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Your data matches 45 different statistics following compositions of up to 3 maps.
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Matching statistic: St000532
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St000532: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 2
[2]
=> 3
[1,1]
=> 3
[3]
=> 4
[2,1]
=> 5
[1,1,1]
=> 4
[4]
=> 5
[3,1]
=> 7
[2,2]
=> 7
[2,1,1]
=> 7
[1,1,1,1]
=> 5
[5]
=> 6
[4,1]
=> 9
[3,2]
=> 10
[3,1,1]
=> 10
[2,2,1]
=> 10
[2,1,1,1]
=> 9
[1,1,1,1,1]
=> 6
[6]
=> 7
[5,1]
=> 11
[4,2]
=> 13
[4,1,1]
=> 13
[3,3]
=> 13
[3,2,1]
=> 15
[3,1,1,1]
=> 13
[2,2,2]
=> 13
[2,2,1,1]
=> 13
[2,1,1,1,1]
=> 11
[1,1,1,1,1,1]
=> 7
[7]
=> 8
[6,1]
=> 13
[5,2]
=> 16
[5,1,1]
=> 16
[4,3]
=> 17
[4,2,1]
=> 20
[4,1,1,1]
=> 17
[3,3,1]
=> 20
[3,2,2]
=> 20
[3,2,1,1]
=> 20
[3,1,1,1,1]
=> 16
[2,2,2,1]
=> 17
[2,2,1,1,1]
=> 16
[2,1,1,1,1,1]
=> 13
[1,1,1,1,1,1,1]
=> 8
Description
The total number of rook placements on a Ferrers board.
Matching statistic: St001658
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(load all 2 compositions to match this statistic)
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001658: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001658: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [[1],[]]
=> 2
[2]
=> [[2],[]]
=> 3
[1,1]
=> [[1,1],[]]
=> 3
[3]
=> [[3],[]]
=> 4
[2,1]
=> [[2,1],[]]
=> 5
[1,1,1]
=> [[1,1,1],[]]
=> 4
[4]
=> [[4],[]]
=> 5
[3,1]
=> [[3,1],[]]
=> 7
[2,2]
=> [[2,2],[]]
=> 7
[2,1,1]
=> [[2,1,1],[]]
=> 7
[1,1,1,1]
=> [[1,1,1,1],[]]
=> 5
[5]
=> [[5],[]]
=> 6
[4,1]
=> [[4,1],[]]
=> 9
[3,2]
=> [[3,2],[]]
=> 10
[3,1,1]
=> [[3,1,1],[]]
=> 10
[2,2,1]
=> [[2,2,1],[]]
=> 10
[2,1,1,1]
=> [[2,1,1,1],[]]
=> 9
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 6
[6]
=> [[6],[]]
=> 7
[5,1]
=> [[5,1],[]]
=> 11
[4,2]
=> [[4,2],[]]
=> 13
[4,1,1]
=> [[4,1,1],[]]
=> 13
[3,3]
=> [[3,3],[]]
=> 13
[3,2,1]
=> [[3,2,1],[]]
=> 15
[3,1,1,1]
=> [[3,1,1,1],[]]
=> 13
[2,2,2]
=> [[2,2,2],[]]
=> 13
[2,2,1,1]
=> [[2,2,1,1],[]]
=> 13
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> 11
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> 7
[7]
=> [[7],[]]
=> 8
[6,1]
=> [[6,1],[]]
=> 13
[5,2]
=> [[5,2],[]]
=> 16
[5,1,1]
=> [[5,1,1],[]]
=> 16
[4,3]
=> [[4,3],[]]
=> 17
[4,2,1]
=> [[4,2,1],[]]
=> 20
[4,1,1,1]
=> [[4,1,1,1],[]]
=> 17
[3,3,1]
=> [[3,3,1],[]]
=> 20
[3,2,2]
=> [[3,2,2],[]]
=> 20
[3,2,1,1]
=> [[3,2,1,1],[]]
=> 20
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> 16
[2,2,2,1]
=> [[2,2,2,1],[]]
=> 17
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> 16
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> 13
[1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> 8
Description
The total number of rook placements on a Ferrers board.
Matching statistic: St000718
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 40%
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 40%
Values
[1]
=> [1,0]
=> [[]]
=> ([(0,1)],2)
=> 2
[2]
=> [1,0,1,0]
=> [[],[]]
=> ([(0,2),(1,2)],3)
=> 3
[1,1]
=> [1,1,0,0]
=> [[[]]]
=> ([(0,2),(1,2)],3)
=> 3
[3]
=> [1,0,1,0,1,0]
=> [[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 4
[2,1]
=> [1,0,1,1,0,0]
=> [[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 5
[1,1,1]
=> [1,1,0,1,0,0]
=> [[[],[]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 4
[4]
=> [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {7,7,7}
[2,2]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {7,7,7}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {7,7,7}
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 6
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {9,9,10,10,10}
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {9,9,10,10,10}
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ? ∊ {9,9,10,10,10}
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {9,9,10,10,10}
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[],[[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {9,9,10,10,10}
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 6
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 7
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[],[[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? ∊ {11,11,13,13,13,13,13,13,15}
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? ∊ {11,11,13,13,13,13,13,13,15}
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[],[],[],[[],[]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ? ∊ {11,11,13,13,13,13,13,13,15}
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {11,11,13,13,13,13,13,13,15}
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ? ∊ {11,11,13,13,13,13,13,13,15}
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[],[],[[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> ? ∊ {11,11,13,13,13,13,13,13,15}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {11,11,13,13,13,13,13,13,15}
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[[]],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {11,11,13,13,13,13,13,13,15}
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [[],[[],[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? ∊ {11,11,13,13,13,13,13,13,15}
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 7
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[],[]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[],[],[[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[],[],[],[[[]]]]
=> ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [[],[],[],[],[[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[],[[[],[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [[],[],[[[]],[]]]
=> ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [[],[],[],[[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[[[],[]],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[],[[[]],[],[]]]
=> ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [[],[],[[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[[[]],[],[],[]]]
=> ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [[],[[],[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[],[],[]]]
=> ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
Description
The largest Laplacian eigenvalue of a graph if it is integral.
This statistic is undefined if the largest Laplacian eigenvalue of the graph is not integral.
Various results are collected in Section 3.9 of [1]
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: 47%
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: 47%
Values
[1]
=> [[1]]
=> [1] => ([],1)
=> 1 = 2 - 1
[2]
=> [[1,2]]
=> [2] => ([],2)
=> ? = 3 - 1
[1,1]
=> [[1],[2]]
=> [1,1] => ([(0,1)],2)
=> 2 = 3 - 1
[3]
=> [[1,2,3]]
=> [3] => ([],3)
=> ? ∊ {4,5} - 1
[2,1]
=> [[1,3],[2]]
=> [1,2] => ([(1,2)],3)
=> ? ∊ {4,5} - 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[4]
=> [[1,2,3,4]]
=> [4] => ([],4)
=> ? ∊ {5,7,7,7} - 1
[3,1]
=> [[1,3,4],[2]]
=> [1,3] => ([(2,3)],4)
=> ? ∊ {5,7,7,7} - 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {5,7,7,7} - 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {5,7,7,7} - 1
[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
[5]
=> [[1,2,3,4,5]]
=> [5] => ([],5)
=> ? ∊ {6,9,9,10,10,10} - 1
[4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => ([(3,4)],5)
=> ? ∊ {6,9,9,10,10,10} - 1
[3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {6,9,9,10,10,10} - 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {6,9,9,10,10,10} - 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {6,9,9,10,10,10} - 1
[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)
=> ? ∊ {6,9,9,10,10,10} - 1
[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
[6]
=> [[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? ∊ {7,11,11,13,13,13,13,13,13,15} - 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => ([(4,5)],6)
=> ? ∊ {7,11,11,13,13,13,13,13,13,15} - 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {7,11,11,13,13,13,13,13,13,15} - 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {7,11,11,13,13,13,13,13,13,15} - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {7,11,11,13,13,13,13,13,13,15} - 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {7,11,11,13,13,13,13,13,13,15} - 1
[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)
=> ? ∊ {7,11,11,13,13,13,13,13,13,15} - 1
[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)
=> ? ∊ {7,11,11,13,13,13,13,13,13,15} - 1
[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)
=> ? ∊ {7,11,11,13,13,13,13,13,13,15} - 1
[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)
=> ? ∊ {7,11,11,13,13,13,13,13,13,15} - 1
[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
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => ([],7)
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [1,6] => ([(5,6)],7)
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [2,5] => ([(4,6),(5,6)],7)
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [3,4] => ([(3,6),(4,6),(5,6)],7)
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 1
[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)
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 1
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 1
[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)
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 1
[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)
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 1
[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)
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 1
[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)
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 1
[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)
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 1
[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)
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 1
[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 = 8 - 1
Description
The pebbling number of a connected graph.
Matching statistic: St001232
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 40%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 40%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0 = 2 - 2
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 3 - 2
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1 = 3 - 2
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {4,4,5} - 2
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {4,4,5} - 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {4,4,5} - 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3 = 5 - 2
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {5,7,7,7} - 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ? ∊ {5,7,7,7} - 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {5,7,7,7} - 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {5,7,7,7} - 2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4 = 6 - 2
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {6,9,9,10,10,10} - 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {6,9,9,10,10,10} - 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {6,9,9,10,10,10} - 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {6,9,9,10,10,10} - 2
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {6,9,9,10,10,10} - 2
[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]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {6,9,9,10,10,10} - 2
[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]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5 = 7 - 2
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {7,11,11,13,13,13,13,13,13,15} - 2
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {7,11,11,13,13,13,13,13,13,15} - 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> ? ∊ {7,11,11,13,13,13,13,13,13,15} - 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {7,11,11,13,13,13,13,13,13,15} - 2
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {7,11,11,13,13,13,13,13,13,15} - 2
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {7,11,11,13,13,13,13,13,13,15} - 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {7,11,11,13,13,13,13,13,13,15} - 2
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {7,11,11,13,13,13,13,13,13,15} - 2
[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]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {7,11,11,13,13,13,13,13,13,15} - 2
[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]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {7,11,11,13,13,13,13,13,13,15} - 2
[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]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6 = 8 - 2
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[1,1,1,1,1,1,1]
=> [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]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? ∊ {8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001582
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 27%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 27%
Values
[1]
=> [1,0]
=> [1,1,0,0]
=> [2,1] => 0 = 2 - 2
[2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1 = 3 - 2
[1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [3,1,2] => 1 = 3 - 2
[3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 3 = 5 - 2
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 2 = 4 - 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 2 = 4 - 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ? ∊ {5,7,7,7} - 2
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => ? ∊ {5,7,7,7} - 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 3 = 5 - 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => ? ∊ {5,7,7,7} - 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => ? ∊ {5,7,7,7} - 2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => ? ∊ {6,6,9,9,10,10,10} - 2
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5] => ? ∊ {6,6,9,9,10,10,10} - 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => ? ∊ {6,6,9,9,10,10,10} - 2
[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]
=> [2,3,5,6,1,4] => ? ∊ {6,6,9,9,10,10,10} - 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => ? ∊ {6,6,9,9,10,10,10} - 2
[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]
=> [2,4,5,6,1,3] => ? ∊ {6,6,9,9,10,10,10} - 2
[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]
=> [3,4,5,6,1,2] => ? ∊ {6,6,9,9,10,10,10} - 2
[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]
=> [2,3,4,5,6,7,1] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15} - 2
[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]
=> [2,3,4,5,7,1,6] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15} - 2
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,1,4,5] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15} - 2
[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]
=> [2,3,4,6,7,1,5] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15} - 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15} - 2
[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]
=> [2,5,1,6,3,4] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15} - 2
[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]
=> [2,3,5,6,7,1,4] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15} - 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15} - 2
[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]
=> [4,1,5,6,2,3] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15} - 2
[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]
=> [2,4,5,6,7,1,3] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15} - 2
[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]
=> [3,4,5,6,7,1,2] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15} - 2
[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]
=> [2,3,4,5,6,7,8,1] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[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]
=> [2,3,4,5,6,8,1,7] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[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]
=> [2,3,4,7,1,5,6] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[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]
=> [2,3,4,5,7,8,1,6] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,5,6,1,3,4] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[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]
=> [2,3,6,1,7,4,5] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[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]
=> [2,3,4,6,7,8,1,5] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[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]
=> [4,5,1,6,2,3] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[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]
=> [2,6,1,3,4,5] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[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]
=> [2,5,1,6,7,3,4] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[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]
=> [2,3,5,6,7,8,1,4] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[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]
=> [5,1,2,6,3,4] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[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]
=> [4,1,5,6,7,2,3] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[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]
=> [2,4,5,6,7,8,1,3] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
[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]
=> [3,4,5,6,7,8,1,2] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20} - 2
Description
The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
Matching statistic: St000080
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000080: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 20%
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000080: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 20%
Values
[1]
=> 10 => 11 => ([(0,2),(2,1)],3)
=> 2
[2]
=> 100 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3
[1,1]
=> 110 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3
[3]
=> 1000 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? ∊ {4,5}
[2,1]
=> 1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,1,1]
=> 1110 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? ∊ {4,5}
[4]
=> 10000 => 11010 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? ∊ {5,5,7,7,7}
[3,1]
=> 10010 => 11000 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? ∊ {5,5,7,7,7}
[2,2]
=> 1100 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? ∊ {5,5,7,7,7}
[2,1,1]
=> 10110 => 11100 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? ∊ {5,5,7,7,7}
[1,1,1,1]
=> 11110 => 10100 => ([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> ? ∊ {5,5,7,7,7}
[5]
=> 100000 => 110101 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ? ∊ {6,6,9,9,10,10,10}
[4,1]
=> 100010 => 110111 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? ∊ {6,6,9,9,10,10,10}
[3,2]
=> 10100 => 11110 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? ∊ {6,6,9,9,10,10,10}
[3,1,1]
=> 100110 => 110011 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> ? ∊ {6,6,9,9,10,10,10}
[2,2,1]
=> 11010 => 10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? ∊ {6,6,9,9,10,10,10}
[2,1,1,1]
=> 101110 => 111011 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? ∊ {6,6,9,9,10,10,10}
[1,1,1,1,1]
=> 111110 => 101011 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ? ∊ {6,6,9,9,10,10,10}
[6]
=> 1000000 => 1101010 => ([(0,2),(0,3),(1,10),(2,14),(2,17),(3,1),(3,14),(3,17),(5,8),(6,5),(7,9),(8,4),(9,4),(10,6),(11,13),(11,16),(12,8),(12,9),(13,7),(13,12),(14,11),(14,15),(15,6),(15,13),(15,16),(16,5),(16,7),(16,12),(17,10),(17,11),(17,15)],18)
=> ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[5,1]
=> 1000010 => 1101000 => ([(0,4),(0,5),(1,13),(2,3),(2,20),(3,7),(4,1),(4,19),(4,21),(5,2),(5,19),(5,21),(7,9),(8,11),(9,12),(10,8),(11,6),(12,6),(13,10),(14,10),(14,17),(15,16),(15,17),(16,9),(16,18),(17,8),(17,18),(18,11),(18,12),(19,14),(19,15),(20,7),(20,16),(21,13),(21,14),(21,15),(21,20)],22)
=> ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[4,2]
=> 100100 => 110001 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[4,1,1]
=> 1000110 => 1101100 => ([(0,3),(0,4),(1,12),(2,14),(2,19),(3,2),(3,18),(3,20),(4,1),(4,18),(4,20),(6,8),(7,10),(8,11),(9,7),(10,5),(11,5),(12,6),(13,9),(13,16),(14,15),(14,16),(15,8),(15,17),(16,7),(16,17),(17,10),(17,11),(18,13),(18,14),(19,6),(19,9),(19,15),(20,12),(20,13),(20,19)],21)
=> ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[3,3]
=> 11000 => 10010 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[3,2,1]
=> 101010 => 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[3,1,1,1]
=> 1001110 => 1100100 => ([(0,3),(0,4),(1,12),(2,14),(2,19),(3,2),(3,18),(3,20),(4,1),(4,18),(4,20),(6,8),(7,10),(8,11),(9,7),(10,5),(11,5),(12,6),(13,9),(13,16),(14,15),(14,16),(15,8),(15,17),(16,7),(16,17),(17,10),(17,11),(18,13),(18,14),(19,6),(19,9),(19,15),(20,12),(20,13),(20,19)],21)
=> ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[2,2,2]
=> 11100 => 10110 => ([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[2,2,1,1]
=> 110110 => 100011 => ([(0,4),(0,5),(1,12),(2,3),(2,13),(2,16),(3,8),(3,14),(4,1),(4,9),(4,15),(5,2),(5,9),(5,15),(7,10),(8,11),(9,13),(10,6),(11,6),(12,7),(13,8),(14,10),(14,11),(15,12),(15,16),(16,7),(16,14)],17)
=> ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[2,1,1,1,1]
=> 1011110 => 1110100 => ([(0,4),(0,5),(1,13),(2,3),(2,20),(3,7),(4,1),(4,19),(4,21),(5,2),(5,19),(5,21),(7,9),(8,11),(9,12),(10,8),(11,6),(12,6),(13,10),(14,10),(14,17),(15,16),(15,17),(16,9),(16,18),(17,8),(17,18),(18,11),(18,12),(19,14),(19,15),(20,7),(20,16),(21,13),(21,14),(21,15),(21,20)],22)
=> ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[1,1,1,1,1,1]
=> 1111110 => 1010100 => ([(0,2),(0,3),(1,10),(2,14),(2,17),(3,1),(3,14),(3,17),(5,8),(6,5),(7,9),(8,4),(9,4),(10,6),(11,13),(11,16),(12,8),(12,9),(13,7),(13,12),(14,11),(14,15),(15,6),(15,13),(15,16),(16,5),(16,7),(16,12),(17,10),(17,11),(17,15)],18)
=> ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[7]
=> 10000000 => 11010101 => ([(0,2),(0,3),(1,11),(2,16),(2,20),(3,1),(3,16),(3,20),(5,9),(6,5),(7,6),(8,10),(9,4),(10,4),(11,7),(12,14),(12,18),(13,9),(13,10),(14,15),(14,19),(15,8),(15,13),(16,12),(16,17),(17,7),(17,14),(17,18),(18,6),(18,15),(18,19),(19,5),(19,8),(19,13),(20,11),(20,12),(20,17)],21)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[6,1]
=> 10000010 => 11010111 => ([(0,3),(0,4),(1,14),(2,1),(2,7),(2,15),(3,23),(3,24),(4,2),(4,23),(4,24),(6,8),(7,6),(8,9),(9,12),(10,13),(11,10),(12,5),(13,5),(14,11),(15,14),(15,19),(16,17),(16,18),(17,8),(17,20),(18,20),(18,21),(19,11),(19,21),(20,9),(20,22),(21,10),(21,22),(22,12),(22,13),(23,7),(23,16),(23,25),(24,15),(24,16),(24,25),(25,6),(25,17),(25,18),(25,19)],26)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[5,2]
=> 1000100 => 1101110 => ([(0,3),(0,4),(1,2),(1,18),(1,19),(2,7),(2,14),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,17),(15,19),(16,17),(16,18),(17,6),(17,13),(18,13),(18,14),(19,6),(19,7)],20)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[5,1,1]
=> 10000110 => 11010011 => ([(0,3),(0,4),(1,24),(1,25),(2,14),(2,15),(3,1),(3,26),(3,27),(4,2),(4,26),(4,27),(6,8),(7,9),(8,10),(9,12),(10,13),(11,7),(12,5),(13,5),(14,11),(15,6),(16,8),(16,22),(17,11),(17,23),(18,19),(18,23),(19,21),(19,22),(20,12),(20,13),(21,9),(21,20),(22,10),(22,20),(23,7),(23,21),(24,16),(24,19),(25,6),(25,16),(26,14),(26,17),(26,18),(26,24),(27,15),(27,17),(27,18),(27,25)],28)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[4,3]
=> 101000 => 111101 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[4,2,1]
=> 1001010 => 1100000 => ([(0,6),(0,7),(1,11),(2,5),(2,15),(3,13),(4,3),(4,17),(5,4),(5,16),(6,2),(6,14),(7,1),(7,14),(9,12),(10,9),(11,10),(12,8),(13,8),(14,11),(14,15),(15,10),(15,16),(16,9),(16,17),(17,12),(17,13)],18)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[4,1,1,1]
=> 10001110 => 11011011 => ([(0,2),(0,3),(1,14),(1,15),(2,19),(2,20),(3,1),(3,19),(3,20),(5,8),(6,7),(7,4),(8,4),(9,16),(9,18),(10,17),(10,18),(11,16),(11,17),(12,7),(12,8),(13,9),(13,10),(14,9),(14,11),(15,10),(15,11),(16,6),(16,12),(17,5),(17,12),(18,5),(18,6),(19,13),(19,14),(20,13),(20,15)],21)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[3,3,1]
=> 110010 => 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[3,2,2]
=> 101100 => 111001 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[3,2,1,1]
=> 1010110 => 1111100 => ([(0,6),(0,7),(1,11),(2,5),(2,15),(3,13),(4,3),(4,17),(5,4),(5,16),(6,2),(6,14),(7,1),(7,14),(9,12),(10,9),(11,10),(12,8),(13,8),(14,11),(14,15),(15,10),(15,16),(16,9),(16,17),(17,12),(17,13)],18)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[3,1,1,1,1]
=> 10011110 => 11001011 => ([(0,3),(0,4),(1,24),(1,25),(2,14),(2,15),(3,1),(3,26),(3,27),(4,2),(4,26),(4,27),(6,8),(7,9),(8,10),(9,12),(10,13),(11,7),(12,5),(13,5),(14,11),(15,6),(16,8),(16,22),(17,11),(17,23),(18,19),(18,23),(19,21),(19,22),(20,12),(20,13),(21,9),(21,20),(22,10),(22,20),(23,7),(23,21),(24,16),(24,19),(25,6),(25,16),(26,14),(26,17),(26,18),(26,24),(27,15),(27,17),(27,18),(27,25)],28)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[2,2,2,1]
=> 111010 => 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[2,2,1,1,1]
=> 1101110 => 1000100 => ([(0,3),(0,4),(1,2),(1,18),(1,19),(2,7),(2,14),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,17),(15,19),(16,17),(16,18),(17,6),(17,13),(18,13),(18,14),(19,6),(19,7)],20)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[2,1,1,1,1,1]
=> 10111110 => 11101011 => ([(0,3),(0,4),(1,14),(2,1),(2,7),(2,15),(3,23),(3,24),(4,2),(4,23),(4,24),(6,8),(7,6),(8,9),(9,12),(10,13),(11,10),(12,5),(13,5),(14,11),(15,14),(15,19),(16,17),(16,18),(17,8),(17,20),(18,20),(18,21),(19,11),(19,21),(20,9),(20,22),(21,10),(21,22),(22,12),(22,13),(23,7),(23,16),(23,25),(24,15),(24,16),(24,25),(25,6),(25,17),(25,18),(25,19)],26)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[1,1,1,1,1,1,1]
=> 11111110 => 10101011 => ([(0,2),(0,3),(1,11),(2,16),(2,20),(3,1),(3,16),(3,20),(5,9),(6,5),(7,6),(8,10),(9,4),(10,4),(11,7),(12,14),(12,18),(13,9),(13,10),(14,15),(14,19),(15,8),(15,13),(16,12),(16,17),(17,7),(17,14),(17,18),(18,6),(18,15),(18,19),(19,5),(19,8),(19,13),(20,11),(20,12),(20,17)],21)
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
Description
The rank of the poset.
Matching statistic: St000570
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000570: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 20%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000570: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 20%
Values
[1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 2
[2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 3
[1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 3
[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] => ? ∊ {4,5}
[2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 4
[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] => ? ∊ {4,5}
[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] => ? ∊ {5,5,7,7,7}
[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] => ? ∊ {5,5,7,7,7}
[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] => ? ∊ {5,5,7,7,7}
[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] => ? ∊ {5,5,7,7,7}
[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] => ? ∊ {5,5,7,7,7}
[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] => ? ∊ {6,6,9,9,10,10,10}
[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] => ? ∊ {6,6,9,9,10,10,10}
[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] => ? ∊ {6,6,9,9,10,10,10}
[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] => ? ∊ {6,6,9,9,10,10,10}
[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] => ? ∊ {6,6,9,9,10,10,10}
[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] => ? ∊ {6,6,9,9,10,10,10}
[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] => ? ∊ {6,6,9,9,10,10,10}
[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] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
Description
The Edelman-Greene number of a permutation.
This is the sum of the coefficients of the expansion of the Stanley symmetric function $F_\omega$ in Schur functions. Equivalently, this is the number of semistandard tableaux whose column words - obtained by reading up columns starting with the leftmost - are reduced words for $\omega$.
Matching statistic: St000572
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000572: Set partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 20%
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000572: Set partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 20%
Values
[1]
=> [1,0,1,0]
=> [[1,3],[2,4]]
=> {{1,3},{2,4}}
=> 2
[2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> {{1,2,5},{3,4,6}}
=> 3
[1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> {{1,3,4},{2,5,6}}
=> 3
[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}}
=> ? ∊ {4,5}
[2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> {{1,3,5},{2,4,6}}
=> 4
[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}}
=> ? ∊ {4,5}
[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}}
=> ? ∊ {5,5,7,7,7}
[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}}
=> ? ∊ {5,5,7,7,7}
[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}}
=> ? ∊ {5,5,7,7,7}
[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}}
=> ? ∊ {5,5,7,7,7}
[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}}
=> ? ∊ {5,5,7,7,7}
[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}}
=> ? ∊ {6,6,9,9,10,10,10}
[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}}
=> ? ∊ {6,6,9,9,10,10,10}
[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}}
=> ? ∊ {6,6,9,9,10,10,10}
[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}}
=> ? ∊ {6,6,9,9,10,10,10}
[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}}
=> ? ∊ {6,6,9,9,10,10,10}
[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}}
=> ? ∊ {6,6,9,9,10,10,10}
[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}}
=> ? ∊ {6,6,9,9,10,10,10}
[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}}
=> ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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}}
=> ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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}}
=> ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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}}
=> ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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}}
=> ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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}}
=> ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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}}
=> ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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}}
=> ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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}}
=> ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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}}
=> ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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}}
=> ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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}}
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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}}
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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}}
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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}}
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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}}
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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}}
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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}}
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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}}
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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}}
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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}}
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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}}
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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}}
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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}}
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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}}
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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}}
=> ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
Description
The dimension exponent of a set partition.
This is
$$\sum_{B\in\pi} (\max(B) - \min(B) + 1) - n$$
where the summation runs over the blocks of the set partition $\pi$ of $\{1,\dots,n\}$.
It is thus equal to the difference [[St000728]] - [[St000211]].
This is also the number of occurrences of the pattern {{1, 3}, {2}}, such that 1 and 3 are consecutive elements in a block.
This is also the number of occurrences of the pattern {{1, 3}, {2}}, such that 1 is the minimal and 3 is the maximal element of the block.
Matching statistic: St001298
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St001298: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 20%
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St001298: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 20%
Values
[1]
=> [1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => 2
[2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => 3
[1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => 3
[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] => ? ∊ {4,5}
[2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => 4
[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] => ? ∊ {4,5}
[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] => ? ∊ {5,5,7,7,7}
[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] => ? ∊ {5,5,7,7,7}
[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] => ? ∊ {5,5,7,7,7}
[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] => ? ∊ {5,5,7,7,7}
[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] => ? ∊ {5,5,7,7,7}
[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] => ? ∊ {6,6,9,9,10,10,10}
[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] => ? ∊ {6,6,9,9,10,10,10}
[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] => ? ∊ {6,6,9,9,10,10,10}
[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] => ? ∊ {6,6,9,9,10,10,10}
[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] => ? ∊ {6,6,9,9,10,10,10}
[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] => ? ∊ {6,6,9,9,10,10,10}
[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] => ? ∊ {6,6,9,9,10,10,10}
[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] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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] => ? ∊ {7,7,11,11,13,13,13,13,13,13,15}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
[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] => ? ∊ {8,8,13,13,16,16,16,16,17,17,17,20,20,20,20}
Description
The number of repeated entries in the Lehmer code of a permutation.
The Lehmer code of a permutation $\pi$ is the sequence $(v_1,\dots,v_n)$, with $v_i=|\{j > i: \pi(j) < \pi(i)\}$. This statistic counts the number of distinct elements in this sequence.
The following 35 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001313The number of Dyck paths above the lattice path given by a binary word. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001760The number of prefix or suffix reversals needed to sort a permutation. St000222The number of alignments in the permutation. St000516The number of stretching pairs of a permutation. St000519The largest length of a factor maximising the subword complexity. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000906The length of the shortest maximal chain in a poset. St000922The minimal number such that all substrings of this length are unique. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001375The pancake length of a permutation. St001535The number of cyclic alignments of a permutation. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001821The sorting index of a signed permutation. St001841The number of inversions of a set partition. St001911A descent variant minus the number of inversions. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001565The number of arithmetic progressions of length 2 in a permutation. St001782The order of rowmotion on the set of order ideals of a poset.
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