,7},{4,6},{5}}
=> [2,3,7,6,5,4,1] => ? = 1
{{1,2,3,6},{4},{5,7}}
=> {{1,2,3,7},{4},{5,6}}
=> [2,3,7,4,6,5,1] => ? = 1
{{1,2,3,6},{4},{5},{7}}
=> {{1,2,3,6},{4},{5},{7}}
=> [2,3,6,4,5,1,7] => ? = 0
{{1,2,3,7},{4,6},{5}}
=> {{1,2,3,6},{4,7},{5}}
=> [2,3,6,7,5,1,4] => ? = 0
{{1,2,3},{4,6,7},{5}}
=> {{1,2,3},{4,6,7},{5}}
=> [2,3,1,6,5,7,4] => ? = 0
{{1,2,3},{4,6},{5,7}}
=> {{1,2,3},{4,7},{5,6}}
=> [2,3,1,7,6,5,4] => ? = 1
{{1,2,3},{4,6},{5},{7}}
=> {{1,2,3},{4,6},{5},{7}}
=> [2,3,1,6,5,4,7] => ? = 0
{{1,2,3,7},{4},{5,6}}
=> {{1,2,3,6},{4},{5,7}}
=> [2,3,6,4,7,1,5] => ? = 0
{{1,2,3},{4,7},{5,6}}
=> {{1,2,3},{4,6},{5,7}}
=> [2,3,1,6,7,4,5] => ? = 0
{{1,2,3},{4},{5,6,7}}
=> {{1,2,3},{4},{5,6,7}}
=> [2,3,1,4,6,7,5] => ? = 0
{{1,2,3},{4},{5,6},{7}}
=> {{1,2,3},{4},{5,6},{7}}
=> [2,3,1,4,6,5,7] => ? = 0
{{1,2,3,7},{4},{5},{6}}
=> {{1,2,3,7},{4},{5},{6}}
=> [2,3,7,4,5,6,1] => ? = 0
{{1,2,3},{4,7},{5},{6}}
=> {{1,2,3},{4,7},{5},{6}}
=> [2,3,1,7,5,6,4] => ? = 0
{{1,2,3},{4},{5,7},{6}}
=> {{1,2,3},{4},{5,7},{6}}
=> [2,3,1,4,7,6,5] => ? = 0
Description
The number of nested exceedences of a permutation.
For a permutation $\pi$, this is the number of pairs $i,j$ such that $i < j < \pi(j) < \pi(i)$. For exceedences, see [[St000155]].
Matching statistic: St001490
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 14%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 14%
Values
{{1}}
=> [1] => [1,0]
=> [[1],[]]
=> 1 = 0 + 1
{{1,2}}
=> [2,1] => [1,1,0,0]
=> [[2],[]]
=> 1 = 0 + 1
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> [[1,1],[]]
=> 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> [[3],[]]
=> 1 = 0 + 1
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 1 = 0 + 1
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [[2,2],[]]
=> 1 = 0 + 1
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> [[2,1],[]]
=> 1 = 0 + 1
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 1 = 0 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [[4],[]]
=> 1 = 0 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 1 = 0 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [[3,3],[1]]
=> 1 = 0 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 1 = 0 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 1 = 0 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [[3,2],[]]
=> 1 = 0 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [[2,2,2],[]]
=> ? = 1 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 1 = 0 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? = 0 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 1 = 0 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 1 = 0 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> ? = 0 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 1 = 0 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1 = 0 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [[5],[]]
=> 1 = 0 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 1 = 0 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [[4,4],[2]]
=> ? = 0 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> 1 = 0 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [[4,3],[1]]
=> ? = 0 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [[3,3,3],[1,1]]
=> ? = 1 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ? = 0 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> ? = 0 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> 1 = 0 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[4,4],[1]]
=> ? = 0 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ? = 0 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> 1 = 0 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [[4,2],[]]
=> ? = 0 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ? = 1 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ? = 0 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2],[]]
=> ? = 2 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2],[]]
=> ? = 1 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ? = 1 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2],[1]]
=> ? = 0 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2],[]]
=> ? = 1 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ? = 0 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ? = 0 + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ? = 0 + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? = 1 + 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? = 0 + 1
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 0 + 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> 1 = 0 + 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> 1 = 0 + 1
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 0 + 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ? = 0 + 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 1 = 0 + 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> 1 = 0 + 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [[4,3],[]]
=> ? = 0 + 1
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [[4,4],[]]
=> ? = 1 + 1
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3],[1]]
=> ? = 1 + 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ? = 0 + 1
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 0 + 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ? = 0 + 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ? = 1 + 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ? = 0 + 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 0 + 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? = 0 + 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> 1 = 0 + 1
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> 1 = 0 + 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> ? = 0 + 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ? = 0 + 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ? = 0 + 1
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 1 = 0 + 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> 1 = 0 + 1
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[6],[]]
=> ? = 0 + 1
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [1,1,0,1,0,1,0,1,0,0,1,0]
=> [[5,5],[4]]
=> ? = 0 + 1
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [1,1,0,1,0,1,0,1,1,0,0,0]
=> [[5,5],[3]]
=> ? = 0 + 1
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [1,1,0,1,0,1,0,0,1,1,0,0]
=> [[5,4],[3]]
=> ? = 0 + 1
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [1,1,0,1,0,1,0,0,1,0,1,0]
=> [[4,4,4],[3,3]]
=> ? = 0 + 1
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [1,1,0,1,0,1,1,0,0,1,0,0]
=> [[5,4],[2]]
=> ? = 0 + 1
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [1,1,0,1,0,1,1,0,1,0,0,0]
=> [[4,4,4],[2,2]]
=> ? = 1 + 1
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [1,1,0,1,0,1,1,0,0,0,1,0]
=> [[4,4,4],[3,2]]
=> ? = 0 + 1
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [[5,5],[2]]
=> ? = 0 + 1
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [1,1,0,1,0,0,1,1,0,1,0,0]
=> [[5,3],[2]]
=> ? = 0 + 1
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [1,1,0,1,0,0,1,1,0,0,1,0]
=> [[4,4,3],[3,2]]
=> ? = 0 + 1
Description
The number of connected components of a skew partition.
Matching statistic: St000068
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 14%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 14%
Values
{{1}}
=> [1] => [1] => ([],1)
=> 1 = 0 + 1
{{1,2}}
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
{{1},{2}}
=> [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1 = 0 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1 = 0 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,1,2,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ? = 0 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,1,2,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 0 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,2,5,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 1 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,1,2] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ? = 0 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 0 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> ? = 1 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 2 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 1 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 0 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,1,4,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 1 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 0 + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,1,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 1 + 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 0 + 1
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 0 + 1
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [3,2,4,5,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0 + 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 + 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [2,3,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 0 + 1
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [3,4,1,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 1 + 1
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 1 + 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [3,4,2,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 0 + 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0 + 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 0 + 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 1 = 0 + 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [6,1,2,3,4,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 1 = 0 + 1
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [5,1,2,3,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 0 + 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 1 = 0 + 1
{{1,6},{2},{3},{4},{5}}
=> [6,2,3,4,5,1] => [2,3,4,5,6,1] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 1 = 0 + 1
{{1},{2},{3},{4},{5,6}}
=> [1,2,3,4,6,5] => [1,2,3,4,6,5] => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 1 = 0 + 1
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
{{1},{2},{3},{4},{5},{6},{7}}
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 0 + 1
Description
The number of minimal elements in a poset.
Matching statistic: St001301
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001301: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 14%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001301: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 14%
Values
{{1}}
=> [1] => [1] => ([],1)
=> 0
{{1,2}}
=> [2,1] => [2,1] => ([(0,1)],2)
=> 0
{{1},{2}}
=> [1,2] => [1,2] => ([(0,1)],2)
=> 0
{{1,2,3}}
=> [2,3,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 0
{{1},{2,3}}
=> [1,3,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 0
{{1},{2,3,4}}
=> [1,3,4,2] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 0
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
{{1},{2},{3,4}}
=> [1,2,4,3] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,2,3,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [2,3,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 0
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,5,2,1,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,4,5,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,5,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 0
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [3,2,4,5,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 0
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [3,4,5,2,1] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,4,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 0
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [4,3,2,5,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,3,1,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [3,5,4,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [3,4,5,1,2] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ? = 0
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [3,5,1,2,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 0
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [4,1,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 1
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [4,3,5,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [4,1,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 0
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ? = 0
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
{{1,6},{2,5},{3,4}}
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
Description
The first Betti number of the order complex associated with the poset.
The order complex of a poset is the simplicial complex whose faces are the chains of the poset. This statistic is the rank of the first homology group of the order complex.
Matching statistic: St000908
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 14%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 14%
Values
{{1}}
=> [1] => [1] => ([],1)
=> 1 = 0 + 1
{{1,2}}
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
{{1},{2}}
=> [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
{{1},{2,3}}
=> [1,3,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1 = 0 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,2,3,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [2,3,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 0 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,5,2,1,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,4,5,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 0 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,5,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [3,2,4,5,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 0 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [3,4,5,2,1] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,4,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 1 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 1 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 0 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0 + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [4,3,2,5,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,3,1,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 1 + 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [3,5,4,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [3,4,5,1,2] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ? = 0 + 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0 + 1
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0 + 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 + 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [3,5,1,2,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 0 + 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [4,1,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 1 + 1
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 1 + 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [4,3,5,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [4,1,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 1 + 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 0 + 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ? = 0 + 1
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
{{1,6},{2,5},{3,4}}
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
Description
The length of the shortest maximal antichain in a poset.
Matching statistic: St001634
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001634: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 14%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001634: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 14%
Values
{{1}}
=> [1] => [1] => ([],1)
=> -1 = 0 - 1
{{1,2}}
=> [2,1] => [2,1] => ([(0,1)],2)
=> -1 = 0 - 1
{{1},{2}}
=> [1,2] => [1,2] => ([(0,1)],2)
=> -1 = 0 - 1
{{1,2,3}}
=> [2,3,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> -1 = 0 - 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> -1 = 0 - 1
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> -1 = 0 - 1
{{1},{2,3}}
=> [1,3,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> -1 = 0 - 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> -1 = 0 - 1
{{1,2,3,4}}
=> [2,3,4,1] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> -1 = 0 - 1
{{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> -1 = 0 - 1
{{1,2,4},{3}}
=> [2,4,3,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> -1 = 0 - 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0 - 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> -1 = 0 - 1
{{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> -1 = 0 - 1
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> -1 = 0 - 1
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> -1 = 0 - 1
{{1},{2,3,4}}
=> [1,3,4,2] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> -1 = 0 - 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> -1 = 0 - 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> -1 = 0 - 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> -1 = 0 - 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> -1 = 0 - 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> -1 = 0 - 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 - 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 - 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,2,3,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 - 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [2,3,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 0 - 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 - 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 - 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,5,2,1,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 - 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 - 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 - 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,4,5,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 0 - 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0 - 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 - 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0 - 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,5,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 0 - 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 - 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [3,2,4,5,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 - 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 0 - 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [3,4,5,2,1] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2 - 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,4,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 1 - 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0 - 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 1 - 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 0 - 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0 - 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [4,3,2,5,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 - 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,3,1,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 1 - 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 - 1
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [3,5,4,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 - 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [3,4,5,1,2] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ? = 0 - 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0 - 1
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0 - 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 - 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [3,5,1,2,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 0 - 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 - 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 - 1
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [4,1,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 1 - 1
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 1 - 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 - 1
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> -1 = 0 - 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [4,3,5,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 - 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [4,1,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 1 - 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 - 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 0 - 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 - 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ? = 0 - 1
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 - 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 - 1
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [3,5,4,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 - 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> -1 = 0 - 1
{{1,6},{2,5},{3,4}}
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> -1 = 0 - 1
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> -1 = 0 - 1
Description
The trace of the Coxeter matrix of the incidence algebra of a poset.
Matching statistic: St000914
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000914: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 14%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000914: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 14%
Values
{{1}}
=> [1] => [1] => ([],1)
=> ? = 0 + 1
{{1,2}}
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
{{1},{2}}
=> [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
{{1,2,3}}
=> [2,3,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
{{1},{2,3}}
=> [1,3,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
{{1,2,3,4}}
=> [2,3,4,1] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1 = 0 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 0 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1 = 0 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [5,2,3,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [2,3,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 0 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [4,2,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,5,2,1,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,4,5,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 0 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,5,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [3,2,4,5,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ? = 0 + 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [3,4,5,2,1] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 2 + 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,4,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 1 + 1
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [3,5,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 0 + 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 1 + 1
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 0 + 1
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0 + 1
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [4,3,2,5,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,3,1,5,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 1 + 1
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [3,5,4,2,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [3,4,5,1,2] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ? = 0 + 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0 + 1
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [3,2,5,4,1] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ? = 0 + 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [5,3,4,1,2] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ? = 0 + 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [3,5,1,2,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 0 + 1
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [3,1,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [4,1,5,3,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ? = 1 + 1
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 1 + 1
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 0 + 1
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [4,3,5,1,2] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [4,1,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 1 + 1
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [5,2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ? = 0 + 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ? = 0 + 1
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [4,1,2,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 0 + 1
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
{{1,6},{2,5},{3,4}}
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
Description
The sum of the values of the Möbius function of a poset.
The Möbius function $\mu$ of a finite poset is defined as
$$\mu (x,y)=\begin{cases} 1& \text{if }x = y\\
-\sum _{z: x\leq z < y}\mu (x,z)& \text{for }x < y\\
0&\text{otherwise}.
\end{cases}
$$
Since $\mu(x,y)=0$ whenever $x\not\leq y$, this statistic is
$$
\sum_{x\leq y} \mu(x,y).
$$
If the poset has a minimal or a maximal element, then the definition implies immediately that the statistic equals $1$. Moreover, the statistic equals the sum of the statistics of the connected components.
This statistic is also called the magnitude of a poset.
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