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Your data matches 20 different statistics following compositions of up to 3 maps.
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Matching statistic: St000147
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,1]
=> [1]
=> []
=> 0
[2,1,1]
=> [1,1]
=> [1]
=> []
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[3,1,1]
=> [1,1]
=> [1]
=> []
=> 0
[2,2,1]
=> [2,1]
=> [1]
=> []
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,1,1]
=> [1,1]
=> [1]
=> []
=> 0
[3,2,1]
=> [2,1]
=> [1]
=> []
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[2,2,2]
=> [2,2]
=> [2]
=> []
=> 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[5,1,1]
=> [1,1]
=> [1]
=> []
=> 0
[4,2,1]
=> [2,1]
=> [1]
=> []
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[3,3,1]
=> [3,1]
=> [1]
=> []
=> 0
[3,2,2]
=> [2,2]
=> [2]
=> []
=> 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[6,1,1]
=> [1,1]
=> [1]
=> []
=> 0
[5,2,1]
=> [2,1]
=> [1]
=> []
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[4,3,1]
=> [3,1]
=> [1]
=> []
=> 0
[4,2,2]
=> [2,2]
=> [2]
=> []
=> 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,3,2]
=> [3,2]
=> [2]
=> []
=> 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[7,1,1]
=> [1,1]
=> [1]
=> []
=> 0
[6,2,1]
=> [2,1]
=> [1]
=> []
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[5,3,1]
=> [3,1]
=> [1]
=> []
=> 0
[5,2,2]
=> [2,2]
=> [2]
=> []
=> 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,4,1]
=> [4,1]
=> [1]
=> []
=> 0
Description
The largest part of an integer partition.
Matching statistic: St001280
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,1]
=> [1]
=> [1]
=> 0
[2,1,1]
=> [1,1]
=> [1]
=> [1]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
[3,1,1]
=> [1,1]
=> [1]
=> [1]
=> 0
[2,2,1]
=> [2,1]
=> [1]
=> [1]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[4,1,1]
=> [1,1]
=> [1]
=> [1]
=> 0
[3,2,1]
=> [2,1]
=> [1]
=> [1]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
[2,2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[5,1,1]
=> [1,1]
=> [1]
=> [1]
=> 0
[4,2,1]
=> [2,1]
=> [1]
=> [1]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
[3,3,1]
=> [3,1]
=> [1]
=> [1]
=> 0
[3,2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1
[6,1,1]
=> [1,1]
=> [1]
=> [1]
=> 0
[5,2,1]
=> [2,1]
=> [1]
=> [1]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
[4,3,1]
=> [3,1]
=> [1]
=> [1]
=> 0
[4,2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[3,3,2]
=> [3,2]
=> [2]
=> [1,1]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2,2]
=> 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 1
[7,1,1]
=> [1,1]
=> [1]
=> [1]
=> 0
[6,2,1]
=> [2,1]
=> [1]
=> [1]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
[5,3,1]
=> [3,1]
=> [1]
=> [1]
=> 0
[5,2,2]
=> [2,2]
=> [2]
=> [1,1]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [2]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[4,4,1]
=> [4,1]
=> [1]
=> [1]
=> 0
[3,3,3,3,3,3,3,3]
=> [3,3,3,3,3,3,3]
=> [3,3,3,3,3,3]
=> [6,6,6]
=> ? = 3
[6,6,6,6,6]
=> [6,6,6,6]
=> [6,6,6]
=> [3,3,3,3,3,3]
=> ? = 6
[5,5,5,5,5,5]
=> [5,5,5,5,5]
=> [5,5,5,5]
=> [4,4,4,4,4]
=> ? = 5
[8,7,6,5,4,3,2,1]
=> [7,6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? = 5
[7,7,6,5,4,3,2,1]
=> [7,6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? = 5
[7,6,6,5,4,3,2,1]
=> [6,6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? = 5
[7,6,5,5,4,3,2,1]
=> [6,5,5,4,3,2,1]
=> [5,5,4,3,2,1]
=> [6,5,4,3,2]
=> ? = 5
[7,6,5,4,4,3,2,1]
=> [6,5,4,4,3,2,1]
=> [5,4,4,3,2,1]
=> [6,5,4,3,1]
=> ? = 4
[7,6,5,4,3,3,2,1]
=> [6,5,4,3,3,2,1]
=> [5,4,3,3,2,1]
=> [6,5,4,2,1]
=> ? = 4
[6,6,6,5,4,3,2,1]
=> [6,6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? = 5
[7,5,5,5,4,3,2,1]
=> [5,5,5,4,3,2,1]
=> [5,5,4,3,2,1]
=> [6,5,4,3,2]
=> ? = 5
[7,6,4,4,4,3,2,1]
=> [6,4,4,4,3,2,1]
=> [4,4,4,3,2,1]
=> [6,5,4,3]
=> ? = 4
[8,7,6,5,4,3,2,1,1]
=> [7,6,5,4,3,2,1,1]
=> [6,5,4,3,2,1,1]
=> [7,5,4,3,2,1]
=> ? = 5
[8,7,6,5,4,3,2]
=> [7,6,5,4,3,2]
=> [6,5,4,3,2]
=> [5,5,4,3,2,1]
=> ? = 5
[8,7,6,5,4,3,1]
=> [7,6,5,4,3,1]
=> [6,5,4,3,1]
=> [5,4,4,3,2,1]
=> ? = 5
[8,7,6,5,4,2,1]
=> [7,6,5,4,2,1]
=> [6,5,4,2,1]
=> [5,4,3,3,2,1]
=> ? = 5
[8,7,6,5,4,3]
=> [7,6,5,4,3]
=> [6,5,4,3]
=> [4,4,4,3,2,1]
=> ? = 5
[9,8,7,6,5,3,2,1]
=> [8,7,6,5,3,2,1]
=> [7,6,5,3,2,1]
=> [6,5,4,3,3,2,1]
=> ? = 6
[9,8,7,6,4,3,2,1]
=> [8,7,6,4,3,2,1]
=> [7,6,4,3,2,1]
=> [6,5,4,3,2,2,1]
=> ? = 6
[9,8,7,5,4,3,2,1]
=> [8,7,5,4,3,2,1]
=> [7,5,4,3,2,1]
=> [6,5,4,3,2,1,1]
=> ? = 5
[9,8,6,5,4,3,2,1]
=> [8,6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? = 5
[9,7,6,5,4,3,2,1]
=> [7,6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ? = 5
[4,4,4,4,4,4,4]
=> [4,4,4,4,4,4]
=> [4,4,4,4,4]
=> [5,5,5,5]
=> ? = 4
[7,6,5,5,4,2,2,1]
=> [6,5,5,4,2,2,1]
=> [5,5,4,2,2,1]
=> [6,5,3,3,2]
=> ? = 5
[7,6,6,5,4,3,1,1]
=> [6,6,5,4,3,1,1]
=> [6,5,4,3,1,1]
=> [6,4,4,3,2,1]
=> ? = 5
[7,7,6,5,4,3,2]
=> [7,6,5,4,3,2]
=> [6,5,4,3,2]
=> [5,5,4,3,2,1]
=> ? = 5
[6,5,5,5,4,3,2,1]
=> [5,5,5,4,3,2,1]
=> [5,5,4,3,2,1]
=> [6,5,4,3,2]
=> ? = 5
[6,5,4,4,4,3,2,1]
=> [5,4,4,4,3,2,1]
=> [4,4,4,3,2,1]
=> [6,5,4,3]
=> ? = 4
[7,6,6,5,4,3,2]
=> [6,6,5,4,3,2]
=> [6,5,4,3,2]
=> [5,5,4,3,2,1]
=> ? = 5
[7,6,6,5,4,3,1]
=> [6,6,5,4,3,1]
=> [6,5,4,3,1]
=> [5,4,4,3,2,1]
=> ? = 5
[7,7,6,5,4,3,1]
=> [7,6,5,4,3,1]
=> [6,5,4,3,1]
=> [5,4,4,3,2,1]
=> ? = 5
[7,7,6,5,4,3,1,1]
=> [7,6,5,4,3,1,1]
=> [6,5,4,3,1,1]
=> [6,4,4,3,2,1]
=> ? = 5
[8,7,6,5,4,1,1,1]
=> [7,6,5,4,1,1,1]
=> [6,5,4,1,1,1]
=> [6,3,3,3,2,1]
=> ? = 5
[8,6,6,5,4,3,2]
=> [6,6,5,4,3,2]
=> [6,5,4,3,2]
=> [5,5,4,3,2,1]
=> ? = 5
[9,8,7,6,5]
=> [8,7,6,5]
=> [7,6,5]
=> [3,3,3,3,3,2,1]
=> ? = 6
[9,8,7,6,5,4]
=> [8,7,6,5,4]
=> [7,6,5,4]
=> [4,4,4,4,3,2,1]
=> ? = 6
[7,7,5,5,4,3,2,1]
=> [7,5,5,4,3,2,1]
=> [5,5,4,3,2,1]
=> [6,5,4,3,2]
=> ? = 5
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001556
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001556: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 38%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001556: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 38%
Values
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 0
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 0
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 1
[7,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[6,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[5,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 0
[5,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[4,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 0
[4,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1
[4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 1
[3,3,3]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 0
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 1
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? = 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? = 1
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 1
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 1
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 1
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 1
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 1
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? = 1
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 1
[2,2,2,2,2]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => ? = 2
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => ? = 2
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? = 1
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? = 1
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? = 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => ? = 1
[6,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 1
[5,5,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0
[5,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 1
[5,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 1
[4,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 1
[4,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 1
[4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? = 1
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 1
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,6,1] => ? = 1
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => ? = 1
[3,2,2,2,2]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => ? = 2
[3,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => ? = 2
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? = 1
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? = 1
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? = 1
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => ? = 2
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => ? = 2
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,1] => ? = 1
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,1] => ? = 1
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => ? = 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => ? = 1
[7,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 1
[6,5,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => ? = 0
Description
The number of inversions of the third entry of a permutation.
This is, for a permutation $\pi$ of length $n$,
$$\# \{3 < k \leq n \mid \pi(3) > \pi(k)\}.$$
The number of inversions of the first entry is [[St000054]] and the number of inversions of the second entry is [[St001557]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.
Matching statistic: St001491
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 25%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 25%
Values
[1,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[2,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[1,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 1 + 1
[3,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[2,2,1]
=> [2,1]
=> 1010 => 0010 => 1 = 0 + 1
[2,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 1 + 1
[1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01110 => ? = 1 + 1
[4,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[3,2,1]
=> [2,1]
=> 1010 => 0010 => 1 = 0 + 1
[3,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 1 + 1
[2,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 0 + 1
[2,2,1,1]
=> [2,1,1]
=> 10110 => 00110 => ? = 1 + 1
[2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01110 => ? = 1 + 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 011110 => ? = 1 + 1
[5,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[4,2,1]
=> [2,1]
=> 1010 => 0010 => 1 = 0 + 1
[4,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 1 + 1
[3,3,1]
=> [3,1]
=> 10010 => 00010 => ? = 0 + 1
[3,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 0 + 1
[3,2,1,1]
=> [2,1,1]
=> 10110 => 00110 => ? = 1 + 1
[3,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01110 => ? = 1 + 1
[2,2,2,1]
=> [2,2,1]
=> 11010 => 01010 => ? = 1 + 1
[2,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 001110 => ? = 1 + 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 011110 => ? = 1 + 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 0111110 => ? = 1 + 1
[6,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[5,2,1]
=> [2,1]
=> 1010 => 0010 => 1 = 0 + 1
[5,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 1 + 1
[4,3,1]
=> [3,1]
=> 10010 => 00010 => ? = 0 + 1
[4,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 0 + 1
[4,2,1,1]
=> [2,1,1]
=> 10110 => 00110 => ? = 1 + 1
[4,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01110 => ? = 1 + 1
[3,3,2]
=> [3,2]
=> 10100 => 00100 => ? = 0 + 1
[3,3,1,1]
=> [3,1,1]
=> 100110 => 000110 => ? = 1 + 1
[3,2,2,1]
=> [2,2,1]
=> 11010 => 01010 => ? = 1 + 1
[3,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 001110 => ? = 1 + 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 011110 => ? = 1 + 1
[2,2,2,2]
=> [2,2,2]
=> 11100 => 01100 => ? = 2 + 1
[2,2,2,1,1]
=> [2,2,1,1]
=> 110110 => 010110 => ? = 1 + 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> 1011110 => 0011110 => ? = 1 + 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 0111110 => ? = 1 + 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 11111110 => 01111110 => ? = 1 + 1
[7,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[6,2,1]
=> [2,1]
=> 1010 => 0010 => 1 = 0 + 1
[6,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 1 + 1
[5,3,1]
=> [3,1]
=> 10010 => 00010 => ? = 0 + 1
[5,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 0 + 1
[5,2,1,1]
=> [2,1,1]
=> 10110 => 00110 => ? = 1 + 1
[5,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01110 => ? = 1 + 1
[4,4,1]
=> [4,1]
=> 100010 => 000010 => ? = 0 + 1
[4,3,2]
=> [3,2]
=> 10100 => 00100 => ? = 0 + 1
[4,3,1,1]
=> [3,1,1]
=> 100110 => 000110 => ? = 1 + 1
[4,2,2,1]
=> [2,2,1]
=> 11010 => 01010 => ? = 1 + 1
[4,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 001110 => ? = 1 + 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 011110 => ? = 1 + 1
[3,3,3]
=> [3,3]
=> 11000 => 01000 => ? = 0 + 1
[3,3,2,1]
=> [3,2,1]
=> 101010 => 001010 => ? = 1 + 1
[3,3,1,1,1]
=> [3,1,1,1]
=> 1001110 => 0001110 => ? = 1 + 1
[3,2,2,2]
=> [2,2,2]
=> 11100 => 01100 => ? = 2 + 1
[3,2,2,1,1]
=> [2,2,1,1]
=> 110110 => 010110 => ? = 1 + 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> 1011110 => 0011110 => ? = 1 + 1
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 0111110 => ? = 1 + 1
[2,2,2,2,1]
=> [2,2,2,1]
=> 111010 => 011010 => ? = 2 + 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> 1101110 => 0101110 => ? = 1 + 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> 10111110 => 00111110 => ? = 1 + 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 11111110 => 01111110 => ? = 1 + 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> 111111110 => 011111110 => ? = 1 + 1
[8,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[7,2,1]
=> [2,1]
=> 1010 => 0010 => 1 = 0 + 1
[7,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 1 + 1
[6,3,1]
=> [3,1]
=> 10010 => 00010 => ? = 0 + 1
[6,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 0 + 1
[6,2,1,1]
=> [2,1,1]
=> 10110 => 00110 => ? = 1 + 1
[6,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01110 => ? = 1 + 1
[5,4,1]
=> [4,1]
=> 100010 => 000010 => ? = 0 + 1
[5,3,2]
=> [3,2]
=> 10100 => 00100 => ? = 0 + 1
[9,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[8,2,1]
=> [2,1]
=> 1010 => 0010 => 1 = 0 + 1
[8,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 1 + 1
[7,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 0 + 1
[10,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[9,2,1]
=> [2,1]
=> 1010 => 0010 => 1 = 0 + 1
[9,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 1 + 1
[8,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 0 + 1
[11,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[10,2,1]
=> [2,1]
=> 1010 => 0010 => 1 = 0 + 1
[10,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 1 + 1
[9,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 0 + 1
[12,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[11,2,1]
=> [2,1]
=> 1010 => 0010 => 1 = 0 + 1
[11,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 1 + 1
[10,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 0 + 1
[13,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[12,2,1]
=> [2,1]
=> 1010 => 0010 => 1 = 0 + 1
[12,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 1 + 1
[11,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 0 + 1
[14,1,1]
=> [1,1]
=> 110 => 010 => 1 = 0 + 1
[13,2,1]
=> [2,1]
=> 1010 => 0010 => 1 = 0 + 1
[13,1,1,1]
=> [1,1,1]
=> 1110 => 0110 => 2 = 1 + 1
[12,2,2]
=> [2,2]
=> 1100 => 0100 => 1 = 0 + 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St001722
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 25%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 25%
Values
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 0 + 1
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 0 + 1
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 2 = 1 + 1
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 0 + 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 2 = 1 + 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => ? = 1 + 1
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 0 + 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 2 = 1 + 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 1 + 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => ? = 1 + 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => ? = 1 + 1
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 0 + 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 2 = 1 + 1
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 0 + 1
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 1 + 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => ? = 1 + 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => ? = 1 + 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 1 + 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => ? = 1 + 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 110101010100 => ? = 1 + 1
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 0 + 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 2 = 1 + 1
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 0 + 1
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 1 + 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => ? = 1 + 1
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0 + 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 1 + 1
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => ? = 1 + 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 1 + 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => ? = 1 + 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => ? = 2 + 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => ? = 1 + 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 101101010100 => ? = 1 + 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 110101010100 => ? = 1 + 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 11010101010100 => ? = 1 + 1
[7,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 0 + 1
[6,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[6,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 2 = 1 + 1
[5,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 0 + 1
[5,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 1 + 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => ? = 1 + 1
[4,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0 + 1
[4,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0 + 1
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => ? = 1 + 1
[4,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => ? = 1 + 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 1 + 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => ? = 1 + 1
[3,3,3]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => ? = 0 + 1
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1 + 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => ? = 1 + 1
[3,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => ? = 2 + 1
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => ? = 1 + 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 101101010100 => ? = 1 + 1
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 110101010100 => ? = 1 + 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => ? = 2 + 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 111001010100 => ? = 1 + 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 10110101010100 => ? = 1 + 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 11010101010100 => ? = 1 + 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1101010101010100 => ? = 1 + 1
[8,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 0 + 1
[7,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[7,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 2 = 1 + 1
[6,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 0 + 1
[6,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[6,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 1 + 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => ? = 1 + 1
[5,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0 + 1
[5,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0 + 1
[9,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 0 + 1
[8,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[8,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 2 = 1 + 1
[7,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[10,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 0 + 1
[9,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[9,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 2 = 1 + 1
[8,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[11,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 0 + 1
[10,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[10,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 2 = 1 + 1
[9,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[12,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 0 + 1
[11,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[11,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 2 = 1 + 1
[10,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[13,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 0 + 1
[12,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[12,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 2 = 1 + 1
[11,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[14,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 0 + 1
[13,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[13,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => 2 = 1 + 1
[12,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
Description
The number of minimal chains with small intervals between a binary word and the top element.
A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. Let $P$ be the lattice on binary words of length $n$, where the covering elements of a word are obtained by replacing a valley with a peak. An interval $[w_1, w_2]$ in $P$ is small if $w_2$ is obtained from $w_1$ by replacing some valleys with peaks.
This statistic counts the number of chains $w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length.
For example, there are two such chains for the word $0110$:
$$ 0110 < 1011 < 1101 < 1110 < 1111 $$
and
$$ 0110 < 1010 < 1101 < 1110 < 1111. $$
Matching statistic: St001136
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001136: Perfect matchings ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 38%
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001136: Perfect matchings ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 38%
Values
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 3 = 0 + 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> 3 = 0 + 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 4 = 1 + 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> 3 = 0 + 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> 3 = 0 + 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> 4 = 1 + 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 1 + 3
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> 3 = 0 + 3
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> 3 = 0 + 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> 4 = 1 + 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> 3 = 0 + 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 4 = 1 + 3
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [(1,12),(2,9),(3,8),(4,7),(5,6),(10,11)]
=> ? = 1 + 3
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7),(13,14)]
=> ? = 1 + 3
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [(1,12),(2,11),(3,4),(5,10),(6,9),(7,8)]
=> ? = 0 + 3
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> 3 = 0 + 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> 4 = 1 + 3
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> 3 = 0 + 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> 3 = 0 + 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> 4 = 1 + 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [(1,12),(2,11),(3,8),(4,7),(5,6),(9,10)]
=> ? = 1 + 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> 4 = 1 + 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,12),(10,11)]
=> ? = 1 + 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [(1,14),(2,11),(3,10),(4,9),(5,8),(6,7),(12,13)]
=> ? = 1 + 3
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,16)]
=> ? = 1 + 3
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [(1,14),(2,13),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> ? = 0 + 3
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
=> 3 = 0 + 3
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,5),(6,9),(7,8)]
=> ? = 1 + 3
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> 3 = 0 + 3
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> 3 = 0 + 3
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> 4 = 1 + 3
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,7),(5,6),(8,9)]
=> ? = 1 + 3
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> 3 = 0 + 3
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> 4 = 1 + 3
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> 4 = 1 + 3
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
=> ? = 1 + 3
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [(1,14),(2,13),(3,10),(4,9),(5,8),(6,7),(11,12)]
=> ? = 1 + 3
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
=> ? = 2 + 3
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,12),(10,11)]
=> ? = 1 + 3
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,14),(12,13)]
=> ? = 1 + 3
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [(1,16),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(14,15)]
=> ? = 1 + 3
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,18)]
=> ? = 1 + 3
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,16),(2,15),(3,4),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ? = 0 + 3
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9),(13,14)]
=> ? = 0 + 3
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,5),(6,11),(7,10),(8,9)]
=> ? = 1 + 3
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
=> 3 = 0 + 3
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8),(11,12)]
=> ? = 0 + 3
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)]
=> ? = 1 + 3
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,6),(7,8)]
=> ? = 1 + 3
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> 3 = 0 + 3
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> 3 = 0 + 3
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> 4 = 1 + 3
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> 4 = 1 + 3
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9),(11,12)]
=> ? = 1 + 3
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,9),(5,8),(6,7),(10,11)]
=> ? = 1 + 3
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4),(7,12),(8,11),(9,10)]
=> ? = 0 + 3
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> 4 = 1 + 3
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [(1,2),(3,12),(4,7),(5,6),(8,11),(9,10)]
=> ? = 1 + 3
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
=> ? = 2 + 3
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10),(11,12)]
=> ? = 1 + 3
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12),(13,14)]
=> ? = 1 + 3
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [(1,16),(2,15),(3,12),(4,11),(5,10),(6,9),(7,8),(13,14)]
=> ? = 1 + 3
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,12),(10,11)]
=> ? = 2 + 3
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7),(11,14),(12,13)]
=> ? = 1 + 3
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8),(13,16),(14,15)]
=> ? = 1 + 3
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [(1,18),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(16,17)]
=> ? = 1 + 3
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10),(19,20)]
=> ? = 1 + 3
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [(1,18),(2,17),(3,4),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11)]
=> ? = 0 + 3
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,2),(3,4),(5,14),(6,13),(7,12),(8,11),(9,10),(15,16)]
=> ? = 0 + 3
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,16),(2,15),(3,14),(4,5),(6,13),(7,12),(8,11),(9,10)]
=> ? = 1 + 3
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [(1,2),(3,4),(5,14),(6,11),(7,10),(8,9),(12,13)]
=> ? = 0 + 3
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [(1,12),(2,5),(3,4),(6,11),(7,10),(8,9),(13,14)]
=> ? = 0 + 3
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [(1,2),(3,12),(4,5),(6,11),(7,10),(8,9),(13,14)]
=> ? = 1 + 3
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> ? = 1 + 3
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,8),(9,10)]
=> 3 = 0 + 3
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [(1,12),(2,5),(3,4),(6,9),(7,8),(10,11)]
=> ? = 0 + 3
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)]
=> ? = 1 + 3
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8),(11,12)]
=> ? = 1 + 3
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8),(11,12)]
=> ? = 1 + 3
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [(1,14),(2,13),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> ? = 1 + 3
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [(1,12),(2,5),(3,4),(6,11),(7,10),(8,9)]
=> ? = 0 + 3
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> 4 = 1 + 3
[3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> 5 = 2 + 3
[4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> 5 = 2 + 3
[5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> 5 = 2 + 3
[4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> 5 = 2 + 3
[5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> 5 = 2 + 3
Description
The largest label with larger sister in the leaf labelled binary unordered tree associated with the perfect matching.
The bijection between perfect matchings of $\{1,\dots,2n\}$ and trees with $n+1$ leaves is described in Example 5.2.6 of [1].
Matching statistic: St001227
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Values
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3 = 0 + 3
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 0 + 3
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 4 = 1 + 3
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 0 + 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 4 = 1 + 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 3
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 3 = 0 + 3
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 0 + 3
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 4 = 1 + 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 3 = 0 + 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 4 = 1 + 3
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 + 3
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 3
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> ? = 0 + 3
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 3 = 0 + 3
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 4 = 1 + 3
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 3 = 0 + 3
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 3 = 0 + 3
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 4 = 1 + 3
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 1 + 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 4 = 1 + 3
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 1 + 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 1 + 3
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 + 3
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0]
=> ? = 0 + 3
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 3
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 3
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 3 = 0 + 3
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 3 = 0 + 3
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 4 = 1 + 3
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 1 + 3
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 3 = 0 + 3
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 4 = 1 + 3
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 4 = 1 + 3
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 1 + 3
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? = 1 + 3
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 3
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 1 + 3
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 1 + 3
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 + 3
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 + 3
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 0 + 3
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
=> ? = 0 + 3
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 1 + 3
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 0 + 3
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> ? = 0 + 3
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 + 3
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 1 + 3
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> ? = 0 + 3
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3 = 0 + 3
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 4 = 1 + 3
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 4 = 1 + 3
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 1 + 3
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> ? = 1 + 3
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? = 0 + 3
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 4 = 1 + 3
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> ? = 1 + 3
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> ? = 2 + 3
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 1 + 3
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 1 + 3
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 1 + 3
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 3
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 3
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 3
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 1 + 3
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1 + 3
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0,0]
=> ? = 0 + 3
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0,0]
=> ? = 0 + 3
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 1 + 3
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> ? = 0 + 3
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,1,0,0]
=> ? = 0 + 3
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,1,0,0]
=> ? = 1 + 3
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 1 + 3
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 0 + 3
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> ? = 0 + 3
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> ? = 1 + 3
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 4 = 1 + 3
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St001816
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 12%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 12%
Values
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? = 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? = 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? = 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> ? = 0
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? = 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 0
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> ? = 0
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> ? = 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? = 1
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 0
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> ? = 0
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 1
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> ? = 0
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> ? = 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> ? = 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> ? = 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> ? = 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8,9],[2,10,11,12,13,14,15,16]]
=> ? = 1
[7,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[6,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? = 1
[5,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 0
[5,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> ? = 0
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 1
[4,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[4,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> ? = 0
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> ? = 1
[4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> ? = 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 1
[3,3,3]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> ? = 0
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> ? = 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 1
[3,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> ? = 2
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> ? = 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 1
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> ? = 1
[8,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[7,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[9,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[8,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[10,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[9,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[11,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[10,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[12,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[11,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[13,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[12,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[14,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[13,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[15,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0
[14,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
Description
Eigenvalues of the top-to-random operator acting on a simple module.
These eigenvalues are given in [1] and [3].
The simple module of the symmetric group indexed by a partition $\lambda$ has dimension equal to the number of standard tableaux of shape $\lambda$. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape $\lambda$; this statistic gives all the eigenvalues of the operator acting on the module.
This statistic bears different names, such as the type in [2] or eig in [3].
Similarly, the eigenvalues of the random-to-random operator acting on a simple module is [[St000508]].
Matching statistic: St000782
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 12%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 12%
Values
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1 + 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1 + 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> ? = 1 + 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1 + 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 0 + 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1 + 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> ? = 1 + 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 1 + 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1 + 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 0 + 1
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 0 + 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1 + 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> ? = 1 + 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 1 + 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> ? = 1 + 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 1 + 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 1 + 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1 + 1
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 0 + 1
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 0 + 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1 + 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> ? = 1 + 1
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> ? = 0 + 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> ? = 1 + 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 1 + 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> ? = 1 + 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 1 + 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> ? = 2 + 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> ? = 1 + 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> ? = 1 + 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [(1,2),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]
=> ? = 1 + 1
[7,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[6,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1 + 1
[5,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 0 + 1
[5,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 0 + 1
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1 + 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> ? = 1 + 1
[4,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> ? = 0 + 1
[4,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> ? = 0 + 1
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> ? = 1 + 1
[4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 1 + 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> ? = 1 + 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 1 + 1
[3,3,3]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> ? = 0 + 1
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? = 1 + 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> ? = 1 + 1
[3,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> ? = 2 + 1
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> ? = 1 + 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)]
=> ? = 1 + 1
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 1 + 1
[8,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[7,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[9,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[8,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[10,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[9,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[11,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[10,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[12,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[11,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[13,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[12,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[14,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[13,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[15,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[14,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
Description
The indicator function of whether a given perfect matching is an L & P matching.
An L&P matching is built inductively as follows:
starting with either a single edge, or a hairpin $([1,3],[2,4])$, insert a noncrossing matching or inflate an edge by a ladder, that is, a number of nested edges.
The number of L&P matchings is (see [thm. 1, 2])
$$\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}$$
Matching statistic: St001928
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 25%
Values
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,5,4,2] => 2 = 0 + 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5,1,4,3,2] => 2 = 0 + 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,6,5,4,2] => 3 = 1 + 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,1,5,4,2] => 2 = 0 + 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,5,3,2] => 2 = 0 + 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [6,1,5,4,3,2] => 3 = 1 + 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,7,6,5,4,2] => ? = 1 + 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [4,3,1,6,5,2] => 2 = 0 + 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,4,3,2] => 2 = 0 + 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [3,1,6,5,4,2] => 3 = 1 + 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [2,6,1,5,4,3] => 2 = 0 + 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [5,1,6,4,3,2] => 3 = 1 + 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [7,1,6,5,4,3,2] => ? = 1 + 2
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => [3,1,8,7,6,5,4,2] => ? = 1 + 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => [5,3,7,1,6,4,2] => ? = 0 + 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [6,4,1,5,3,2] => 2 = 0 + 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,6,5,4,2] => 3 = 1 + 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [5,3,1,6,4,2] => 2 = 0 + 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [2,6,1,5,4,3] => 2 = 0 + 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [6,1,5,4,3,2] => 3 = 1 + 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [3,1,7,6,5,4,2] => ? = 1 + 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,1,6,5,3,2] => 3 = 1 + 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [6,1,7,5,4,3,2] => ? = 1 + 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => [8,1,7,6,5,4,3,2] => ? = 1 + 2
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => [3,1,9,8,7,6,5,4,2] => ? = 1 + 2
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,4,5,1,8,2,7] => [6,3,8,7,1,5,4,2] => ? = 0 + 2
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => [7,5,4,1,6,3,2] => ? = 0 + 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [4,3,1,7,6,5,2] => ? = 1 + 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [6,3,1,5,4,2] => 2 = 0 + 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [2,6,1,5,4,3] => 2 = 0 + 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [6,1,5,4,3,2] => 3 = 1 + 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => [3,1,7,6,5,4,2] => ? = 1 + 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [2,6,1,5,4,3] => 2 = 0 + 2
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [3,1,6,5,4,2] => 3 = 1 + 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [6,1,5,4,3,2] => 3 = 1 + 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [7,1,6,5,4,3,2] => ? = 1 + 2
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => [3,1,8,7,6,5,4,2] => ? = 1 + 2
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => [2,7,1,6,5,4,3] => ? = 2 + 2
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [5,1,4,2,6,7,3] => [5,1,7,6,4,3,2] => ? = 1 + 2
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => [7,1,8,6,5,4,3,2] => ? = 1 + 2
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => [9,1,8,7,6,5,4,3,2] => ? = 1 + 2
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,2] => [3,1,10,9,8,7,6,5,4,2] => ? = 1 + 2
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [7,3,4,5,6,1,9,2,8] => [7,3,9,8,6,1,5,4,2] => ? = 0 + 2
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [8,6,4,5,1,2,3,7] => [8,6,4,7,1,5,3,2] => ? = 0 + 2
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [5,3,4,1,6,8,2,7] => [5,3,8,1,7,6,4,2] => ? = 1 + 2
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [7,3,6,1,5,4,2] => ? = 0 + 2
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [2,5,4,1,7,3,6] => [2,7,6,1,5,4,3] => ? = 0 + 2
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [7,4,1,6,5,3,2] => ? = 1 + 2
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [3,1,4,5,7,2,6] => [3,1,7,6,5,4,2] => ? = 1 + 2
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => [6,3,7,1,5,4,2] => ? = 0 + 2
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [2,6,1,5,4,3] => 2 = 0 + 2
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [3,1,6,5,4,2] => 3 = 1 + 2
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [4,1,6,5,3,2] => 3 = 1 + 2
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [7,1,4,6,2,3,5] => [7,1,6,5,4,3,2] => ? = 1 + 2
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [3,1,4,8,6,7,2,5] => [3,1,8,7,6,5,4,2] => ? = 1 + 2
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [2,7,6,1,5,4,3] => ? = 0 + 2
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [5,1,6,4,3,2] => 3 = 1 + 2
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,1,6,5,2,7,4] => [3,1,7,6,5,4,2] => ? = 1 + 2
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,7,1,5,6,3,4] => [2,7,1,6,5,4,3] => ? = 2 + 2
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [7,1,5,2,6,3,4] => [7,1,6,5,4,3,2] => ? = 1 + 2
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [8,1,7,5,6,2,3,4] => [8,1,7,6,5,4,3,2] => ? = 1 + 2
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,9,5,6,7,8,2,4] => [3,1,9,8,7,6,5,4,2] => ? = 1 + 2
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,1,2,5,6,7,3] => [4,1,7,6,5,3,2] => ? = 2 + 2
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,1,4,5,2,7,8,3] => [6,1,8,7,5,4,3,2] => ? = 1 + 2
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [8,1,4,5,6,7,2,9,3] => [8,1,9,7,6,5,4,3,2] => ? = 1 + 2
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [10,1,4,5,6,7,8,9,2,3] => [10,1,9,8,7,6,5,4,3,2] => ? = 1 + 2
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,11,2] => [3,1,11,10,9,8,7,6,5,4,2] => ? = 1 + 2
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [8,3,4,5,6,7,1,10,2,9] => [8,3,10,9,7,6,1,5,4,2] => ? = 0 + 2
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [9,7,4,5,6,1,2,3,8] => [9,7,4,8,6,1,5,3,2] => ? = 0 + 2
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [6,3,4,5,1,7,9,2,8] => [6,3,9,8,1,7,5,4,2] => ? = 1 + 2
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [8,3,6,5,1,2,4,7] => [8,3,7,6,1,5,4,2] => ? = 0 + 2
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [2,6,4,5,1,8,3,7] => [2,8,7,6,1,5,4,3] => ? = 0 + 2
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [8,5,4,1,6,2,3,7] => [8,5,4,1,7,6,3,2] => ? = 1 + 2
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [4,3,1,5,6,8,2,7] => [4,3,1,8,7,6,5,2] => ? = 1 + 2
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,3,4,1,2,5,6] => [7,3,6,1,5,4,2] => ? = 0 + 2
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [2,7,5,1,3,4,6] => [2,7,6,1,5,4,3] => ? = 0 + 2
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,3,1,5,2,4,6] => [7,3,1,6,5,4,2] => ? = 1 + 2
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [6,1,5,4,3,2] => 3 = 1 + 2
Description
The number of non-overlapping descents in a permutation.
In other words, any maximal descending subsequence $\pi_i,\pi_{i+1},\dots,\pi_k$ contributes $\lfloor\frac{k-i+1}{2}\rfloor$ to the total count.
The following 10 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000133The "bounce" of a permutation. St000653The last descent of a permutation. St001480The number of simple summands of the module J^2/J^3. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000740The last entry of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
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