Your data matches 12 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000532
(load all 3 compositions to match this statistic)
Mapping
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
(load all 4 compositions to match this statistic)
Mapping
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%
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
Mapping
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%
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
[2,2]
 => [1,1,1,0,0,0]
 => [[[[]]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 7
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 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
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [[],[[[]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 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)
 => ? = 10
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 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
[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
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[],[],[[[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 13
[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)
 => ? = 13
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 13
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[],[[[]],[]]]
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ? = 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)
 => ? = 13
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [[[[[]]]]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ? = 13
[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)
 => ? = 13
[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
[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
[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)
 => ? = 13
[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)
 => ? = 16
[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)
 => ? = 16
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[],[[[],[]]]]
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ? = 17
[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)
 => ? = 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)
 => ? = 17
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[[[],[]],[]]]
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 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)
 => ? = 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)
 => ? = 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)
 => ? = 16
[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)
 => ? = 17
[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)
 => ? = 16
[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)
 => ? = 13
[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
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)
Mapping
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%
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 - 1
[2,1]
 => [[1,3],[2]]
 => [1,2] => ([(1,2)],3)
 => ? = 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 - 1
[3,1]
 => [[1,3,4],[2]]
 => [1,3] => ([(2,3)],4)
 => ? = 7 - 1
[2,2]
 => [[1,2],[3,4]]
 => [2,2] => ([(1,3),(2,3)],4)
 => ? = 7 - 1
[2,1,1]
 => [[1,4],[2],[3]]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 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 - 1
[4,1]
 => [[1,3,4,5],[2]]
 => [1,4] => ([(3,4)],5)
 => ? = 9 - 1
[3,2]
 => [[1,2,5],[3,4]]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 10 - 1
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 10 - 1
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 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)
 => ? = 9 - 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 - 1
[5,1]
 => [[1,3,4,5,6],[2]]
 => [1,5] => ([(4,5)],6)
 => ? = 11 - 1
[4,2]
 => [[1,2,5,6],[3,4]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 13 - 1
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ? = 13 - 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? = 13 - 1
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 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)
 => ? = 13 - 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)
 => ? = 13 - 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)
 => ? = 13 - 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)
 => ? = 11 - 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 - 1
[6,1]
 => [[1,3,4,5,6,7],[2]]
 => [1,6] => ([(5,6)],7)
 => ? = 13 - 1
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => [2,5] => ([(4,6),(5,6)],7)
 => ? = 16 - 1
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => ? = 16 - 1
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = 17 - 1
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 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)
 => ? = 17 - 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)
 => ? = 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)
 => ? = 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)
 => ? = 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)
 => ? = 16 - 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)
 => ? = 17 - 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)
 => ? = 16 - 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)
 => ? = 13 - 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 12 compositions to match this statistic)
Mapping
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%
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 - 2
[2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 5 - 2
[1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,0,1,0,1,0]
 => ? = 4 - 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]
 => ? = 7 - 2
[2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => ? = 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]
 => ? = 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 - 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]
 => ? = 9 - 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 9 - 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 - 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]
 => ? = 11 - 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]
 => ? = 13 - 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]
 => ? = 13 - 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]
 => ? = 13 - 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]
 => ? = 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]
 => ? = 13 - 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]
 => ? = 13 - 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]
 => ? = 13 - 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]
 => ? = 11 - 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 - 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]
 => ? = 13 - 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]
 => ? = 16 - 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]
 => ? = 16 - 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]
 => ? = 17 - 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]
 => ? = 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]
 => ? = 17 - 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]
 => ? = 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]
 => ? = 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]
 => ? = 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]
 => ? = 16 - 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]
 => ? = 17 - 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]
 => ? = 16 - 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]
 => ? = 13 - 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 - 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001880
(load all 10 compositions to match this statistic)
Mapping
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: 33%
Values
[1]
 => [1,0]
 => ([],1)
 => ? = 2 - 1
[2]
 => [1,0,1,0]
 => ([(0,1)],2)
 => ? = 3 - 1
[1,1]
 => [1,1,0,0]
 => ([],2)
 => ? = 3 - 1
[3]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 3 = 4 - 1
[2,1]
 => [1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => ? = 5 - 1
[1,1,1]
 => [1,1,0,1,0,0]
 => ([(1,2)],3)
 => ? = 4 - 1
[4]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 4 = 5 - 1
[3,1]
 => [1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 7 - 1
[2,2]
 => [1,1,1,0,0,0]
 => ([],3)
 => ? = 7 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 7 - 1
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ? = 5 - 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5 = 6 - 1
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 9 - 1
[3,2]
 => [1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => ? = 10 - 1
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 10 - 1
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => ? = 10 - 1
[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)
 => ? = 9 - 1
[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)
 => ? = 6 - 1
[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 = 7 - 1
[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)
 => ? = 11 - 1
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ? = 13 - 1
[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)
 => ? = 13 - 1
[3,3]
 => [1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => ? = 13 - 1
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 15 - 1
[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)
 => ? = 13 - 1
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => ([],4)
 => ? = 13 - 1
[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)
 => ? = 13 - 1
[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)
 => ? = 11 - 1
[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)
 => ? = 7 - 1
[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 = 8 - 1
[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)
 => ? = 13 - 1
[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)
 => ? = 16 - 1
[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)
 => ? = 16 - 1
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 17 - 1
[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)
 => ? = 20 - 1
[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)
 => ? = 17 - 1
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ? = 20 - 1
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 20 - 1
[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)
 => ? = 20 - 1
[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)
 => ? = 16 - 1
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => ([(1,2),(1,3),(1,4)],5)
 => ? = 17 - 1
[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)
 => ? = 16 - 1
[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)
 => ? = 13 - 1
[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)
 => ? = 8 - 1
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001879
(load all 10 compositions to match this statistic)
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%
Values
[1]
 => [1,0]
 => ([],1)
 => ? = 2 - 2
[2]
 => [1,0,1,0]
 => ([(0,1)],2)
 => ? = 3 - 2
[1,1]
 => [1,1,0,0]
 => ([],2)
 => ? = 3 - 2
[3]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 2 = 4 - 2
[2,1]
 => [1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => ? = 5 - 2
[1,1,1]
 => [1,1,0,1,0,0]
 => ([(1,2)],3)
 => ? = 4 - 2
[4]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 3 = 5 - 2
[3,1]
 => [1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => ? = 7 - 2
[2,2]
 => [1,1,1,0,0,0]
 => ([],3)
 => ? = 7 - 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => ? = 7 - 2
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ? = 5 - 2
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 4 = 6 - 2
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ? = 9 - 2
[3,2]
 => [1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => ? = 10 - 2
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ? = 10 - 2
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => ? = 10 - 2
[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)
 => ? = 9 - 2
[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)
 => ? = 6 - 2
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 5 = 7 - 2
[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)
 => ? = 11 - 2
[4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ? = 13 - 2
[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)
 => ? = 13 - 2
[3,3]
 => [1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => ? = 13 - 2
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ? = 15 - 2
[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)
 => ? = 13 - 2
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => ([],4)
 => ? = 13 - 2
[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)
 => ? = 13 - 2
[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)
 => ? = 11 - 2
[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)
 => ? = 7 - 2
[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)
 => 6 = 8 - 2
[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)
 => ? = 13 - 2
[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)
 => ? = 16 - 2
[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)
 => ? = 16 - 2
[4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ? = 17 - 2
[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)
 => ? = 20 - 2
[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)
 => ? = 17 - 2
[3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => ([(0,4),(1,2),(1,3),(1,4)],5)
 => ? = 20 - 2
[3,2,2]
 => [1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ? = 20 - 2
[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)
 => ? = 20 - 2
[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)
 => ? = 16 - 2
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => ([(1,2),(1,3),(1,4)],5)
 => ? = 17 - 2
[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)
 => ? = 16 - 2
[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)
 => ? = 13 - 2
[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)
 => ? = 8 - 2
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001313
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001313: Binary words ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 20%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 110100 => 2
[2]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 11100100 => 3
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 11011000 => 3
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1111000100 => ? = 4
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 11010100 => 5
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => ? = 4
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 111110000100 => ? = 5
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1110100100 => ? = 7
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1110011000 => ? = 7
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => ? = 7
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 110111100000 => ? = 5
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => 11111100000100 => ? = 6
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 111101000100 => ? = 9
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1110010100 => ? = 10
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => ? = 10
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => ? = 10
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 110111010000 => ? = 9
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => 11011111000000 => ? = 6
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
 => 1111111000000100 => ? = 7
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
 => 11111010000100 => ? = 11
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,1,0,0]
 => 111100100100 => ? = 13
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 111011000100 => ? = 13
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 111100011000 => ? = 13
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => ? = 15
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 110111001000 => ? = 13
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 111001110000 => ? = 13
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 110110110000 => ? = 13
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => 11011110100000 => ? = 11
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 1101111110000000 => ? = 7
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
 => 111111110000000100 => ? = 8
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
 => 1111110100000100 => ? = 13
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => 11111001000100 => ? = 16
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
 => 11110110000100 => ? = 16
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => 111100010100 => ? = 17
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 111010100100 => ? = 20
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 110111000100 => ? = 17
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => 111010011000 => ? = 20
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => 111001101000 => ? = 20
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => 110110101000 => ? = 20
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => 11011110010000 => ? = 16
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 110101110000 => ? = 17
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => 11011101100000 => ? = 16
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 1101111101000000 => ? = 13
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => 110111111100000000 => ? = 8
Description
The number of Dyck paths above the lattice path given by a binary word. One may treat a binary word as a lattice path starting at the origin and treating $1$'s as steps $(1,0)$ and $0$'s as steps $(0,1)$. Given a binary word $w$, this statistic counts the number of lattice paths from the origin to the same endpoint as $w$ that stay weakly above $w$. See [[St001312]] for this statistic on compositions treated as bounce paths.
Matching statistic: St001583
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 20%
Values
[1]
 => [1,0,1,0]
 => [3,1,2] => [1,3,2] => 2
[2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => [2,4,1,3] => 3
[1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => [3,1,4,2] => 3
[3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [2,5,1,3,4] => ? = 4
[2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => [1,2,4,3] => 5
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [3,1,4,5,2] => ? = 4
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => [2,6,1,3,4,5] => ? = 5
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,3,5,4,2] => ? = 7
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [2,1,4,3,5] => ? = 7
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,5,2,4,3] => ? = 7
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [3,1,4,5,6,2] => ? = 5
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => [2,7,1,3,4,5,6] => ? = 6
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => [3,4,6,1,5,2] => ? = 9
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [2,3,5,1,4] => ? = 10
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [1,3,2,5,4] => ? = 10
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [4,1,2,5,3] => ? = 10
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [4,6,1,2,5,3] => ? = 9
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => [3,1,4,5,6,7,2] => ? = 6
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,8,1,7] => [2,8,1,3,4,5,6,7] => ? = 7
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => [3,4,7,1,5,6,2] => ? = 11
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => [2,4,6,5,1,3] => ? = 13
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [1,4,3,6,5,2] => ? = 13
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => [2,1,5,3,4,6] => ? = 13
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,2,3,5,4] => ? = 15
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [1,6,3,2,5,4] => ? = 13
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [2,1,4,5,3,6] => ? = 13
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [5,1,6,2,4,3] => ? = 13
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => [4,7,1,2,5,6,3] => ? = 11
[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] => [3,1,4,5,6,7,8,2] => ? = 7
[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] => [2,9,1,3,4,5,6,7,8] => ? = 8
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,4,5,6,1,2,7] => [3,4,8,1,5,6,7,2] => ? = 13
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [2,7,4,5,1,3,6] => [2,4,7,1,5,3,6] => ? = 16
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [5,3,4,1,7,2,6] => [3,5,1,7,4,6,2] => ? = 16
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => [2,3,6,1,4,5] => ? = 17
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [1,2,4,6,5,3] => ? = 20
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [3,4,1,2,6,5] => ? = 17
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [5,1,3,6,4,2] => ? = 20
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [2,6,3,5,1,4] => ? = 20
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [1,2,6,3,5,4] => ? = 20
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,7,5,6,2,4] => [3,7,1,5,2,6,4] => ? = 16
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [4,1,2,5,6,3] => ? = 17
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => [4,1,6,2,5,7,3] => ? = 16
[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] => [4,8,1,2,5,6,7,3] => ? = 13
[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] => [3,1,4,5,6,7,8,9,2] => ? = 8
Description
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
Matching statistic: St001582
(load all 4 compositions to match this statistic)
Mapping
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 20%
Values
[1]
 => [1]
 => [1,0,1,0]
 => [3,1,2] => 1 = 2 - 1
[2]
 => [2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 2 = 3 - 1
[1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2 = 3 - 1
[3]
 => [2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 3 = 4 - 1
[2,1]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ? = 5 - 1
[1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ? = 4 - 1
[4]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ? = 5 - 1
[3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ? = 7 - 1
[2,2]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => ? = 7 - 1
[2,1,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ? = 7 - 1
[1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ? = 5 - 1
[5]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => ? = 6 - 1
[4,1]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ? = 9 - 1
[3,2]
 => [4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ? = 10 - 1
[3,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ? = 10 - 1
[2,2,1]
 => [5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => ? = 10 - 1
[2,1,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => ? = 9 - 1
[1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ? = 6 - 1
[6]
 => [2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ? = 7 - 1
[5,1]
 => [2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ? = 11 - 1
[4,2]
 => [4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ? = 13 - 1
[4,1,1]
 => [4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ? = 13 - 1
[3,3]
 => [3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ? = 13 - 1
[3,2,1]
 => [3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ? = 15 - 1
[3,1,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [7,1,4,5,6,2,3] => ? = 13 - 1
[2,2,2]
 => [6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [2,3,4,5,6,8,1,7] => ? = 13 - 1
[2,2,1,1]
 => [5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [7,3,4,5,1,2,6] => ? = 13 - 1
[2,1,1,1,1]
 => [3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ? = 11 - 1
[1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [3,1,4,5,6,7,8,2] => ? = 7 - 1
[7]
 => [2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ? = 8 - 1
[6,1]
 => [3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => ? = 13 - 1
[5,2]
 => [4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => ? = 16 - 1
[5,1,1]
 => [2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [6,1,4,5,2,7,3] => ? = 16 - 1
[4,3]
 => [4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ? = 17 - 1
[4,2,1]
 => [5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [2,7,4,5,1,3,6] => ? = 20 - 1
[4,1,1,1]
 => [4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => ? = 17 - 1
[3,3,1]
 => [3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => ? = 20 - 1
[3,2,2]
 => [6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [8,3,4,5,6,1,2,7] => ? = 20 - 1
[3,2,1,1]
 => [3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => ? = 20 - 1
[3,1,1,1,1]
 => [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] => ? = 16 - 1
[2,2,2,1]
 => [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] => ? = 17 - 1
[2,2,1,1,1]
 => [5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [5,3,4,1,7,2,6] => ? = 16 - 1
[2,1,1,1,1,1]
 => [3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [3,1,7,5,6,2,4] => ? = 13 - 1
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [3,1,4,5,6,7,8,9,2] => ? = 8 - 1
Description
The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
The following 2 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001821The sorting index of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.