searching the database
Your data matches 9 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St001127
St001127: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 4
[1,1]
=> 2
[3]
=> 9
[2,1]
=> 5
[1,1,1]
=> 3
[4]
=> 16
[3,1]
=> 10
[2,2]
=> 8
[2,1,1]
=> 6
[1,1,1,1]
=> 4
[5]
=> 25
[4,1]
=> 17
[3,2]
=> 13
[3,1,1]
=> 11
[2,2,1]
=> 9
[2,1,1,1]
=> 7
[1,1,1,1,1]
=> 5
[6]
=> 36
[5,1]
=> 26
[4,2]
=> 20
[4,1,1]
=> 18
[3,3]
=> 18
[3,2,1]
=> 14
[3,1,1,1]
=> 12
[2,2,2]
=> 12
[2,2,1,1]
=> 10
[2,1,1,1,1]
=> 8
[1,1,1,1,1,1]
=> 6
[7]
=> 49
[6,1]
=> 37
[5,2]
=> 29
[5,1,1]
=> 27
[4,3]
=> 25
[4,2,1]
=> 21
[4,1,1,1]
=> 19
[3,3,1]
=> 19
[3,2,2]
=> 17
[3,2,1,1]
=> 15
[3,1,1,1,1]
=> 13
[2,2,2,1]
=> 13
[2,2,1,1,1]
=> 11
[2,1,1,1,1,1]
=> 9
[1,1,1,1,1,1,1]
=> 7
Description
The sum of the squares of the parts of a partition.
Matching statistic: St001688
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001688: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001688: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 4
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 9
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 5
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 16
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 10
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 8
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 6
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 25
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 17
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 13
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 11
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 9
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 7
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 36
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 26
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 20
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 18
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 12
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 14
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 12
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 18
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 10
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 8
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 49
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 37
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 29
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> 27
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 17
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 21
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 19
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 13
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 25
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 15
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> 13
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 19
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 11
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> 9
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
Description
The sum of the squares of the heights of the peaks of a Dyck path.
Matching statistic: St001645
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 25%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 25%
Values
[1]
=> [[1]]
=> [1] => ([],1)
=> 1
[2]
=> [[1,2]]
=> [2] => ([],2)
=> ? = 4
[1,1]
=> [[1],[2]]
=> [1,1] => ([(0,1)],2)
=> 2
[3]
=> [[1,2,3]]
=> [3] => ([],3)
=> ? ∊ {5,9}
[2,1]
=> [[1,3],[2]]
=> [1,2] => ([(1,2)],3)
=> ? ∊ {5,9}
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[4]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> ? ∊ {6,8,10,16}
[3,1]
=> [[1,3,4],[2]]
=> [1,3] => ([(2,3)],4)
=> ? ∊ {6,8,10,16}
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {6,8,10,16}
[2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {6,8,10,16}
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[5]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? ∊ {7,9,11,13,17,25}
[4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => ([(3,4)],5)
=> ? ∊ {7,9,11,13,17,25}
[3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {7,9,11,13,17,25}
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {7,9,11,13,17,25}
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {7,9,11,13,17,25}
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {7,9,11,13,17,25}
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5
[6]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => ([(4,5)],6)
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => ([],7)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [1,6] => ([(5,6)],7)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [2,5] => ([(4,6),(5,6)],7)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
Description
The pebbling number of a connected graph.
Matching statistic: St001232
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 21%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 21%
Values
[1]
=> [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 4
[3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {5,9}
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {5,9}
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? ∊ {6,8,10,16}
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {6,8,10,16}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ? ∊ {6,8,10,16}
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {6,8,10,16}
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? ∊ {7,9,11,13,17,25}
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {7,9,11,13,17,25}
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> ? ∊ {7,9,11,13,17,25}
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {7,9,11,13,17,25}
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> ? ∊ {7,9,11,13,17,25}
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {7,9,11,13,17,25}
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001880
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 18%
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 18%
Values
[1]
=> [1,0]
=> ([],1)
=> ? = 1
[2]
=> [1,0,1,0]
=> ([(0,1)],2)
=> ? ∊ {2,4}
[1,1]
=> [1,1,0,0]
=> ([],2)
=> ? ∊ {2,4}
[3]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3
[2,1]
=> [1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> ? ∊ {5,9}
[1,1,1]
=> [1,1,0,1,0,0]
=> ([(1,2)],3)
=> ? ∊ {5,9}
[4]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {6,8,10,16}
[2,2]
=> [1,1,1,0,0,0]
=> ([],3)
=> ? ∊ {6,8,10,16}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {6,8,10,16}
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {6,8,10,16}
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? ∊ {7,9,11,13,17,25}
[3,2]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {7,9,11,13,17,25}
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {7,9,11,13,17,25}
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> ? ∊ {7,9,11,13,17,25}
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {7,9,11,13,17,25}
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ? ∊ {7,9,11,13,17,25}
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[3,3]
=> [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
=> ? ∊ {8,10,12,12,14,18,18,20,26,36}
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> ([(0,5),(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(1,4)],5)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,6),(1,5),(1,6),(3,5),(4,2),(5,4),(6,4)],7)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4)],5)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4)],6)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,6),(1,5),(1,6),(2,4),(2,5),(4,3),(5,3),(6,4)],7)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,6),(1,5),(1,6),(2,4),(2,5),(5,3),(6,3),(6,4)],7)
=> ? ∊ {9,11,13,13,15,17,19,19,21,25,27,29,37,49}
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000605
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000605: Set partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 14%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000605: Set partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 14%
Values
[1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> {{1,3},{2,4}}
=> 1
[2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> {{1,2,5},{3,4,6}}
=> 2
[1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> {{1,3,4},{2,5,6}}
=> 4
[3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> {{1,2,3,7},{4,5,6,8}}
=> ? ∊ {5,9}
[2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> {{1,3,5},{2,4,6}}
=> 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> {{1,3,4,5},{2,6,7,8}}
=> ? ∊ {5,9}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> {{1,2,3,4,9},{5,6,7,8,10}}
=> ? ∊ {4,6,8,10,16}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> {{1,2,4,7},{3,5,6,8}}
=> ? ∊ {4,6,8,10,16}
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> {{1,2,5,6},{3,4,7,8}}
=> ? ∊ {4,6,8,10,16}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> {{1,3,4,6},{2,5,7,8}}
=> ? ∊ {4,6,8,10,16}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> {{1,3,4,5,6},{2,7,8,9,10}}
=> ? ∊ {4,6,8,10,16}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,11],[6,7,8,9,10,12]]
=> {{1,2,3,4,5,11},{6,7,8,9,10,12}}
=> ? ∊ {5,7,9,11,13,17,25}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> {{1,2,3,5,9},{4,6,7,8,10}}
=> ? ∊ {5,7,9,11,13,17,25}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> {{1,2,5,7},{3,4,6,8}}
=> ? ∊ {5,7,9,11,13,17,25}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> {{1,3,4,7},{2,5,6,8}}
=> ? ∊ {5,7,9,11,13,17,25}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> {{1,3,5,6},{2,4,7,8}}
=> ? ∊ {5,7,9,11,13,17,25}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> {{1,3,4,5,7},{2,6,8,9,10}}
=> ? ∊ {5,7,9,11,13,17,25}
[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]]
=> {{1,3,4,5,6,7},{2,8,9,10,11,12}}
=> ? ∊ {5,7,9,11,13,17,25}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12,14]]
=> {{1,2,3,4,5,6,13},{7,8,9,10,11,12,14}}
=> ? ∊ {6,8,10,12,12,14,18,18,20,26,36}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[1,2,3,4,6,11],[5,7,8,9,10,12]]
=> {{1,2,3,4,6,11},{5,7,8,9,10,12}}
=> ? ∊ {6,8,10,12,12,14,18,18,20,26,36}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> {{1,2,3,6,9},{4,5,7,8,10}}
=> ? ∊ {6,8,10,12,12,14,18,18,20,26,36}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> {{1,2,4,5,9},{3,6,7,8,10}}
=> ? ∊ {6,8,10,12,12,14,18,18,20,26,36}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[1,2,3,7,8],[4,5,6,9,10]]
=> {{1,2,3,7,8},{4,5,6,9,10}}
=> ? ∊ {6,8,10,12,12,14,18,18,20,26,36}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> {{1,3,5,7},{2,4,6,8}}
=> ? ∊ {6,8,10,12,12,14,18,18,20,26,36}
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> {{1,3,4,5,8},{2,6,7,9,10}}
=> ? ∊ {6,8,10,12,12,14,18,18,20,26,36}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[1,2,5,6,7],[3,4,8,9,10]]
=> {{1,2,5,6,7},{3,4,8,9,10}}
=> ? ∊ {6,8,10,12,12,14,18,18,20,26,36}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[1,3,4,6,7],[2,5,8,9,10]]
=> {{1,3,4,6,7},{2,5,8,9,10}}
=> ? ∊ {6,8,10,12,12,14,18,18,20,26,36}
[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]]
=> {{1,3,4,5,6,8},{2,7,9,10,11,12}}
=> ? ∊ {6,8,10,12,12,14,18,18,20,26,36}
[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]]
=> {{1,3,4,5,6,7,8},{2,9,10,11,12,13,14}}
=> ? ∊ {6,8,10,12,12,14,18,18,20,26,36}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,6,7,15],[8,9,10,11,12,13,14,16]]
=> {{1,2,3,4,5,6,7,15},{8,9,10,11,12,13,14,16}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[1,2,3,4,5,7,13],[6,8,9,10,11,12,14]]
=> {{1,2,3,4,5,7,13},{6,8,9,10,11,12,14}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[1,2,3,4,7,11],[5,6,8,9,10,12]]
=> {{1,2,3,4,7,11},{5,6,8,9,10,12}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[1,2,3,5,6,11],[4,7,8,9,10,12]]
=> {{1,2,3,5,6,11},{4,7,8,9,10,12}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[1,2,3,7,9],[4,5,6,8,10]]
=> {{1,2,3,7,9},{4,5,6,8,10}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> {{1,2,4,6,9},{3,5,7,8,10}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> {{1,3,4,5,9},{2,6,7,8,10}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[1,2,4,7,8],[3,5,6,9,10]]
=> {{1,2,4,7,8},{3,5,6,9,10}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> {{1,2,5,6,8},{3,4,7,9,10}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> {{1,3,4,6,8},{2,5,7,9,10}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[1,3,4,5,6,9],[2,7,8,10,11,12]]
=> {{1,3,4,5,6,9},{2,7,8,10,11,12}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> {{1,3,5,6,7},{2,4,8,9,10}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[1,3,4,5,7,8],[2,6,9,10,11,12]]
=> {{1,3,4,5,7,8},{2,6,9,10,11,12}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[1,3,4,5,6,7,9],[2,8,10,11,12,13,14]]
=> {{1,3,4,5,6,7,9},{2,8,10,11,12,13,14}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[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]]
=> {{1,3,4,5,6,7,8,9},{2,10,11,12,13,14,15,16}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
Description
The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block.
Matching statistic: St001583
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 14%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 14%
Values
[1]
=> [1,0,1,0]
=> [3,1,2] => [2,3,1] => 1
[2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => [1,3,4,2] => 4
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,2,4,5,3] => ? ∊ {5,9}
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,5,2,4] => ? ∊ {5,9}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [1,2,3,5,6,4] => ? ∊ {4,6,8,10,16}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,4,2,5,1] => ? ∊ {4,6,8,10,16}
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [5,1,3,2,4] => ? ∊ {4,6,8,10,16}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [4,5,2,3,1] => ? ∊ {4,6,8,10,16}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [3,1,4,6,2,5] => ? ∊ {4,6,8,10,16}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => [1,2,3,4,6,7,5] => ? ∊ {5,7,9,11,13,17,25}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [4,2,5,3,6,1] => ? ∊ {5,7,9,11,13,17,25}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,3,4,5,2] => ? ∊ {5,7,9,11,13,17,25}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [4,1,5,2,3] => ? ∊ {5,7,9,11,13,17,25}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [5,2,3,1,4] => ? ∊ {5,7,9,11,13,17,25}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [5,3,6,2,4,1] => ? ∊ {5,7,9,11,13,17,25}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => [3,1,4,5,7,2,6] => ? ∊ {5,7,9,11,13,17,25}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => [1,2,3,4,5,7,8,6] => ? ∊ {6,8,10,12,12,14,18,18,20,26,36}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => [2,5,3,6,4,7,1] => ? ∊ {6,8,10,12,12,14,18,18,20,26,36}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [4,5,1,3,6,2] => ? ∊ {6,8,10,12,12,14,18,18,20,26,36}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [5,2,1,6,3,4] => ? ∊ {6,8,10,12,12,14,18,18,20,26,36}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [6,1,2,4,3,5] => ? ∊ {6,8,10,12,12,14,18,18,20,26,36}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => ? ∊ {6,8,10,12,12,14,18,18,20,26,36}
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,5,6,4,2,3] => ? ∊ {6,8,10,12,12,14,18,18,20,26,36}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [4,1,2,6,3,5] => ? ∊ {6,8,10,12,12,14,18,18,20,26,36}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [6,4,2,3,1,5] => ? ∊ {6,8,10,12,12,14,18,18,20,26,36}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => [3,6,4,7,2,5,1] => ? ∊ {6,8,10,12,12,14,18,18,20,26,36}
[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] => [3,1,4,5,6,8,2,7] => ? ∊ {6,8,10,12,12,14,18,18,20,26,36}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,9,1,8] => [1,2,3,4,5,6,8,9,7] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,1,2,7] => [2,3,6,4,7,5,8,1] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => [5,3,6,1,4,7,2] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => [2,6,3,1,7,4,5] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [1,2,4,5,6,3] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [3,4,5,2,6,1] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [3,1,5,6,2,4] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [6,3,2,4,1,5] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [5,6,1,3,4,2] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [4,5,6,2,3,1] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => [6,1,4,7,5,2,3] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [4,6,2,1,3,5] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => [3,7,5,2,4,1,6] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[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] => [3,4,7,5,8,2,6,1] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
[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] => [3,1,4,5,6,7,9,2,8] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49}
Description
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
Matching statistic: St000615
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00092: Perfect matchings —to set partition⟶ Set partitions
St000615: Set partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 14%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00092: Perfect matchings —to set partition⟶ Set partitions
St000615: Set partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 14%
Values
[1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> {{1,2},{3,4}}
=> 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> {{1,4},{2,3},{5,6}}
=> 1 = 2 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> {{1,2},{3,6},{4,5}}
=> 3 = 4 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> {{1,6},{2,5},{3,4},{7,8}}
=> ? ∊ {5,9} - 1
[2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> {{1,2},{3,4},{5,6}}
=> 2 = 3 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> {{1,2},{3,8},{4,7},{5,6}}
=> ? ∊ {5,9} - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> {{1,8},{2,7},{3,6},{4,5},{9,10}}
=> ? ∊ {4,6,8,10,16} - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> {{1,6},{2,3},{4,5},{7,8}}
=> ? ∊ {4,6,8,10,16} - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> {{1,4},{2,3},{5,8},{6,7}}
=> ? ∊ {4,6,8,10,16} - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> {{1,2},{3,8},{4,5},{6,7}}
=> ? ∊ {4,6,8,10,16} - 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)]
=> {{1,2},{3,10},{4,9},{5,8},{6,7}}
=> ? ∊ {4,6,8,10,16} - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> {{1,10},{2,9},{3,8},{4,7},{5,6},{11,12}}
=> ? ∊ {5,7,9,11,13,17,25} - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> {{1,8},{2,7},{3,4},{5,6},{9,10}}
=> ? ∊ {5,7,9,11,13,17,25} - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> {{1,4},{2,3},{5,6},{7,8}}
=> ? ∊ {5,7,9,11,13,17,25} - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> {{1,2},{3,6},{4,5},{7,8}}
=> ? ∊ {5,7,9,11,13,17,25} - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> {{1,2},{3,4},{5,8},{6,7}}
=> ? ∊ {5,7,9,11,13,17,25} - 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)]
=> {{1,2},{3,10},{4,9},{5,6},{7,8}}
=> ? ∊ {5,7,9,11,13,17,25} - 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)]
=> {{1,2},{3,12},{4,11},{5,10},{6,9},{7,8}}
=> ? ∊ {5,7,9,11,13,17,25} - 1
[6]
=> [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)]
=> {{1,12},{2,11},{3,10},{4,9},{5,8},{6,7},{13,14}}
=> ? ∊ {6,8,10,12,12,14,18,18,20,26,36} - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> {{1,10},{2,9},{3,8},{4,5},{6,7},{11,12}}
=> ? ∊ {6,8,10,12,12,14,18,18,20,26,36} - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> {{1,8},{2,5},{3,4},{6,7},{9,10}}
=> ? ∊ {6,8,10,12,12,14,18,18,20,26,36} - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> {{1,8},{2,3},{4,7},{5,6},{9,10}}
=> ? ∊ {6,8,10,12,12,14,18,18,20,26,36} - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> {{1,6},{2,5},{3,4},{7,10},{8,9}}
=> ? ∊ {6,8,10,12,12,14,18,18,20,26,36} - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? ∊ {6,8,10,12,12,14,18,18,20,26,36} - 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)]
=> {{1,2},{3,10},{4,7},{5,6},{8,9}}
=> ? ∊ {6,8,10,12,12,14,18,18,20,26,36} - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> {{1,4},{2,3},{5,10},{6,9},{7,8}}
=> ? ∊ {6,8,10,12,12,14,18,18,20,26,36} - 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)]
=> {{1,2},{3,10},{4,5},{6,9},{7,8}}
=> ? ∊ {6,8,10,12,12,14,18,18,20,26,36} - 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)]
=> {{1,2},{3,12},{4,11},{5,10},{6,7},{8,9}}
=> ? ∊ {6,8,10,12,12,14,18,18,20,26,36} - 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)]
=> {{1,2},{3,14},{4,13},{5,12},{6,11},{7,10},{8,9}}
=> ? ∊ {6,8,10,12,12,14,18,18,20,26,36} - 1
[7]
=> [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)]
=> {{1,14},{2,13},{3,12},{4,11},{5,10},{6,9},{7,8},{15,16}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,6),(7,8),(13,14)]
=> {{1,12},{2,11},{3,10},{4,9},{5,6},{7,8},{13,14}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> {{1,10},{2,9},{3,6},{4,5},{7,8},{11,12}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> {{1,10},{2,9},{3,4},{5,8},{6,7},{11,12}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> {{1,6},{2,5},{3,4},{7,8},{9,10}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> {{1,8},{2,3},{4,5},{6,7},{9,10}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> {{1,2},{3,8},{4,7},{5,6},{9,10}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> {{1,6},{2,3},{4,5},{7,10},{8,9}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> {{1,4},{2,3},{5,10},{6,7},{8,9}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> {{1,2},{3,10},{4,5},{6,7},{8,9}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> {{1,2},{3,12},{4,11},{5,8},{6,7},{9,10}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> {{1,2},{3,4},{5,10},{6,9},{7,8}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> {{1,2},{3,12},{4,11},{5,6},{7,10},{8,9}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> {{1,2},{3,14},{4,13},{5,12},{6,11},{7,8},{9,10}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} - 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)]
=> {{1,2},{3,16},{4,15},{5,14},{6,13},{7,12},{8,11},{9,10}}
=> ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} - 1
Description
The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal.
Matching statistic: St000868
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000868: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 14%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000868: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 14%
Values
[1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 3 = 1 + 2
[2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 6 = 4 + 2
[1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 4 = 2 + 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => ? ∊ {5,9} + 2
[2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 5 = 3 + 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => ? ∊ {5,9} + 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [5,6,7,8,4,3,2,1,10,9] => ? ∊ {4,6,8,10,16} + 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => ? ∊ {4,6,8,10,16} + 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => ? ∊ {4,6,8,10,16} + 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => ? ∊ {4,6,8,10,16} + 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,7,8,9,10,6,5,4,3] => ? ∊ {4,6,8,10,16} + 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> [6,7,8,9,10,5,4,3,2,1,12,11] => ? ∊ {5,7,9,11,13,17,25} + 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [4,6,7,3,8,5,2,1,10,9] => ? ∊ {5,7,9,11,13,17,25} + 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => ? ∊ {5,7,9,11,13,17,25} + 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => ? ∊ {5,7,9,11,13,17,25} + 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => ? ∊ {5,7,9,11,13,17,25} + 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,6,8,9,5,10,7,4,3] => ? ∊ {5,7,9,11,13,17,25} + 2
[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)]
=> [2,1,8,9,10,11,12,7,6,5,4,3] => ? ∊ {5,7,9,11,13,17,25} + 2
[6]
=> [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)]
=> [7,8,9,10,11,12,6,5,4,3,2,1,14,13] => ? ∊ {6,8,10,12,12,14,18,18,20,26,36} + 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7),(11,12)]
=> [5,7,8,9,4,10,6,3,2,1,12,11] => ? ∊ {6,8,10,12,12,14,18,18,20,26,36} + 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [4,5,7,3,2,8,6,1,10,9] => ? ∊ {6,8,10,12,12,14,18,18,20,26,36} + 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [3,6,2,7,8,5,4,1,10,9] => ? ∊ {6,8,10,12,12,14,18,18,20,26,36} + 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [4,5,6,3,2,1,9,10,8,7] => ? ∊ {6,8,10,12,12,14,18,18,20,26,36} + 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => ? ∊ {6,8,10,12,12,14,18,18,20,26,36} + 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,6,7,9,5,4,10,8,3] => ? ∊ {6,8,10,12,12,14,18,18,20,26,36} + 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [3,4,2,1,8,9,10,7,6,5] => ? ∊ {6,8,10,12,12,14,18,18,20,26,36} + 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,5,8,4,9,10,7,6,3] => ? ∊ {6,8,10,12,12,14,18,18,20,26,36} + 2
[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)]
=> [2,1,7,9,10,11,6,12,8,5,4,3] => ? ∊ {6,8,10,12,12,14,18,18,20,26,36} + 2
[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)]
=> [2,1,9,10,11,12,13,14,8,7,6,5,4,3] => ? ∊ {6,8,10,12,12,14,18,18,20,26,36} + 2
[7]
=> [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)]
=> [8,9,10,11,12,13,14,7,6,5,4,3,2,1,16,15] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} + 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,6),(7,8),(13,14)]
=> [6,8,9,10,11,5,12,7,4,3,2,1,14,13] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} + 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8),(11,12)]
=> [5,6,8,9,4,3,10,7,2,1,12,11] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} + 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7),(11,12)]
=> [4,7,8,3,9,10,6,5,2,1,12,11] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} + 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [4,5,6,3,2,1,8,7,10,9] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} + 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [3,5,2,7,4,8,6,1,10,9] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} + 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,6,7,8,5,4,3,10,9] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} + 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [3,5,2,6,4,1,9,10,8,7] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} + 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [3,4,2,1,7,9,6,10,8,5] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} + 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,5,7,4,9,6,10,8,3] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} + 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)]
=> [2,1,7,8,10,11,6,5,12,9,4,3] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} + 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,8,9,10,7,6,5] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} + 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,6),(7,10),(8,9)]
=> [2,1,6,9,10,5,11,12,8,7,4,3] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} + 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,8),(9,10)]
=> [2,1,8,10,11,12,13,7,14,9,6,5,4,3] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} + 2
[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)]
=> [2,1,10,11,12,13,14,15,16,9,8,7,6,5,4,3] => ? ∊ {7,9,11,13,13,15,17,19,19,21,25,27,29,37,49} + 2
Description
The aid statistic in the sense of Shareshian-Wachs.
This is the number of admissible inversions [[St000866]] plus the number of descents [[St000021]]. This statistic was introduced by John Shareshian and Michelle L. Wachs in [1]. Theorem 4.1 states that the aid statistic together with the descent statistic is Euler-Mahonian.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!