Your data matches 31 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000058
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000058: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [[1]]
 => {{1}}
 => [1] => 1
[2]
 => [[1,2]]
 => {{1,2}}
 => [2,1] => 2
[1,1]
 => [[1],[2]]
 => {{1},{2}}
 => [1,2] => 1
[3]
 => [[1,2,3]]
 => {{1,2,3}}
 => [2,3,1] => 3
[2,1]
 => [[1,3],[2]]
 => {{1,3},{2}}
 => [3,2,1] => 2
[1,1,1]
 => [[1],[2],[3]]
 => {{1},{2},{3}}
 => [1,2,3] => 1
[4]
 => [[1,2,3,4]]
 => {{1,2,3,4}}
 => [2,3,4,1] => 4
[3,1]
 => [[1,3,4],[2]]
 => {{1,3,4},{2}}
 => [3,2,4,1] => 3
[2,2]
 => [[1,2],[3,4]]
 => {{1,2},{3,4}}
 => [2,1,4,3] => 2
[2,1,1]
 => [[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => [4,2,3,1] => 2
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => [1,2,3,4] => 1
[5]
 => [[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => [2,3,4,5,1] => 5
[4,1]
 => [[1,3,4,5],[2]]
 => {{1,3,4,5},{2}}
 => [3,2,4,5,1] => 4
[3,2]
 => [[1,2,5],[3,4]]
 => {{1,2,5},{3,4}}
 => [2,5,4,3,1] => 6
[3,1,1]
 => [[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => [4,2,3,5,1] => 3
[2,2,1]
 => [[1,3],[2,5],[4]]
 => {{1,3},{2,5},{4}}
 => [3,5,1,4,2] => 2
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => {{1,5},{2},{3},{4}}
 => [5,2,3,4,1] => 2
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => {{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => 1
[6]
 => [[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => [2,3,4,5,6,1] => 6
[5,1]
 => [[1,3,4,5,6],[2]]
 => {{1,3,4,5,6},{2}}
 => [3,2,4,5,6,1] => 5
[4,2]
 => [[1,2,5,6],[3,4]]
 => {{1,2,5,6},{3,4}}
 => [2,5,4,3,6,1] => 4
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => {{1,4,5,6},{2},{3}}
 => [4,2,3,5,6,1] => 4
[3,3]
 => [[1,2,3],[4,5,6]]
 => {{1,2,3},{4,5,6}}
 => [2,3,1,5,6,4] => 3
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => {{1,3,6},{2,5},{4}}
 => [3,5,6,4,2,1] => 6
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => {{1,5,6},{2},{3},{4}}
 => [5,2,3,4,6,1] => 3
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => {{1,2},{3,4},{5,6}}
 => [2,1,4,3,6,5] => 2
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => {{1,4},{2,6},{3},{5}}
 => [4,6,3,1,5,2] => 2
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => {{1,6},{2},{3},{4},{5}}
 => [6,2,3,4,5,1] => 2
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => {{1},{2},{3},{4},{5},{6}}
 => [1,2,3,4,5,6] => 1
[7]
 => [[1,2,3,4,5,6,7]]
 => {{1,2,3,4,5,6,7}}
 => [2,3,4,5,6,7,1] => 7
[6,1]
 => [[1,3,4,5,6,7],[2]]
 => {{1,3,4,5,6,7},{2}}
 => [3,2,4,5,6,7,1] => 6
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => {{1,2,5,6,7},{3,4}}
 => [2,5,4,3,6,7,1] => 10
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => {{1,4,5,6,7},{2},{3}}
 => [4,2,3,5,6,7,1] => 5
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => {{1,2,3,7},{4,5,6}}
 => [2,3,7,5,6,4,1] => 12
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => {{1,3,6,7},{2,5},{4}}
 => [3,5,6,4,2,7,1] => 4
[4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => {{1,5,6,7},{2},{3},{4}}
 => [5,2,3,4,6,7,1] => 4
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => {{1,2,7},{3,4},{5,6}}
 => [2,7,4,3,6,5,1] => 6
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => {{1,4,7},{2,6},{3},{5}}
 => [4,6,3,7,5,2,1] => 6
[3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => {{1,6,7},{2},{3},{4},{5}}
 => [6,2,3,4,5,7,1] => 3
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => {{1,3},{2,5},{4,7},{6}}
 => [3,5,1,7,2,6,4] => 2
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => {{1,5},{2,7},{3},{4},{6}}
 => [5,7,3,4,1,6,2] => 2
[2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => {{1,7},{2},{3},{4},{5},{6}}
 => [7,2,3,4,5,6,1] => 2
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => {{1},{2},{3},{4},{5},{6},{7}}
 => [1,2,3,4,5,6,7] => 1
Description
The order of a permutation. $\operatorname{ord}(\pi)$ is given by the minimial $k$ for which $\pi^k$ is the identity permutation.
Matching statistic: St000668
(load all 3 compositions to match this statistic)
Mapping
St000668: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1]
 => ? = 1
[2]
 => 2
[1,1]
 => 1
[3]
 => 3
[2,1]
 => 2
[1,1,1]
 => 1
[4]
 => 4
[3,1]
 => 3
[2,2]
 => 2
[2,1,1]
 => 2
[1,1,1,1]
 => 1
[5]
 => 5
[4,1]
 => 4
[3,2]
 => 6
[3,1,1]
 => 3
[2,2,1]
 => 2
[2,1,1,1]
 => 2
[1,1,1,1,1]
 => 1
[6]
 => 6
[5,1]
 => 5
[4,2]
 => 4
[4,1,1]
 => 4
[3,3]
 => 3
[3,2,1]
 => 6
[3,1,1,1]
 => 3
[2,2,2]
 => 2
[2,2,1,1]
 => 2
[2,1,1,1,1]
 => 2
[1,1,1,1,1,1]
 => 1
[7]
 => 7
[6,1]
 => 6
[5,2]
 => 10
[5,1,1]
 => 5
[4,3]
 => 12
[4,2,1]
 => 4
[4,1,1,1]
 => 4
[3,2,2]
 => 6
[3,2,1,1]
 => 6
[3,1,1,1,1]
 => 3
[2,2,2,1]
 => 2
[2,2,1,1,1]
 => 2
[2,1,1,1,1,1]
 => 2
[1,1,1,1,1,1,1]
 => 1
Description
The least common multiple of the parts of the partition.
Matching statistic: St001232
(load all 8 compositions to match this statistic)
Mapping
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 78%
Values
[1]
 => [1]
 => [1,0]
 => [1,0]
 => 0 = 1 - 1
[2]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
[1,1]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0 = 1 - 1
[3]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
[2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0 = 1 - 1
[4]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 4 - 1
[3,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
[2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ? = 2 - 1
[2,1,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[5]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 4 = 5 - 1
[4,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 3 = 4 - 1
[3,2]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => ? = 6 - 1
[3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 2 = 3 - 1
[2,2,1]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ? = 2 - 1
[2,1,1,1]
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,1,1]
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[6]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 5 = 6 - 1
[5,1]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 4 = 5 - 1
[4,2]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ? = 4 - 1
[4,1,1]
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => 3 = 4 - 1
[3,3]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ? = 3 - 1
[3,2,1]
 => [3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ? = 6 - 1
[3,1,1,1]
 => [4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 2 = 3 - 1
[2,2,2]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => ? = 2 - 1
[2,2,1,1]
 => [4,2]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ? = 2 - 1
[2,1,1,1,1]
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,1,1,1]
 => [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]
 => 0 = 1 - 1
[7]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 6 = 7 - 1
[6,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => 5 = 6 - 1
[5,2]
 => [2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => ? = 10 - 1
[5,1,1]
 => [3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
 => 4 = 5 - 1
[4,3]
 => [2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ? = 12 - 1
[4,2,1]
 => [3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => ? = 4 - 1
[4,1,1,1]
 => [4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
 => 3 = 4 - 1
[3,2,2]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ? = 6 - 1
[3,2,1,1]
 => [4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => ? = 6 - 1
[3,1,1,1,1]
 => [5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
 => 2 = 3 - 1
[2,2,2,1]
 => [4,3]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ? = 2 - 1
[2,2,1,1,1]
 => [5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2 - 1
[2,1,1,1,1,1]
 => [6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 1 = 2 - 1
[1,1,1,1,1,1,1]
 => [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]
 => 0 = 1 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000260
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 22% ●values known / values provided: 28%●distinct values known / distinct values provided: 22%
Values
[1]
 => [[1]]
 => [1] => ([],1)
 => 0 = 1 - 1
[2]
 => [[1,2]]
 => [2] => ([],2)
 => ? = 2 - 1
[1,1]
 => [[1],[2]]
 => [2] => ([],2)
 => ? = 1 - 1
[3]
 => [[1,2,3]]
 => [3] => ([],3)
 => ? = 3 - 1
[2,1]
 => [[1,3],[2]]
 => [2,1] => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
[1,1,1]
 => [[1],[2],[3]]
 => [3] => ([],3)
 => ? = 1 - 1
[4]
 => [[1,2,3,4]]
 => [4] => ([],4)
 => ? = 4 - 1
[3,1]
 => [[1,3,4],[2]]
 => [2,2] => ([(1,3),(2,3)],4)
 => ? = 3 - 1
[2,2]
 => [[1,2],[3,4]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 2 - 1
[2,1,1]
 => [[1,4],[2],[3]]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 2 - 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4] => ([],4)
 => ? = 1 - 1
[5]
 => [[1,2,3,4,5]]
 => [5] => ([],5)
 => ? = 5 - 1
[4,1]
 => [[1,3,4,5],[2]]
 => [2,3] => ([(2,4),(3,4)],5)
 => ? = 4 - 1
[3,2]
 => [[1,2,5],[3,4]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 6 - 1
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 3 - 1
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1 = 2 - 1
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5] => ([],5)
 => ? = 1 - 1
[6]
 => [[1,2,3,4,5,6]]
 => [6] => ([],6)
 => ? = 6 - 1
[5,1]
 => [[1,3,4,5,6],[2]]
 => [2,4] => ([(3,5),(4,5)],6)
 => ? = 5 - 1
[4,2]
 => [[1,2,5,6],[3,4]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? = 4 - 1
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ? = 4 - 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 3 - 1
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 6 - 1
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ? = 3 - 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1 = 2 - 1
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 2 - 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6] => ([],6)
 => ? = 1 - 1
[7]
 => [[1,2,3,4,5,6,7]]
 => [7] => ([],7)
 => ? = 7 - 1
[6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,5] => ([(4,6),(5,6)],7)
 => ? = 6 - 1
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = 10 - 1
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ? = 5 - 1
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 12 - 1
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 - 1
[4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 4 - 1
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6 - 1
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 6 - 1
[3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ? = 3 - 1
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 2 - 1
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1 = 2 - 1
[2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1 = 2 - 1
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7] => ([],7)
 => ? = 1 - 1
Description
The radius of a connected graph. This is the minimum eccentricity of any vertex.
Matching statistic: St000942
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St000942: Parking functions ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 44%
Values
[1]
 => [[1]]
 => [1] => [1] => 1
[2]
 => [[1,2]]
 => [1,2] => [1,2] => 2
[1,1]
 => [[1],[2]]
 => [2,1] => [2,1] => 1
[3]
 => [[1,2,3]]
 => [1,2,3] => [1,2,3] => 3
[2,1]
 => [[1,3],[2]]
 => [2,1,3] => [2,1,3] => 2
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => [3,2,1] => 1
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => 4
[3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3,4] => 3
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => [3,4,1,2] => 2
[2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => 2
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => 1
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => ? = 5
[4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => [2,1,3,4,5] => ? = 4
[3,2]
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => [3,4,1,2,5] => ? = 6
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [3,2,1,4,5] => ? = 3
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [4,2,5,1,3] => ? = 2
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [4,3,2,1,5] => ? = 2
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 6
[5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [2,1,3,4,5,6] => ? = 5
[4,2]
 => [[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => [3,4,1,2,5,6] => ? = 4
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => [3,2,1,4,5,6] => ? = 4
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? = 3
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [4,2,5,1,3,6] => [4,2,5,1,3,6] => ? = 6
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => [4,3,2,1,5,6] => ? = 3
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => [5,6,3,4,1,2] => ? = 2
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [5,3,2,6,1,4] => [5,3,2,6,1,4] => ? = 2
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [5,4,3,2,1,6] => [5,4,3,2,1,6] => ? = 2
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ? = 1
[7]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
[6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 6
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => [3,4,1,2,5,6,7] => ? = 10
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => ? = 5
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => [4,5,6,1,2,3,7] => ? = 12
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [4,2,5,1,3,6,7] => [4,2,5,1,3,6,7] => ? = 4
[4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => ? = 4
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [5,6,3,4,1,2,7] => [5,6,3,4,1,2,7] => ? = 6
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => [5,3,2,6,1,4,7] => [5,3,2,6,1,4,7] => ? = 6
[3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => ? = 3
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3] => [6,4,7,2,5,1,3] => ? = 2
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => [6,4,3,2,7,1,5] => [6,4,3,2,7,1,5] => ? = 2
[2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ? = 2
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
Description
The number of critical left to right maxima of the parking functions. An entry $p$ in a parking function is critical, if there are exactly $p-1$ entries smaller than $p$ and $n-p$ entries larger than $p$. It is a left to right maximum, if there are no larger entries before it. This statistic allows the computation of the Tutte polynomial of the complete graph $K_{n+1}$, via $$ \sum_{P} x^{st(P)}y^{\binom{n+1}{2}-\sum P}, $$ where the sum is over all parking functions of length $n$, see [1, thm.13.5.16].
Matching statistic: St001937
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001937: Parking functions ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 44%
Values
[1]
 => [[1]]
 => [1] => [1] => 1
[2]
 => [[1,2]]
 => [1,2] => [1,2] => 2
[1,1]
 => [[1],[2]]
 => [2,1] => [2,1] => 1
[3]
 => [[1,2,3]]
 => [1,2,3] => [1,2,3] => 3
[2,1]
 => [[1,2],[3]]
 => [3,1,2] => [3,1,2] => 2
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => [3,2,1] => 1
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => [1,2,3,4] => 4
[3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => [4,1,2,3] => 3
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => [3,4,1,2] => 2
[2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => [4,3,1,2] => 2
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => 1
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,2,3,4,5] => ? = 5
[4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => [5,1,2,3,4] => ? = 4
[3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => [4,5,1,2,3] => ? = 6
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [5,4,1,2,3] => ? = 3
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [5,3,4,1,2] => ? = 2
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [5,4,3,1,2] => ? = 2
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5,4,3,2,1] => ? = 1
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ? = 6
[5,1]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? = 5
[4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => [5,6,1,2,3,4] => ? = 4
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => [6,5,1,2,3,4] => ? = 4
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => [4,5,6,1,2,3] => ? = 3
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => [6,4,5,1,2,3] => ? = 6
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => [6,5,4,1,2,3] => ? = 3
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => [5,6,3,4,1,2] => ? = 2
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => [6,5,3,4,1,2] => ? = 2
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => [6,5,4,3,1,2] => ? = 2
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ? = 1
[7]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 7
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 6
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => [6,7,1,2,3,4,5] => ? = 10
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [7,6,1,2,3,4,5] => [7,6,1,2,3,4,5] => ? = 5
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => [5,6,7,1,2,3,4] => ? = 12
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [7,5,6,1,2,3,4] => [7,5,6,1,2,3,4] => ? = 4
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [7,6,5,1,2,3,4] => [7,6,5,1,2,3,4] => ? = 4
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => [6,7,4,5,1,2,3] => ? = 6
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [7,6,4,5,1,2,3] => [7,6,4,5,1,2,3] => ? = 6
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [7,6,5,4,1,2,3] => [7,6,5,4,1,2,3] => ? = 3
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => [7,5,6,3,4,1,2] => ? = 2
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [7,6,5,3,4,1,2] => [7,6,5,3,4,1,2] => ? = 2
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,1,2] => [7,6,5,4,3,1,2] => ? = 2
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
Description
The size of the center of a parking function. The center of a parking function $p_1,\dots,p_n$ is the longest subsequence $a_1,\dots,a_k$ such that $a_i\leq i$.
Matching statistic: St000031
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 67%
Values
[1]
 => [1,0]
 => [(1,2)]
 => [2,1] => 1
[2]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 2
[1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => [3,4,2,1] => 1
[3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 3
[2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => 2
[1,1,1]
 => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => 4
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,7,8,6,5] => ? = 3
[2,2]
 => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,5,7,4,8,6,3] => ? = 2
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => 5
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [2,1,4,3,6,5,9,10,8,7] => ? = 4
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,6,7,8,5,4,3] => ? = 6
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => [2,1,4,3,7,9,6,10,8,5] => ? = 3
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => ? = 2
[2,1,1,1]
 => [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] => ? = 2
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [3,5,2,7,4,9,6,10,8,1] => ? = 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => 6
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
 => [2,1,4,3,6,5,8,7,11,12,10,9] => ? = 5
[4,2]
 => [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] => ? = 4
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
 => [2,1,4,3,6,5,9,11,8,12,10,7] => ? = 4
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => ? = 3
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,6,7,9,5,4,10,8,3] => ? = 6
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
 => [2,1,4,3,7,9,6,11,8,12,10,5] => ? = 3
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => ? = 2
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [4,5,7,3,2,9,6,10,8,1] => ? = 2
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => [2,1,5,7,4,9,6,11,8,12,10,3] => ? = 2
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
 => [3,5,2,7,4,9,6,11,8,12,10,1] => ? = 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
 => [2,1,4,3,6,5,8,7,10,9,12,11,14,13] => ? = 7
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,14),(12,13)]
 => [2,1,4,3,6,5,8,7,10,9,13,14,12,11] => ? = 6
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
 => [2,1,4,3,6,5,10,11,12,9,8,7] => ? = 10
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,14),(10,11),(12,13)]
 => [2,1,4,3,6,5,8,7,11,13,10,14,12,9] => ? = 5
[4,3]
 => [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] => ? = 12
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
 => [2,1,4,3,8,9,11,7,6,12,10,5] => ? = 4
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,14),(8,9),(10,11),(12,13)]
 => [2,1,4,3,6,5,9,11,8,13,10,14,12,7] => ? = 4
[3,2,2]
 => [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] => ? = 6
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
 => [2,1,6,7,9,5,4,11,8,12,10,3] => ? = 6
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,14),(6,7),(8,9),(10,11),(12,13)]
 => [2,1,4,3,7,9,6,11,8,13,10,14,12,5] => ? = 3
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => [5,6,7,9,4,3,2,10,8,1] => ? = 2
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
 => [4,5,7,3,2,9,6,11,8,12,10,1] => ? = 2
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,14),(4,5),(6,7),(8,9),(10,11),(12,13)]
 => [2,1,5,7,4,9,6,11,8,13,10,14,12,3] => ? = 2
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
 => [3,5,2,7,4,9,6,11,8,13,10,14,12,1] => ? = 1
Description
The number of cycles in the cycle decomposition of a permutation.
Matching statistic: St001207
(load all 7 compositions to match this statistic)
Mapping
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 44%
Values
[1]
 => [[1]]
 => [1] => [1] => ? = 1 - 1
[2]
 => [[1,2]]
 => [1,2] => [2,1] => 1 = 2 - 1
[1,1]
 => [[1],[2]]
 => [2,1] => [1,2] => 0 = 1 - 1
[3]
 => [[1,2,3]]
 => [1,2,3] => [3,2,1] => 2 = 3 - 1
[2,1]
 => [[1,2],[3]]
 => [3,1,2] => [1,3,2] => 1 = 2 - 1
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => [1,2,3] => 0 = 1 - 1
[4]
 => [[1,2,3,4]]
 => [1,2,3,4] => [4,3,2,1] => 3 = 4 - 1
[3,1]
 => [[1,2,3],[4]]
 => [4,1,2,3] => [1,4,3,2] => 2 = 3 - 1
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => [2,1,4,3] => 1 = 2 - 1
[2,1,1]
 => [[1,2],[3],[4]]
 => [4,3,1,2] => [1,2,4,3] => 1 = 2 - 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[5]
 => [[1,2,3,4,5]]
 => [1,2,3,4,5] => [5,4,3,2,1] => ? = 5 - 1
[4,1]
 => [[1,2,3,4],[5]]
 => [5,1,2,3,4] => [1,5,4,3,2] => ? = 4 - 1
[3,2]
 => [[1,2,3],[4,5]]
 => [4,5,1,2,3] => [2,1,5,4,3] => ? = 6 - 1
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,2,5,4,3] => ? = 3 - 1
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [1,3,2,5,4] => ? = 2 - 1
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [1,2,3,5,4] => ? = 2 - 1
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,2,3,4,5] => ? = 1 - 1
[6]
 => [[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ? = 6 - 1
[5,1]
 => [[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [1,6,5,4,3,2] => ? = 5 - 1
[4,2]
 => [[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => [2,1,6,5,4,3] => ? = 4 - 1
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => [1,2,6,5,4,3] => ? = 4 - 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => [3,2,1,6,5,4] => ? = 3 - 1
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => [1,3,2,6,5,4] => ? = 6 - 1
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => [1,2,3,6,5,4] => ? = 3 - 1
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => [2,1,4,3,6,5] => ? = 2 - 1
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => [1,2,4,3,6,5] => ? = 2 - 1
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [6,5,4,3,1,2] => [1,2,3,4,6,5] => ? = 2 - 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ? = 1 - 1
[7]
 => [[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7 - 1
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => ? = 6 - 1
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [6,7,1,2,3,4,5] => [2,1,7,6,5,4,3] => ? = 10 - 1
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [7,6,1,2,3,4,5] => [1,2,7,6,5,4,3] => ? = 5 - 1
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [5,6,7,1,2,3,4] => [3,2,1,7,6,5,4] => ? = 12 - 1
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [7,5,6,1,2,3,4] => [1,3,2,7,6,5,4] => ? = 4 - 1
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => ? = 4 - 1
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [6,7,4,5,1,2,3] => [2,1,4,3,7,6,5] => ? = 6 - 1
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [7,6,4,5,1,2,3] => [1,2,4,3,7,6,5] => ? = 6 - 1
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [7,6,5,4,1,2,3] => [1,2,3,4,7,6,5] => ? = 3 - 1
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [7,5,6,3,4,1,2] => [1,3,2,5,4,7,6] => ? = 2 - 1
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [7,6,5,3,4,1,2] => [1,2,3,5,4,7,6] => ? = 2 - 1
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,1,2] => [1,2,3,4,5,7,6] => ? = 2 - 1
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ? = 1 - 1
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St000162
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St000162: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%
Values
[1]
 => [1,0]
 => [(1,2)]
 => [2,1] => 1
[2]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 2
[1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => [3,4,2,1] => 1
[3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 3
[2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => 2
[1,1,1]
 => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => ? = 4
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,7,8,6,5] => ? = 3
[2,2]
 => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,5,7,4,8,6,3] => ? = 2
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => ? = 5
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [2,1,4,3,6,5,9,10,8,7] => ? = 4
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,6,7,8,5,4,3] => ? = 6
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => [2,1,4,3,7,9,6,10,8,5] => ? = 3
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => ? = 2
[2,1,1,1]
 => [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] => ? = 2
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [3,5,2,7,4,9,6,10,8,1] => ? = 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => ? = 6
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
 => [2,1,4,3,6,5,8,7,11,12,10,9] => ? = 5
[4,2]
 => [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] => ? = 4
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
 => [2,1,4,3,6,5,9,11,8,12,10,7] => ? = 4
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => ? = 3
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,6,7,9,5,4,10,8,3] => ? = 6
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
 => [2,1,4,3,7,9,6,11,8,12,10,5] => ? = 3
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => ? = 2
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [4,5,7,3,2,9,6,10,8,1] => ? = 2
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => [2,1,5,7,4,9,6,11,8,12,10,3] => ? = 2
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
 => [3,5,2,7,4,9,6,11,8,12,10,1] => ? = 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
 => [2,1,4,3,6,5,8,7,10,9,12,11,14,13] => ? = 7
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,14),(12,13)]
 => [2,1,4,3,6,5,8,7,10,9,13,14,12,11] => ? = 6
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
 => [2,1,4,3,6,5,10,11,12,9,8,7] => ? = 10
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,14),(10,11),(12,13)]
 => [2,1,4,3,6,5,8,7,11,13,10,14,12,9] => ? = 5
[4,3]
 => [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] => ? = 12
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
 => [2,1,4,3,8,9,11,7,6,12,10,5] => ? = 4
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,14),(8,9),(10,11),(12,13)]
 => [2,1,4,3,6,5,9,11,8,13,10,14,12,7] => ? = 4
[3,2,2]
 => [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] => ? = 6
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
 => [2,1,6,7,9,5,4,11,8,12,10,3] => ? = 6
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,14),(6,7),(8,9),(10,11),(12,13)]
 => [2,1,4,3,7,9,6,11,8,13,10,14,12,5] => ? = 3
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => [5,6,7,9,4,3,2,10,8,1] => ? = 2
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
 => [4,5,7,3,2,9,6,11,8,12,10,1] => ? = 2
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,14),(4,5),(6,7),(8,9),(10,11),(12,13)]
 => [2,1,5,7,4,9,6,11,8,13,10,14,12,3] => ? = 2
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
 => [3,5,2,7,4,9,6,11,8,13,10,14,12,1] => ? = 1
Description
The number of nontrivial cycles in the cycle decomposition of a permutation. This statistic is equal to the difference of the number of cycles of $\pi$ (see [[St000031]]) and the number of fixed points of $\pi$ (see [[St000022]]).
Matching statistic: St001114
Mapping
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
St001114: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 33%
Values
[1]
 => [1,0]
 => [(1,2)]
 => [2,1] => 1
[2]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 2
[1,1]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => [3,4,2,1] => 1
[3]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 3
[2,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => 2
[1,1,1]
 => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => 1
[4]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => ? = 4
[3,1]
 => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,7,8,6,5] => ? = 3
[2,2]
 => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => [2,1,5,7,4,8,6,3] => ? = 2
[1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => [3,5,2,7,4,8,6,1] => ? = 1
[5]
 => [1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => ? = 5
[4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [2,1,4,3,6,5,9,10,8,7] => ? = 4
[3,2]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,6,7,8,5,4,3] => ? = 6
[3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,10),(6,7),(8,9)]
 => [2,1,4,3,7,9,6,10,8,5] => ? = 3
[2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => [4,5,7,3,2,8,6,1] => ? = 2
[2,1,1,1]
 => [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] => ? = 2
[1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [(1,10),(2,3),(4,5),(6,7),(8,9)]
 => [3,5,2,7,4,9,6,10,8,1] => ? = 1
[6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => ? = 6
[5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
 => [2,1,4,3,6,5,8,7,11,12,10,9] => ? = 5
[4,2]
 => [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] => ? = 4
[4,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
 => [2,1,4,3,6,5,9,11,8,12,10,7] => ? = 4
[3,3]
 => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => [4,6,7,3,8,5,2,1] => ? = 3
[3,2,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,10),(4,7),(5,6),(8,9)]
 => [2,1,6,7,9,5,4,10,8,3] => ? = 6
[3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,7),(8,9),(10,11)]
 => [2,1,4,3,7,9,6,11,8,12,10,5] => ? = 3
[2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [5,6,7,8,4,3,2,1] => ? = 2
[2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [(1,10),(2,5),(3,4),(6,7),(8,9)]
 => [4,5,7,3,2,9,6,10,8,1] => ? = 2
[2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)]
 => [2,1,5,7,4,9,6,11,8,12,10,3] => ? = 2
[1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)]
 => [3,5,2,7,4,9,6,11,8,12,10,1] => ? = 1
[7]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14)]
 => [2,1,4,3,6,5,8,7,10,9,12,11,14,13] => ? = 7
[6,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,14),(12,13)]
 => [2,1,4,3,6,5,8,7,10,9,13,14,12,11] => ? = 6
[5,2]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
 => [2,1,4,3,6,5,10,11,12,9,8,7] => ? = 10
[5,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,14),(10,11),(12,13)]
 => [2,1,4,3,6,5,8,7,11,13,10,14,12,9] => ? = 5
[4,3]
 => [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] => ? = 12
[4,2,1]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [(1,2),(3,4),(5,12),(6,9),(7,8),(10,11)]
 => [2,1,4,3,8,9,11,7,6,12,10,5] => ? = 4
[4,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,6),(7,14),(8,9),(10,11),(12,13)]
 => [2,1,4,3,6,5,9,11,8,13,10,14,12,7] => ? = 4
[3,2,2]
 => [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] => ? = 6
[3,2,1,1]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [(1,2),(3,12),(4,7),(5,6),(8,9),(10,11)]
 => [2,1,6,7,9,5,4,11,8,12,10,3] => ? = 6
[3,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,4),(5,14),(6,7),(8,9),(10,11),(12,13)]
 => [2,1,4,3,7,9,6,11,8,13,10,14,12,5] => ? = 3
[2,2,2,1]
 => [1,1,1,1,0,0,0,1,0,0]
 => [(1,10),(2,7),(3,6),(4,5),(8,9)]
 => [5,6,7,9,4,3,2,10,8,1] => ? = 2
[2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [(1,12),(2,5),(3,4),(6,7),(8,9),(10,11)]
 => [4,5,7,3,2,9,6,11,8,12,10,1] => ? = 2
[2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,2),(3,14),(4,5),(6,7),(8,9),(10,11),(12,13)]
 => [2,1,5,7,4,9,6,11,8,13,10,14,12,3] => ? = 2
[1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [(1,14),(2,3),(4,5),(6,7),(8,9),(10,11),(12,13)]
 => [3,5,2,7,4,9,6,11,8,13,10,14,12,1] => ? = 1
Description
The number of odd descents of a permutation.
The following 21 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001151The number of blocks with odd minimum. St001645The pebbling number of a connected graph. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001879The 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. St000023The number of inner peaks of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000497The lcb statistic of a set partition. St001469The holeyness of a permutation. St001520The number of strict 3-descents. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001935The number of ascents in a parking function. St000075The orbit size of a standard tableau under promotion. St000099The number of valleys of a permutation, including the boundary. St000166The depth minus 1 of an ordered tree. St000522The number of 1-protected nodes of a rooted tree. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001557The number of inversions of the second entry of a permutation. St001948The number of augmented double ascents of a permutation. St000521The number of distinct subtrees of an ordered tree.