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Your data matches 127 different statistics following compositions of up to 3 maps.
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Matching statistic: St001657
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
St001657: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 1
[1,1]
=> 0
[3]
=> 0
[2,1]
=> 1
[1,1,1]
=> 0
[4]
=> 0
[3,1]
=> 0
[2,2]
=> 2
[2,1,1]
=> 1
[1,1,1,1]
=> 0
[5]
=> 0
[4,1]
=> 0
[3,2]
=> 1
[3,1,1]
=> 0
[2,2,1]
=> 2
[2,1,1,1]
=> 1
[1,1,1,1,1]
=> 0
[6]
=> 0
[5,1]
=> 0
[4,2]
=> 1
[4,1,1]
=> 0
[3,3]
=> 0
[3,2,1]
=> 1
[3,1,1,1]
=> 0
[2,2,2]
=> 3
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 0
[7]
=> 0
[6,1]
=> 0
[5,2]
=> 1
[5,1,1]
=> 0
[4,3]
=> 0
[4,2,1]
=> 1
[4,1,1,1]
=> 0
[3,3,1]
=> 0
[3,2,2]
=> 2
[3,2,1,1]
=> 1
[3,1,1,1,1]
=> 0
[2,2,2,1]
=> 3
[2,2,1,1,1]
=> 2
[2,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> 0
[8]
=> 0
[7,1]
=> 0
[6,2]
=> 1
[6,1,1]
=> 0
[5,3]
=> 0
[5,2,1]
=> 1
Description
The number of twos in an integer partition.
The total number of twos in all partitions of $n$ is equal to the total number of singletons [[St001484]] in all partitions of $n-1$, see [1].
Matching statistic: St000475
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 98%●distinct values known / distinct values provided: 89%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 98%●distinct values known / distinct values provided: 89%
Values
[1]
=> []
=> []
=> ?
=> ? = 0
[2]
=> []
=> []
=> ?
=> ? = 1
[1,1]
=> [1]
=> [1]
=> []
=> 0
[3]
=> []
=> []
=> ?
=> ? = 1
[2,1]
=> [1]
=> [1]
=> []
=> 0
[1,1,1]
=> [1,1]
=> [2]
=> []
=> 0
[4]
=> []
=> []
=> ?
=> ? = 2
[3,1]
=> [1]
=> [1]
=> []
=> 0
[2,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[2,1,1]
=> [1,1]
=> [2]
=> []
=> 0
[1,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> 0
[5]
=> []
=> []
=> ?
=> ? = 2
[4,1]
=> [1]
=> [1]
=> []
=> 0
[3,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[3,1,1]
=> [1,1]
=> [2]
=> []
=> 0
[2,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> 0
[6]
=> []
=> []
=> ?
=> ? = 3
[5,1]
=> [1]
=> [1]
=> []
=> 0
[4,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[4,1,1]
=> [1,1]
=> [2]
=> []
=> 0
[3,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> 0
[2,2,2]
=> [2,2]
=> [2,2]
=> [2]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> []
=> 0
[7]
=> []
=> []
=> ?
=> ? = 3
[6,1]
=> [1]
=> [1]
=> []
=> 0
[5,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[5,1,1]
=> [1,1]
=> [2]
=> []
=> 0
[4,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[4,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> 0
[3,3,1]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[3,2,2]
=> [2,2]
=> [2,2]
=> [2]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> 0
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> [2]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> [1]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> []
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> []
=> 0
[8]
=> []
=> []
=> ?
=> ? = 4
[7,1]
=> [1]
=> [1]
=> []
=> 0
[6,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[6,1,1]
=> [1,1]
=> [2]
=> []
=> 0
[5,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[5,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> 0
[4,4]
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[4,3,1]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[4,2,2]
=> [2,2]
=> [2,2]
=> [2]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> 0
[3,3,2]
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2
[9]
=> []
=> []
=> ?
=> ? = 4
[10]
=> []
=> []
=> ?
=> ? = 5
[11]
=> []
=> []
=> ?
=> ? = 5
[12]
=> []
=> []
=> ?
=> ? = 6
[13]
=> []
=> []
=> ?
=> ? = 6
[14]
=> []
=> []
=> ?
=> ? = 7
[15]
=> []
=> []
=> ?
=> ? = 7
[16]
=> []
=> []
=> ?
=> ? = 8
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000674
(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
St000674: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> ? ∊ {0,1}
[1,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1}
[3]
=> []
=> []
=> ? ∊ {0,1}
[2,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1}
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[4]
=> []
=> []
=> ? ∊ {0,1}
[3,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1}
[2,2]
=> [2]
=> [1,0,1,0]
=> 2
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[5]
=> []
=> []
=> ? ∊ {0,1}
[4,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1}
[3,2]
=> [2]
=> [1,0,1,0]
=> 2
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[6]
=> []
=> []
=> ? ∊ {0,1}
[5,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1}
[4,2]
=> [2]
=> [1,0,1,0]
=> 2
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[7]
=> []
=> []
=> ? ∊ {0,1}
[6,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1}
[5,2]
=> [2]
=> [1,0,1,0]
=> 2
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[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]
=> 0
[8]
=> []
=> []
=> ? ∊ {0,1}
[7,1]
=> [1]
=> [1,0]
=> ? ∊ {0,1}
[6,2]
=> [2]
=> [1,0,1,0]
=> 2
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 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]
=> 0
[9]
=> []
=> []
=> ? ∊ {0,0,1}
[8,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,1}
[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]
=> ? ∊ {0,0,1}
[10]
=> []
=> []
=> ? ∊ {0,0,0,1}
[9,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,1}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,1}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,1}
[11]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,1}
[10,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,1,1}
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1}
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1}
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1}
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1}
[12]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2}
[11,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2}
[4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2}
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2}
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2}
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2}
[3,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2}
[2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2}
[2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2}
[2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2}
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,2}
[13]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3}
[12,1]
=> [1]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3}
[5,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3}
[4,4,1,1,1,1,1]
=> [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3}
[4,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3}
[4,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3}
[4,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3}
[4,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3}
[3,3,3,1,1,1,1]
=> [3,3,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,3}
Description
The number of hills of a Dyck path.
A hill is a peak with up step starting and down step ending at height zero.
Matching statistic: St001498
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 56% ●values known / values provided: 72%●distinct values known / distinct values provided: 56%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 56% ●values known / values provided: 72%●distinct values known / distinct values provided: 56%
Values
[1]
=> []
=> []
=> ?
=> ? = 0
[2]
=> []
=> []
=> ?
=> ? ∊ {0,1}
[1,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {0,1}
[3]
=> []
=> []
=> ?
=> ? ∊ {0,1}
[2,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {0,1}
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[4]
=> []
=> []
=> ?
=> ? ∊ {0,2}
[3,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {0,2}
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[5]
=> []
=> []
=> ?
=> ? ∊ {0,1,2}
[4,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {0,1,2}
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {0,1,2}
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[6]
=> []
=> []
=> ?
=> ? ∊ {0,2,3}
[5,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {0,2,3}
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {0,2,3}
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[7]
=> []
=> []
=> ?
=> ? ∊ {1,2,3}
[6,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {1,2,3}
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,2,3}
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[8]
=> []
=> []
=> ?
=> ? ∊ {2,2,3,4}
[7,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {2,2,3,4}
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {2,2,3,4}
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 0
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,2,3,4}
[9]
=> []
=> []
=> ?
=> ? ∊ {1,2,2,3,3,4}
[8,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {1,2,2,3,3,4}
[7,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[7,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[6,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[6,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,2,2,3,3,4}
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2,2,3,3,4}
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,3,3,4}
[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]
=> ?
=> ? ∊ {1,2,2,3,3,4}
[10]
=> []
=> []
=> ?
=> ? ∊ {0,1,2,3,3,3,4,5}
[9,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {0,1,2,3,3,3,4,5}
[7,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {0,1,2,3,3,3,4,5}
[4,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,1,2,3,3,3,4,5}
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,1,2,3,3,3,4,5}
[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]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? ∊ {0,1,2,3,3,3,4,5}
[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]
=> ?
=> ? ∊ {0,1,2,3,3,3,4,5}
[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]
=> ?
=> ? ∊ {0,1,2,3,3,3,4,5}
[11]
=> []
=> []
=> ?
=> ? ∊ {1,1,1,2,2,3,3,3,4,4,5}
[10,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {1,1,1,2,2,3,3,3,4,4,5}
[8,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {1,1,1,2,2,3,3,3,4,4,5}
[5,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,2,2,3,3,3,4,4,5}
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,2,2,3,3,3,4,4,5}
[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]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? ∊ {1,1,1,2,2,3,3,3,4,4,5}
[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]
=> ?
=> ? ∊ {1,1,1,2,2,3,3,3,4,4,5}
[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]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? ∊ {1,1,1,2,2,3,3,3,4,4,5}
[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]
=> ?
=> ? ∊ {1,1,1,2,2,3,3,3,4,4,5}
[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]
=> ?
=> ? ∊ {1,1,1,2,2,3,3,3,4,4,5}
[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]
=> ?
=> ? ∊ {1,1,1,2,2,3,3,3,4,4,5}
[12]
=> []
=> []
=> ?
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,3,4,4,4,5,6}
[11,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,3,4,4,4,5,6}
[9,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,3,4,4,4,5,6}
[6,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,3,4,4,4,5,6}
[5,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,1,1,2,2,3,3,3,3,3,4,4,4,5,6}
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St001274
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001274: Dyck paths ⟶ ℤResult quality: 56% ●values known / values provided: 71%●distinct values known / distinct values provided: 56%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001274: Dyck paths ⟶ ℤResult quality: 56% ●values known / values provided: 71%●distinct values known / distinct values provided: 56%
Values
[1]
=> []
=> ?
=> ?
=> ? = 0
[2]
=> []
=> ?
=> ?
=> ? ∊ {0,1}
[1,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1}
[3]
=> []
=> ?
=> ?
=> ? ∊ {0,1}
[2,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1}
[1,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[4]
=> []
=> ?
=> ?
=> ? ∊ {0,1,2}
[3,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,2}
[2,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,2}
[2,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[5]
=> []
=> ?
=> ?
=> ? ∊ {0,1,1}
[4,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,1}
[3,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,1}
[3,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[2,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[6]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,3}
[5,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,1,3}
[4,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,1,3}
[4,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[3,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,1,3}
[3,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[7]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,3}
[6,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,1,3}
[5,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,1,3}
[5,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[4,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,1,3}
[4,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,3,1]
=> [3,1]
=> [1]
=> [1,0]
=> 0
[3,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[8]
=> []
=> ?
=> ?
=> ? ∊ {0,1,2,3,4}
[7,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,2,3,4}
[6,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,2,3,4}
[6,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[5,3]
=> [3]
=> []
=> []
=> ? ∊ {0,1,2,3,4}
[5,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[4,4]
=> [4]
=> []
=> []
=> ? ∊ {0,1,2,3,4}
[4,3,1]
=> [3,1]
=> [1]
=> [1,0]
=> 0
[4,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[3,3,2]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[9]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,3,3,4}
[8,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,1,3,3,4}
[7,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,1,3,3,4}
[7,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 0
[6,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,1,3,3,4}
[6,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[5,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,1,3,3,4}
[5,3,1]
=> [3,1]
=> [1]
=> [1,0]
=> 0
[5,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[4,4,1]
=> [4,1]
=> [1]
=> [1,0]
=> 0
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,1,3,3,4}
[10]
=> []
=> ?
=> ?
=> ? ∊ {1,2,2,3,3,3,4,5}
[9,1]
=> [1]
=> []
=> []
=> ? ∊ {1,2,2,3,3,3,4,5}
[8,2]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,3,3,3,4,5}
[7,3]
=> [3]
=> []
=> []
=> ? ∊ {1,2,2,3,3,3,4,5}
[6,4]
=> [4]
=> []
=> []
=> ? ∊ {1,2,2,3,3,3,4,5}
[5,5]
=> [5]
=> []
=> []
=> ? ∊ {1,2,2,3,3,3,4,5}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,3,3,3,4,5}
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,2,2,3,3,3,4,5}
[11]
=> []
=> ?
=> ?
=> ? ∊ {0,1,1,2,3,3,3,3,4,4,5}
[10,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,1,2,3,3,3,3,4,4,5}
[9,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,1,2,3,3,3,3,4,4,5}
[8,3]
=> [3]
=> []
=> []
=> ? ∊ {0,1,1,2,3,3,3,3,4,4,5}
[7,4]
=> [4]
=> []
=> []
=> ? ∊ {0,1,1,2,3,3,3,3,4,4,5}
[6,5]
=> [5]
=> []
=> []
=> ? ∊ {0,1,1,2,3,3,3,3,4,4,5}
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,1,1,2,3,3,3,3,4,4,5}
[2,2,2,1,1,1,1,1]
=> [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]
=> ? ∊ {0,1,1,2,3,3,3,3,4,4,5}
[2,2,1,1,1,1,1,1,1]
=> [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]
=> ? ∊ {0,1,1,2,3,3,3,3,4,4,5}
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,1,1,2,3,3,3,3,4,4,5}
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,1,1,2,3,3,3,3,4,4,5}
[12]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,4,4,5,6}
Description
The number of indecomposable injective modules with projective dimension equal to two.
Matching statistic: St001107
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 89%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 89%
Values
[1]
=> []
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> []
=> ? ∊ {0,1}
[1,1]
=> [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,1}
[3]
=> []
=> []
=> []
=> ? ∊ {0,1}
[2,1]
=> [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,1}
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[4]
=> []
=> []
=> []
=> ? ∊ {0,1}
[3,1]
=> [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,1}
[2,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 0
[5]
=> []
=> []
=> []
=> ? ∊ {0,1}
[4,1]
=> [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,1}
[3,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[6]
=> []
=> []
=> []
=> ? ∊ {0,1}
[5,1]
=> [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,1}
[4,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 0
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[7]
=> []
=> []
=> []
=> ? ∊ {0,1}
[6,1]
=> [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,1}
[5,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 0
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[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]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[8]
=> []
=> []
=> []
=> ? ∊ {0,1}
[7,1]
=> [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,1}
[6,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 0
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 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]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[9]
=> []
=> []
=> []
=> ? ∊ {0,1}
[8,1]
=> [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,1}
[10]
=> []
=> []
=> []
=> ? ∊ {0,0,1,1}
[9,1]
=> [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,1,1}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1}
[11]
=> []
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2}
[10,1]
=> [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,1,1,1,2}
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,1,1,1,2}
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,1,1,1,2}
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,1,1,1,2}
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,1,1,1,2}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,1,1,1,2}
[12]
=> []
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,3}
[11,1]
=> [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,3}
[4,4,1,1,1,1]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,3}
[4,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,3}
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,3}
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,3}
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,3}
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,3}
[3,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,3}
[2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,3}
[2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,3}
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,3}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,3}
[13]
=> []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,4}
[12,1]
=> [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,4}
[5,5,1,1,1]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,4}
[5,4,1,1,1,1]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,4}
[5,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,4}
[5,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,4}
[4,4,2,1,1,1]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,4}
[4,4,1,1,1,1,1]
=> [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,4}
[4,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,3,3,4}
Description
The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path.
In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.
Matching statistic: St000234
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 78%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 78%
Values
[1]
=> []
=> []
=> [] => ? = 0
[2]
=> []
=> []
=> [] => ? = 1
[1,1]
=> [1]
=> [1,0]
=> [1] => 0
[3]
=> []
=> []
=> [] => ? = 1
[2,1]
=> [1]
=> [1,0]
=> [1] => 0
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 0
[4]
=> []
=> []
=> [] => ? = 2
[3,1]
=> [1]
=> [1,0]
=> [1] => 0
[2,2]
=> [2]
=> [1,0,1,0]
=> [1,2] => 1
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 0
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[5]
=> []
=> []
=> [] => ? = 2
[4,1]
=> [1]
=> [1,0]
=> [1] => 0
[3,2]
=> [2]
=> [1,0,1,0]
=> [1,2] => 1
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 0
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[6]
=> []
=> []
=> [] => ? = 3
[5,1]
=> [1]
=> [1,0]
=> [1] => 0
[4,2]
=> [2]
=> [1,0,1,0]
=> [1,2] => 1
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 0
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 2
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[7]
=> []
=> []
=> [] => ? = 3
[6,1]
=> [1]
=> [1,0]
=> [1] => 0
[5,2]
=> [2]
=> [1,0,1,0]
=> [1,2] => 1
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 0
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 2
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[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]
=> [2,3,4,5,6,1] => 0
[8]
=> []
=> []
=> [] => ? = 4
[7,1]
=> [1]
=> [1,0]
=> [1] => 0
[6,2]
=> [2]
=> [1,0,1,0]
=> [1,2] => 1
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 0
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 2
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 3
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[9]
=> []
=> []
=> [] => ? ∊ {0,4}
[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]
=> [2,3,4,5,6,7,8,1] => ? ∊ {0,4}
[10]
=> []
=> []
=> [] => ? ∊ {0,0,1,5}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => ? ∊ {0,0,1,5}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {0,0,1,5}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? ∊ {0,0,1,5}
[11]
=> []
=> []
=> [] => ? ∊ {0,0,0,0,1,1,2,5}
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,3] => ? ∊ {0,0,0,0,1,1,2,5}
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => ? ∊ {0,0,0,0,1,1,2,5}
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {0,0,0,0,1,1,2,5}
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,8,2] => ? ∊ {0,0,0,0,1,1,2,5}
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,2] => ? ∊ {0,0,0,0,1,1,2,5}
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? ∊ {0,0,0,0,1,1,2,5}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => ? ∊ {0,0,0,0,1,1,2,5}
[12]
=> []
=> []
=> [] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,3,6}
[4,4,1,1,1,1]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,3,6}
[4,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,3] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,3,6}
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,3,6}
[4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,3,6}
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,2,5,6,7,8,3] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,3,6}
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,9,3] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,3,6}
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,8,2] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,3,6}
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,2] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,3,6}
[3,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,3,6}
[2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [4,1,2,5,6,7,8,3] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,3,6}
[2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,8,9,2] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,3,6}
[2,2,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,10,2] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,3,6}
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,3,6}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,3,6}
[13]
=> []
=> []
=> [] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,4,6}
[5,5,1,1,1]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,4,6}
[5,4,1,1,1,1]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,4,6}
[5,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,4,6}
[5,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,4,6}
[5,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,4,6}
[4,4,2,1,1,1]
=> [4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,2,5,3,6,7,8,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,4,6}
[4,4,1,1,1,1,1]
=> [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,9,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,4,6}
[4,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,2,5,6,7,8,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,4,6}
[4,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,9,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,4,6}
[4,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,7,8,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,4,6}
[4,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,9,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,4,6}
[4,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,4,6}
Description
The number of global ascents of a permutation.
The global ascents are the integers $i$ such that
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$$
Equivalently, by the pigeonhole principle,
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$$
For $n > 1$ it can also be described as an occurrence of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives [[oeis:A003319]].
Matching statistic: St001744
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001744: Permutations ⟶ ℤResult quality: 44% ●values known / values provided: 65%●distinct values known / distinct values provided: 44%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001744: Permutations ⟶ ℤResult quality: 44% ●values known / values provided: 65%●distinct values known / distinct values provided: 44%
Values
[1]
=> []
=> []
=> [] => ? = 0
[2]
=> []
=> []
=> [] => ? = 1
[1,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[3]
=> []
=> []
=> [] => ? = 1
[2,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[4]
=> []
=> []
=> [] => ? = 2
[3,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[5]
=> []
=> []
=> [] => ? = 1
[4,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 2
[6]
=> []
=> []
=> [] => ? = 1
[5,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => 3
[7]
=> []
=> []
=> [] => ? ∊ {1,2}
[6,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 0
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 2
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => 3
[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]
=> [1,7,2,3,4,5,6] => ? ∊ {1,2}
[8]
=> []
=> []
=> [] => ? ∊ {0,1,4}
[7,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 2
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 0
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 2
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[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]
=> [1,7,2,3,4,5,6] => ? ∊ {0,1,4}
[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]
=> [1,8,2,3,4,5,6,7] => ? ∊ {0,1,4}
[9]
=> []
=> []
=> [] => ? ∊ {0,0,0,3,4}
[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]
=> [1,7,2,3,4,5,6] => ? ∊ {0,0,0,3,4}
[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]
=> [1,6,7,2,3,4,5] => ? ∊ {0,0,0,3,4}
[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]
=> [1,8,2,3,4,5,6,7] => ? ∊ {0,0,0,3,4}
[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]
=> [1,9,2,3,4,5,6,7,8] => ? ∊ {0,0,0,3,4}
[10]
=> []
=> []
=> [] => ? ∊ {0,0,0,0,1,3,4,5}
[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]
=> [1,7,2,3,4,5,6] => ? ∊ {0,0,0,0,1,3,4,5}
[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]
=> [1,6,7,2,3,4,5] => ? ∊ {0,0,0,0,1,3,4,5}
[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]
=> [1,8,2,3,4,5,6,7] => ? ∊ {0,0,0,0,1,3,4,5}
[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]
=> [1,5,7,2,3,4,6] => ? ∊ {0,0,0,0,1,3,4,5}
[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]
=> [1,7,8,2,3,4,5,6] => ? ∊ {0,0,0,0,1,3,4,5}
[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]
=> [1,9,2,3,4,5,6,7,8] => ? ∊ {0,0,0,0,1,3,4,5}
[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]
=> [1,10,2,3,4,5,6,7,8,9] => ? ∊ {0,0,0,0,1,3,4,5}
[11]
=> []
=> []
=> [] => ? ∊ {0,0,0,0,0,0,1,1,3,3,4,4,5}
[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]
=> [1,7,2,3,4,5,6] => ? ∊ {0,0,0,0,0,0,1,1,3,3,4,4,5}
[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]
=> [1,6,7,2,3,4,5] => ? ∊ {0,0,0,0,0,0,1,1,3,3,4,4,5}
[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]
=> [1,8,2,3,4,5,6,7] => ? ∊ {0,0,0,0,0,0,1,1,3,3,4,4,5}
[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]
=> [1,6,2,7,3,4,5] => ? ∊ {0,0,0,0,0,0,1,1,3,3,4,4,5}
[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]
=> [1,5,7,2,3,4,6] => ? ∊ {0,0,0,0,0,0,1,1,3,3,4,4,5}
[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]
=> [1,7,8,2,3,4,5,6] => ? ∊ {0,0,0,0,0,0,1,1,3,3,4,4,5}
[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]
=> [1,9,2,3,4,5,6,7,8] => ? ∊ {0,0,0,0,0,0,1,1,3,3,4,4,5}
[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]
=> [1,4,7,2,3,5,6] => ? ∊ {0,0,0,0,0,0,1,1,3,3,4,4,5}
[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]
=> [1,6,8,2,3,4,5,7] => ? ∊ {0,0,0,0,0,0,1,1,3,3,4,4,5}
[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]
=> [1,8,9,2,3,4,5,6,7] => ? ∊ {0,0,0,0,0,0,1,1,3,3,4,4,5}
[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]
=> [1,10,2,3,4,5,6,7,8,9] => ? ∊ {0,0,0,0,0,0,1,1,3,3,4,4,5}
[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]
=> [1,11,2,3,4,5,6,7,8,9,10] => ? ∊ {0,0,0,0,0,0,1,1,3,3,4,4,5}
[12]
=> []
=> []
=> [] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,3,3,3,4,4,4,5,6}
[6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,3,3,3,4,4,4,5,6}
[6,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,2,3,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,3,3,3,4,4,4,5,6}
[5,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,2,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,3,3,3,4,4,4,5,6}
[5,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,2,3,4,5,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,3,3,3,4,4,4,5,6}
[4,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,2,7,3,4,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,3,3,3,4,4,4,5,6}
[4,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,2,3,4,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,3,3,3,4,4,4,5,6}
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,2,3,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,3,3,3,4,4,4,5,6}
[4,1,1,1,1,1,1,1,1]
=> [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]
=> [1,9,2,3,4,5,6,7,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,3,3,3,4,4,4,5,6}
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,5,6,7,2,3,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,3,3,3,4,4,4,5,6}
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,7,2,8,3,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,3,3,3,4,4,4,5,6}
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,2,3,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,3,3,3,4,4,4,5,6}
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,6,8,2,3,4,5,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1,1,3,3,3,4,4,4,5,6}
Description
The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation.
Let $\nu$ be a (partial) permutation of $[k]$ with $m$ letters together with dashes between some of its letters. An occurrence of $\nu$ in a permutation $\tau$ is a subsequence $\tau_{a_1},\dots,\tau_{a_m}$
such that $a_i + 1 = a_{i+1}$ whenever there is a dash between the $i$-th and the $(i+1)$-st letter of $\nu$, which is order isomorphic to $\nu$.
Thus, $\nu$ is a vincular pattern, except that it is not required to be a permutation.
An arrow pattern of size $k$ consists of such a generalized vincular pattern $\nu$ and arrows $b_1\to c_1, b_2\to c_2,\dots$, such that precisely the numbers $1,\dots,k$ appear in the vincular pattern and the arrows.
Let $\Phi$ be the map [[Mp00087]]. Let $\tau$ be a permutation and $\sigma = \Phi(\tau)$. Then a subsequence $w = (x_{a_1},\dots,x_{a_m})$ of $\tau$ is an occurrence of the arrow pattern if $w$ is an occurrence of $\nu$, for each arrow $b\to c$ we have $\sigma(x_b) = x_c$ and $x_1 < x_2 < \dots < x_k$.
Matching statistic: St000954
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000954: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 67%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000954: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 67%
Values
[1]
=> []
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> []
=> ? = 1
[1,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[3]
=> []
=> []
=> []
=> ? = 0
[2,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[4]
=> []
=> []
=> []
=> ? = 0
[3,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[5]
=> []
=> []
=> []
=> ? = 1
[4,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[6]
=> []
=> []
=> []
=> ? = 1
[5,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 0
[7]
=> []
=> []
=> []
=> ? ∊ {0,2}
[6,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 0
[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]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {0,2}
[8]
=> []
=> []
=> []
=> ? ∊ {2,2,4}
[7,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 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]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {2,2,4}
[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]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? ∊ {2,2,4}
[9]
=> []
=> []
=> []
=> ? ∊ {1,2,2,3,4}
[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]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {1,2,2,3,4}
[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]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? ∊ {1,2,2,3,4}
[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]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? ∊ {1,2,2,3,4}
[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]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> ? ∊ {1,2,2,3,4}
[10]
=> []
=> []
=> []
=> ? ∊ {1,1,1,2,3,3,4,5}
[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]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {1,1,1,2,3,3,4,5}
[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]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? ∊ {1,1,1,2,3,3,4,5}
[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]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? ∊ {1,1,1,2,3,3,4,5}
[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]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> ? ∊ {1,1,1,2,3,3,4,5}
[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]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? ∊ {1,1,1,2,3,3,4,5}
[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]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> ? ∊ {1,1,1,2,3,3,4,5}
[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]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> ? ∊ {1,1,1,2,3,3,4,5}
[11]
=> []
=> []
=> []
=> ? ∊ {0,1,1,1,1,1,2,3,3,3,4,4,5}
[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]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,2,3,3,3,4,4,5}
[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]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,2,3,3,3,4,4,5}
[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]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,2,3,3,3,4,4,5}
[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]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,2,3,3,3,4,4,5}
[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]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,2,3,3,3,4,4,5}
[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]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,2,3,3,3,4,4,5}
[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]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,2,3,3,3,4,4,5}
[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]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,2,3,3,3,4,4,5}
[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]
=> [1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,2,3,3,3,4,4,5}
[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]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0]
=> ? ∊ {0,1,1,1,1,1,2,3,3,3,4,4,5}
[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]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,2,3,3,3,4,4,5}
[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]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
=> ? ∊ {0,1,1,1,1,1,2,3,3,3,4,4,5}
[12]
=> []
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,6}
[6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,6}
[6,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,6}
[5,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,6}
[5,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,6}
[4,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,6}
[4,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,6}
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,6}
[4,1,1,1,1,1,1,1,1]
=> [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]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,6}
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,6}
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,6}
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,6}
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,2,2,2,3,3,3,3,3,4,4,5,6}
Description
Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$.
Matching statistic: St001130
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001130: Permutations ⟶ ℤResult quality: 44% ●values known / values provided: 62%●distinct values known / distinct values provided: 44%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001130: Permutations ⟶ ℤResult quality: 44% ●values known / values provided: 62%●distinct values known / distinct values provided: 44%
Values
[1]
=> []
=> []
=> [] => ? = 0
[2]
=> []
=> []
=> [] => ? = 1
[1,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[3]
=> []
=> []
=> [] => ? = 1
[2,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[4]
=> []
=> []
=> [] => ? = 2
[3,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 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
[5]
=> []
=> []
=> [] => ? = 1
[4,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0
[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]
=> [2,1,3] => 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] => 2
[6]
=> []
=> []
=> [] => ? = 1
[5,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,1,3] => 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]
=> [2,3,1,4] => 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 2
[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] => 3
[7]
=> []
=> []
=> [] => ? ∊ {1,2}
[6,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,1,3] => 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]
=> [3,1,2,4] => 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]
=> [2,3,1,4] => 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 2
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 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] => 3
[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] => ? ∊ {1,2}
[8]
=> []
=> []
=> [] => ? ∊ {1,1,4}
[7,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,1,3] => 0
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 2
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => 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]
=> [2,3,1,4] => 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 2
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 0
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 0
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 0
[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] => ? ∊ {1,1,4}
[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] => ? ∊ {1,1,4}
[9]
=> []
=> []
=> [] => ? ∊ {0,1,2,3,4}
[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] => ? ∊ {0,1,2,3,4}
[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]
=> [2,3,4,5,6,1,7] => ? ∊ {0,1,2,3,4}
[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] => ? ∊ {0,1,2,3,4}
[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] => ? ∊ {0,1,2,3,4}
[10]
=> []
=> []
=> [] => ? ∊ {1,1,1,2,2,3,4,5}
[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] => ? ∊ {1,1,1,2,2,3,4,5}
[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]
=> [2,3,4,5,6,1,7] => ? ∊ {1,1,1,2,2,3,4,5}
[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] => ? ∊ {1,1,1,2,2,3,4,5}
[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]
=> [2,3,4,5,7,1,6] => ? ∊ {1,1,1,2,2,3,4,5}
[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]
=> [2,3,4,5,6,7,1,8] => ? ∊ {1,1,1,2,2,3,4,5}
[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] => ? ∊ {1,1,1,2,2,3,4,5}
[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] => ? ∊ {1,1,1,2,2,3,4,5}
[11]
=> []
=> []
=> [] => ? ∊ {0,1,1,1,1,2,2,2,3,3,4,4,5}
[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] => ? ∊ {0,1,1,1,1,2,2,2,3,3,4,4,5}
[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]
=> [2,3,4,5,6,1,7] => ? ∊ {0,1,1,1,1,2,2,2,3,3,4,4,5}
[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] => ? ∊ {0,1,1,1,1,2,2,2,3,3,4,4,5}
[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]
=> [2,3,4,5,1,6,7] => ? ∊ {0,1,1,1,1,2,2,2,3,3,4,4,5}
[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]
=> [2,3,4,5,7,1,6] => ? ∊ {0,1,1,1,1,2,2,2,3,3,4,4,5}
[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]
=> [2,3,4,5,6,7,1,8] => ? ∊ {0,1,1,1,1,2,2,2,3,3,4,4,5}
[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] => ? ∊ {0,1,1,1,1,2,2,2,3,3,4,4,5}
[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]
=> [2,3,4,6,7,1,5] => ? ∊ {0,1,1,1,1,2,2,2,3,3,4,4,5}
[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]
=> [2,3,4,5,6,8,1,7] => ? ∊ {0,1,1,1,1,2,2,2,3,3,4,4,5}
[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]
=> [2,3,4,5,6,7,8,1,9] => ? ∊ {0,1,1,1,1,2,2,2,3,3,4,4,5}
[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] => ? ∊ {0,1,1,1,1,2,2,2,3,3,4,4,5}
[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] => ? ∊ {0,1,1,1,1,2,2,2,3,3,4,4,5}
[12]
=> []
=> []
=> [] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,6}
[6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,6}
[6,1,1,1,1,1,1]
=> [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] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,6}
[5,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,7] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,6}
[5,1,1,1,1,1,1,1]
=> [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] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,6}
[4,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,3,4,5,1,6,7] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,6}
[4,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,3,4,5,7,1,6] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,6}
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1,8] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,6}
[4,1,1,1,1,1,1,1,1]
=> [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] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,6}
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,3,4,5,1,7,6] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,6}
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [2,3,4,5,6,1,7,8] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,6}
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,3,4,6,7,1,5] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,6}
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,8,1,7] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,6}
Description
The number of two successive successions in a permutation.
The following 117 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St000022The number of fixed points of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000873The aix statistic of a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000895The number of ones on the main diagonal of an alternating sign matrix. St000051The size of the left subtree of a binary tree. St000117The number of centered tunnels of a Dyck path. St000221The number of strong fixed points of a permutation. St000241The number of cyclical small excedances. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001596The number of two-by-two squares inside a skew partition. St000150The floored half-sum of the multiplicities of a partition. St001557The number of inversions of the second entry of a permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000011The number of touch points (or returns) of a Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000678The number of up steps after the last double rise of a Dyck path. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000675The number of centered multitunnels of a Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000932The number of occurrences of the pattern UDU in a Dyck path. St001520The number of strict 3-descents. St000925The number of topologically connected components of a set partition. St000654The first descent of a permutation. St001176The size of a partition minus its first part. St001525The number of symmetric hooks on the diagonal of a partition. St000478Another weight of a partition according to Alladi. St000260The radius of a connected graph. St000237The number of small exceedances. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St000025The number of initial rises of a Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000502The number of successions of a set partitions. St000363The number of minimal vertex covers of a graph. St000546The number of global descents of a permutation. St000007The number of saliances of the permutation. St000883The number of longest increasing subsequences of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000909The number of maximal chains of maximal size in a poset. St000911The number of maximal antichains of maximal size in a poset. St000441The number of successions of a permutation. St000214The number of adjacencies of a permutation. St000717The number of ordinal summands of a poset. St001651The Frankl number of a lattice. St000456The monochromatic index of a connected graph. St000884The number of isolated descents of a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St000843The decomposition number of a perfect matching. St000990The first ascent of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000504The cardinality of the first block of a set partition. St001052The length of the exterior of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000335The difference of lower and upper interactions. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000056The decomposition (or block) number of a permutation. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000084The number of subtrees. St000287The number of connected components of a graph. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St000061The number of nodes on the left branch of a binary tree. St000731The number of double exceedences of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000665The number of rafts of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000741The Colin de Verdière graph invariant. St000647The number of big descents of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St000989The number of final rises of a permutation. St000028The number of stack-sorts needed to sort a permutation. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St000366The number of double descents of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000732The number of double deficiencies of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001810The number of fixed points of a permutation smaller than its largest moved point. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St000542The number of left-to-right-minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001487The number of inner corners of a skew partition. St000066The column of the unique '1' in the first row of the alternating sign matrix. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000455The second largest eigenvalue of a graph if it is integral.
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