searching the database
Your data matches 50 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000925
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000925: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => {{1,2}}
=> 1
[2,1] => [2,1] => [1,2] => {{1},{2}}
=> 2
[1,2,3] => [1,3,2] => [3,1,2] => {{1,2,3}}
=> 1
[1,3,2] => [1,3,2] => [3,1,2] => {{1,2,3}}
=> 1
[2,1,3] => [2,1,3] => [2,3,1] => {{1,2,3}}
=> 1
[2,3,1] => [2,3,1] => [2,1,3] => {{1,2},{3}}
=> 2
[3,1,2] => [3,1,2] => [1,3,2] => {{1},{2,3}}
=> 2
[3,2,1] => [3,2,1] => [1,2,3] => {{1},{2},{3}}
=> 3
[1,2,3,4] => [1,4,3,2] => [4,1,2,3] => {{1,2,3,4}}
=> 1
[1,2,4,3] => [1,4,3,2] => [4,1,2,3] => {{1,2,3,4}}
=> 1
[1,3,2,4] => [1,4,3,2] => [4,1,2,3] => {{1,2,3,4}}
=> 1
[1,3,4,2] => [1,4,3,2] => [4,1,2,3] => {{1,2,3,4}}
=> 1
[1,4,2,3] => [1,4,3,2] => [4,1,2,3] => {{1,2,3,4}}
=> 1
[1,4,3,2] => [1,4,3,2] => [4,1,2,3] => {{1,2,3,4}}
=> 1
[2,1,3,4] => [2,1,4,3] => [3,4,1,2] => {{1,3},{2,4}}
=> 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => {{1,3},{2,4}}
=> 1
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => {{1,2,3,4}}
=> 1
[2,3,4,1] => [2,4,3,1] => [3,1,2,4] => {{1,2,3},{4}}
=> 2
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => {{1,2,3,4}}
=> 1
[2,4,3,1] => [2,4,3,1] => [3,1,2,4] => {{1,2,3},{4}}
=> 2
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => {{1,2,3,4}}
=> 1
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => {{1,2,3,4}}
=> 1
[3,2,1,4] => [3,2,1,4] => [2,3,4,1] => {{1,2,3,4}}
=> 1
[3,2,4,1] => [3,2,4,1] => [2,3,1,4] => {{1,2,3},{4}}
=> 2
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
[3,4,2,1] => [3,4,2,1] => [2,1,3,4] => {{1,2},{3},{4}}
=> 3
[4,1,2,3] => [4,1,3,2] => [1,4,2,3] => {{1},{2,3,4}}
=> 2
[4,1,3,2] => [4,1,3,2] => [1,4,2,3] => {{1},{2,3,4}}
=> 2
[4,2,1,3] => [4,2,1,3] => [1,3,4,2] => {{1},{2,3,4}}
=> 2
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => {{1},{2,3},{4}}
=> 3
[4,3,1,2] => [4,3,1,2] => [1,2,4,3] => {{1},{2},{3,4}}
=> 3
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4
[1,2,3,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1
[1,2,3,5,4] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1
[1,2,4,3,5] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1
[1,2,4,5,3] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1
[1,2,5,3,4] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1
[1,2,5,4,3] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1
[1,3,2,4,5] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1
[1,3,2,5,4] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1
[1,3,4,2,5] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1
[1,3,4,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1
[1,3,5,2,4] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1
[1,4,2,3,5] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1
[1,4,2,5,3] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1
[1,4,3,2,5] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1
[1,4,3,5,2] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1
[1,4,5,3,2] => [1,5,4,3,2] => [5,1,2,3,4] => {{1,2,3,4,5}}
=> 1
Description
The number of topologically connected components of a set partition.
For example, the set partition $\{\{1,5\},\{2,3\},\{4,6\}\}$ has the two connected components $\{1,4,5,6\}$ and $\{2,3\}$.
The number of set partitions with only one block is [[oeis:A099947]].
Matching statistic: St000011
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => [1,1,0,0]
=> 1
[2,1] => [2,1] => [1,2] => [1,0,1,0]
=> 2
[1,2,3] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[1,3,2] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[2,3,1] => [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[3,1,2] => [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[3,2,1] => [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[1,2,3,4] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[1,2,4,3] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[1,3,2,4] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[1,3,4,2] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[1,4,2,3] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[2,1,3,4] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 1
[2,3,4,1] => [2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 1
[2,4,3,1] => [2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[3,2,4,1] => [3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[3,4,2,1] => [3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3
[4,1,2,3] => [4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[4,1,3,2] => [4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[4,2,1,3] => [4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 2
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 3
[4,3,1,2] => [4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 3
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,2,3,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,3,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,4,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,4,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,5,3,4] => [1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,2,5,4,3] => [1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,3,2,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,3,2,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,3,4,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,3,4,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,3,5,2,4] => [1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,4,2,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,4,2,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,4,3,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,4,3,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,4,5,3,2] => [1,5,4,3,2] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[8,5,6,4,3,2,7,1] => [8,5,7,4,3,2,6,1] => [1,6,2,3,4,7,5,8] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 3
[8,6,4,5,3,2,7,1] => [8,6,4,7,3,2,5,1] => [1,5,2,3,7,4,6,8] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0,1,0]
=> ? = 3
[8,6,5,3,4,2,7,1] => [8,6,5,3,7,2,4,1] => [1,4,2,7,3,5,6,8] => [1,0,1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> ? = 3
[8,6,3,4,5,2,7,1] => [8,6,3,7,5,2,4,1] => [1,4,2,5,7,3,6,8] => [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> ? = 3
[8,5,4,3,6,2,7,1] => [8,5,4,3,7,2,6,1] => [1,6,2,7,3,4,5,8] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 3
[8,3,4,5,6,2,7,1] => [8,3,7,6,5,2,4,1] => [1,4,2,5,6,7,3,8] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
[8,6,5,4,2,3,7,1] => [8,6,5,4,2,7,3,1] => [1,3,7,2,4,5,6,8] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 3
[8,5,6,4,2,3,7,1] => [8,5,7,4,2,6,3,1] => [1,3,6,2,4,7,5,8] => [1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 3
[8,6,4,5,2,3,7,1] => [8,6,4,7,2,5,3,1] => [1,3,5,2,7,4,6,8] => [1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> ? = 3
[8,6,4,3,2,5,7,1] => [8,6,4,3,2,7,5,1] => [1,5,7,2,3,4,6,8] => [1,0,1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 3
[8,6,2,3,4,5,7,1] => [8,6,2,7,5,4,3,1] => [1,3,4,5,7,2,6,8] => [1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> ? = 3
[8,4,5,3,2,6,7,1] => [8,4,7,3,2,6,5,1] => [1,5,6,2,3,7,4,8] => ?
=> ? = 3
[8,5,4,2,3,6,7,1] => [8,5,4,2,7,6,3,1] => [1,3,6,7,2,4,5,8] => ?
=> ? = 3
[8,3,4,2,5,6,7,1] => [8,3,7,2,6,5,4,1] => [1,4,5,6,2,7,3,8] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 3
[8,2,3,4,5,6,7,1] => [8,2,7,6,5,4,3,1] => [1,3,4,5,6,7,2,8] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? = 3
[6,7,5,4,3,2,8,1] => [6,8,5,4,3,2,7,1] => [1,7,2,3,4,5,8,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2
[7,5,6,4,3,2,8,1] => [7,5,8,4,3,2,6,1] => [1,6,2,3,4,8,5,7] => [1,0,1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> ? = 2
[7,6,5,3,4,2,8,1] => [7,6,5,3,8,2,4,1] => [1,4,2,8,3,5,6,7] => [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[7,5,6,3,4,2,8,1] => [7,5,8,3,6,2,4,1] => [1,4,2,6,3,8,5,7] => [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 2
[5,6,7,3,4,2,8,1] => [5,8,7,3,6,2,4,1] => [1,4,2,6,3,7,8,5] => ?
=> ? = 2
[7,6,3,4,5,2,8,1] => [7,6,3,8,5,2,4,1] => [1,4,2,5,8,3,6,7] => [1,0,1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> ? = 2
[7,5,4,3,6,2,8,1] => [7,5,4,3,8,2,6,1] => [1,6,2,8,3,4,5,7] => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> ? = 2
[7,3,4,5,6,2,8,1] => [7,3,8,6,5,2,4,1] => [1,4,2,5,6,8,3,7] => [1,0,1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> ? = 2
[6,5,3,4,7,2,8,1] => [6,5,3,8,7,2,4,1] => [1,4,2,7,8,3,5,6] => [1,0,1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> ? = 2
[4,5,3,6,7,2,8,1] => [4,8,3,7,6,2,5,1] => [1,5,2,6,7,3,8,4] => ?
=> ? = 2
[7,6,5,4,2,3,8,1] => [7,6,5,4,2,8,3,1] => [1,3,8,2,4,5,6,7] => [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[5,6,7,4,2,3,8,1] => [5,8,7,4,2,6,3,1] => [1,3,6,2,4,7,8,5] => [1,0,1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
[6,7,4,5,2,3,8,1] => [6,8,4,7,2,5,3,1] => [1,3,5,2,7,4,8,6] => [1,0,1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 2
[7,4,5,6,2,3,8,1] => [7,4,8,6,2,5,3,1] => [1,3,5,2,6,8,4,7] => [1,0,1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 2
[5,4,6,7,2,3,8,1] => [5,4,8,7,2,6,3,1] => [1,3,6,2,7,8,4,5] => [1,0,1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 2
[7,6,5,2,3,4,8,1] => [7,6,5,2,8,4,3,1] => [1,3,4,8,2,5,6,7] => [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[5,6,7,2,3,4,8,1] => [5,8,7,2,6,4,3,1] => [1,3,4,6,2,7,8,5] => ?
=> ? = 2
[7,6,2,3,4,5,8,1] => [7,6,2,8,5,4,3,1] => [1,3,4,5,8,2,6,7] => [1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 2
[6,7,2,3,4,5,8,1] => [6,8,2,7,5,4,3,1] => [1,3,4,5,7,2,8,6] => [1,0,1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 2
[5,4,6,3,2,7,8,1] => [5,4,8,3,2,7,6,1] => [1,6,7,2,3,8,4,5] => [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> ? = 2
[4,5,6,3,2,7,8,1] => [4,8,7,3,2,6,5,1] => [1,5,6,2,3,7,8,4] => [1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2
[5,3,4,2,6,7,8,1] => [5,3,8,2,7,6,4,1] => [1,4,6,7,2,8,3,5] => [1,0,1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> ? = 2
[4,5,2,3,6,7,8,1] => [4,8,2,7,6,5,3,1] => [1,3,5,6,7,2,8,4] => [1,0,1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 2
[5,2,3,4,6,7,8,1] => [5,2,8,7,6,4,3,1] => [1,3,4,6,7,8,2,5] => ?
=> ? = 2
[7,5,6,4,3,8,1,2] => [7,5,8,4,3,6,1,2] => [2,1,6,3,4,8,5,7] => [1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 2
[5,6,7,4,3,8,1,2] => [5,8,7,4,3,6,1,2] => [2,1,6,3,4,7,8,5] => [1,1,0,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 2
[6,7,4,5,3,8,1,2] => [6,8,4,7,3,5,1,2] => [2,1,5,3,7,4,8,6] => [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 2
[5,4,6,7,3,8,1,2] => [5,4,8,7,3,6,1,2] => [2,1,6,3,7,8,4,5] => [1,1,0,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 2
[7,5,6,3,4,8,1,2] => [7,5,8,3,6,4,1,2] => [2,1,4,6,3,8,5,7] => [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 2
[5,6,7,3,4,8,1,2] => [5,8,7,3,6,4,1,2] => [2,1,4,6,3,7,8,5] => [1,1,0,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 2
[7,3,4,5,6,8,1,2] => [7,3,8,6,5,4,1,2] => [2,1,4,5,6,8,3,7] => [1,1,0,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 2
[5,6,4,3,7,8,1,2] => [5,8,4,3,7,6,1,2] => [2,1,6,7,3,4,8,5] => [1,1,0,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 2
[6,5,3,4,7,8,1,2] => [6,5,3,8,7,4,1,2] => [2,1,4,7,8,3,5,6] => [1,1,0,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 2
[4,5,3,6,7,8,1,2] => [4,8,3,7,6,5,1,2] => [2,1,5,6,7,3,8,4] => [1,1,0,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 2
[5,3,4,6,7,8,1,2] => [5,3,8,7,6,4,1,2] => [2,1,4,6,7,8,3,5] => [1,1,0,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 2
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St001462
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St001462: Standard tableaux ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => [2,1] => [[1],[2]]
=> 1
[2,1] => [1,2] => [1,2] => [[1,2]]
=> 2
[1,2,3] => [3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 1
[1,3,2] => [3,1,2] => [3,2,1] => [[1],[2],[3]]
=> 1
[2,1,3] => [2,3,1] => [3,2,1] => [[1],[2],[3]]
=> 1
[2,3,1] => [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 2
[3,1,2] => [1,3,2] => [1,3,2] => [[1,2],[3]]
=> 2
[3,2,1] => [1,2,3] => [1,2,3] => [[1,2,3]]
=> 3
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
[1,2,4,3] => [4,3,1,2] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
[1,3,2,4] => [4,2,3,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
[1,3,4,2] => [4,2,1,3] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
[1,4,2,3] => [4,1,3,2] => [4,2,3,1] => [[1,3],[2],[4]]
=> 1
[1,4,3,2] => [4,1,2,3] => [4,2,3,1] => [[1,3],[2],[4]]
=> 1
[2,1,3,4] => [3,4,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
[2,1,4,3] => [3,4,1,2] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
[2,3,1,4] => [3,2,4,1] => [4,2,3,1] => [[1,3],[2],[4]]
=> 1
[2,3,4,1] => [3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2
[2,4,1,3] => [3,1,4,2] => [4,2,3,1] => [[1,3],[2],[4]]
=> 1
[2,4,3,1] => [3,1,2,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2
[3,1,2,4] => [2,4,3,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
[3,1,4,2] => [2,4,1,3] => [3,4,1,2] => [[1,2],[3,4]]
=> 1
[3,2,1,4] => [2,3,4,1] => [4,2,3,1] => [[1,3],[2],[4]]
=> 1
[3,2,4,1] => [2,3,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[3,4,2,1] => [2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 3
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[4,1,3,2] => [1,4,2,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[4,2,1,3] => [1,3,4,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[4,2,3,1] => [1,3,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 3
[4,3,1,2] => [1,2,4,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 3
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 4
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
[1,2,3,5,4] => [5,4,3,1,2] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
[1,2,4,3,5] => [5,4,2,3,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
[1,2,4,5,3] => [5,4,2,1,3] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
[1,2,5,3,4] => [5,4,1,3,2] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
[1,2,5,4,3] => [5,4,1,2,3] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
[1,3,2,4,5] => [5,3,4,2,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
[1,3,2,5,4] => [5,3,4,1,2] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
[1,3,4,2,5] => [5,3,2,4,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
[1,3,4,5,2] => [5,3,2,1,4] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
[1,3,5,2,4] => [5,3,1,4,2] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
[1,3,5,4,2] => [5,3,1,2,4] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
[1,4,2,3,5] => [5,2,4,3,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
[1,4,2,5,3] => [5,2,4,1,3] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
[1,4,3,2,5] => [5,2,3,4,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
[1,4,3,5,2] => [5,2,3,1,4] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 1
[1,4,5,2,3] => [5,2,1,4,3] => [5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> 1
[1,4,5,3,2] => [5,2,1,3,4] => [5,3,2,4,1] => [[1,4],[2],[3],[5]]
=> 1
[8,6,4,5,3,2,7,1] => [1,3,5,4,6,7,2,8] => [1,7,4,3,5,6,2,8] => ?
=> ? = 3
[8,6,3,4,5,2,7,1] => [1,3,6,5,4,7,2,8] => [1,7,5,4,3,6,2,8] => ?
=> ? = 3
[8,5,4,3,6,2,7,1] => [1,4,5,6,3,7,2,8] => [1,7,5,4,3,6,2,8] => ?
=> ? = 3
[8,3,4,5,6,2,7,1] => [1,6,5,4,3,7,2,8] => ? => ?
=> ? = 3
[8,6,5,4,2,3,7,1] => [1,3,4,5,7,6,2,8] => [1,7,3,4,6,5,2,8] => ?
=> ? = 3
[8,5,6,4,2,3,7,1] => [1,4,3,5,7,6,2,8] => [1,7,3,4,6,5,2,8] => ?
=> ? = 3
[8,6,4,5,2,3,7,1] => [1,3,5,4,7,6,2,8] => [1,7,4,3,6,5,2,8] => ?
=> ? = 3
[8,5,4,2,3,6,7,1] => [1,4,5,7,6,3,2,8] => ? => ?
=> ? = 3
[7,5,6,4,3,2,8,1] => [2,4,3,5,6,7,1,8] => [7,3,2,4,5,6,1,8] => ?
=> ? = 2
[5,6,7,4,3,2,8,1] => [4,3,2,5,6,7,1,8] => [7,3,2,4,5,6,1,8] => ?
=> ? = 2
[6,5,4,7,3,2,8,1] => [3,4,5,2,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? = 2
[4,5,6,7,3,2,8,1] => [5,4,3,2,6,7,1,8] => [7,4,3,2,5,6,1,8] => ?
=> ? = 2
[7,6,3,4,5,2,8,1] => [2,3,6,5,4,7,1,8] => ? => ?
=> ? = 2
[7,3,4,5,6,2,8,1] => [2,6,5,4,3,7,1,8] => ? => ?
=> ? = 2
[5,6,7,4,2,3,8,1] => [4,3,2,5,7,6,1,8] => [7,3,2,4,6,5,1,8] => ?
=> ? = 2
[7,4,5,6,2,3,8,1] => [2,5,4,3,7,6,1,8] => ? => ?
=> ? = 2
[5,6,7,2,3,4,8,1] => [4,3,2,7,6,5,1,8] => ? => ?
=> ? = 2
[5,4,6,3,2,7,8,1] => [4,5,3,6,7,2,1,8] => ? => ?
=> ? = 2
[7,5,6,4,3,8,1,2] => [2,4,3,5,6,1,8,7] => [6,3,2,4,5,1,8,7] => ?
=> ? = 2
[5,6,7,4,3,8,1,2] => [4,3,2,5,6,1,8,7] => [6,3,2,4,5,1,8,7] => ?
=> ? = 2
[7,5,4,3,6,8,1,2] => [2,4,5,6,3,1,8,7] => ? => ?
=> ? = 2
[7,8,3,4,5,2,1,6] => [2,1,6,5,4,7,8,3] => ? => ?
=> ? = 2
[8,7,4,5,2,3,1,6] => [1,2,5,4,7,6,8,3] => [1,2,8,4,6,5,7,3] => ?
=> ? = 3
[8,7,5,4,3,1,2,6] => [1,2,4,5,6,8,7,3] => [1,2,8,4,5,7,6,3] => ?
=> ? = 3
[8,7,5,3,4,1,2,6] => [1,2,4,6,5,8,7,3] => [1,2,8,5,4,7,6,3] => ?
=> ? = 3
[8,7,3,4,5,1,2,6] => [1,2,6,5,4,8,7,3] => [1,2,8,5,4,7,6,3] => ?
=> ? = 3
[8,4,3,5,6,2,1,7] => [1,5,6,4,3,7,8,2] => ? => ?
=> ? = 2
[8,5,6,4,2,3,1,7] => [1,4,3,5,7,6,8,2] => ? => ?
=> ? = 2
[8,4,5,6,2,3,1,7] => [1,5,4,3,7,6,8,2] => ? => ?
=> ? = 2
[8,5,6,2,3,4,1,7] => [1,4,3,7,6,5,8,2] => ? => ?
=> ? = 2
[8,6,3,2,4,5,1,7] => [1,3,6,7,5,4,8,2] => ? => ?
=> ? = 2
[8,5,4,2,3,6,1,7] => [1,4,5,7,6,3,8,2] => ? => ?
=> ? = 2
[8,6,5,4,3,1,2,7] => [1,3,4,5,6,8,7,2] => [1,8,3,4,5,7,6,2] => ?
=> ? = 2
[8,6,4,5,3,1,2,7] => [1,3,5,4,6,8,7,2] => ? => ?
=> ? = 2
[8,5,6,4,1,2,3,7] => [1,4,3,5,8,7,6,2] => [1,8,3,4,7,6,5,2] => ?
=> ? = 2
[8,6,4,5,1,2,3,7] => [1,3,5,4,8,7,6,2] => [1,8,4,3,7,6,5,2] => ?
=> ? = 2
[8,4,5,6,1,2,3,7] => [1,5,4,3,8,7,6,2] => [1,8,4,3,7,6,5,2] => ?
=> ? = 2
[8,6,5,2,1,3,4,7] => [1,3,4,7,8,6,5,2] => ? => ?
=> ? = 2
[7,6,4,5,3,2,1,8] => [2,3,5,4,6,7,8,1] => [8,2,4,3,5,6,7,1] => ?
=> ? = 1
[6,7,4,5,3,2,1,8] => [3,2,5,4,6,7,8,1] => [8,2,4,3,5,6,7,1] => ?
=> ? = 1
[6,7,5,3,4,2,1,8] => [3,2,4,6,5,7,8,1] => ? => ?
=> ? = 1
[7,5,6,3,4,2,1,8] => [2,4,3,6,5,7,8,1] => [8,3,2,5,4,6,7,1] => ?
=> ? = 1
[6,5,7,3,4,2,1,8] => [3,4,2,6,5,7,8,1] => [8,3,2,5,4,6,7,1] => ?
=> ? = 1
[5,6,7,3,4,2,1,8] => [4,3,2,6,5,7,8,1] => [8,3,2,5,4,6,7,1] => ?
=> ? = 1
[5,4,6,3,7,2,1,8] => [4,5,3,6,2,7,8,1] => ? => ?
=> ? = 1
[7,6,5,4,2,3,1,8] => [2,3,4,5,7,6,8,1] => [8,2,3,4,6,5,7,1] => ?
=> ? = 1
[6,7,5,4,2,3,1,8] => [3,2,4,5,7,6,8,1] => [8,2,3,4,6,5,7,1] => ?
=> ? = 1
[5,4,6,7,2,3,1,8] => [4,5,3,2,7,6,8,1] => ? => ?
=> ? = 1
[7,5,6,2,3,4,1,8] => [2,4,3,7,6,5,8,1] => [8,3,2,6,5,4,7,1] => ?
=> ? = 1
[5,6,7,2,3,4,1,8] => [4,3,2,7,6,5,8,1] => [8,3,2,6,5,4,7,1] => ?
=> ? = 1
Description
The number of factors of a standard tableaux under concatenation.
The concatenation of two standard Young tableaux $T_1$ and $T_2$ is obtained by adding the largest entry of $T_1$ to each entry of $T_2$, and then appending the rows of the result to $T_1$, see [1, dfn 2.10].
This statistic returns the maximal number of standard tableaux such that their concatenation is the given tableau.
Matching statistic: St000439
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[2,1] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 3 = 2 + 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,3] => [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,3,1] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 3 = 2 + 1
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 2 + 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4 = 3 + 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[2,4,3,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,1,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,2,4,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 3 = 2 + 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[3,4,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 3 + 1
[4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[4,1,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[4,2,1,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 3 + 1
[4,3,1,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 4 + 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,4,3,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,5,3,4] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2,4,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,2,5,4] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,4,2,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,5,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,3,5,4,2] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,4,2,5,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,4,3,5,2] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,4,5,2,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,4,5,3,2] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[8,6,5,4,3,2,7,1] => [1,3,4,5,6,7,2,8] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
[8,5,6,4,3,2,7,1] => [1,4,3,5,6,7,2,8] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 3 + 1
[8,6,4,5,3,2,7,1] => [1,3,5,4,6,7,2,8] => [1,0,1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 3 + 1
[8,6,5,3,4,2,7,1] => [1,3,4,6,5,7,2,8] => [1,0,1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
[8,5,6,3,4,2,7,1] => [1,4,3,6,5,7,2,8] => [1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 3 + 1
[8,6,3,4,5,2,7,1] => [1,3,6,5,4,7,2,8] => [1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,1,0,1,0,0,0]
=> ? = 3 + 1
[8,5,4,3,6,2,7,1] => [1,4,5,6,3,7,2,8] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 3 + 1
[8,3,4,5,6,2,7,1] => [1,6,5,4,3,7,2,8] => ?
=> ?
=> ? = 3 + 1
[8,6,5,4,2,3,7,1] => [1,3,4,5,7,6,2,8] => [1,0,1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 3 + 1
[8,5,6,4,2,3,7,1] => [1,4,3,5,7,6,2,8] => [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 3 + 1
[8,6,4,5,2,3,7,1] => [1,3,5,4,7,6,2,8] => [1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 3 + 1
[8,6,4,3,2,5,7,1] => [1,3,5,6,7,4,2,8] => [1,0,1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[8,6,2,3,4,5,7,1] => [1,3,7,6,5,4,2,8] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 3 + 1
[8,4,5,3,2,6,7,1] => [1,5,4,6,7,3,2,8] => [1,0,1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,1,0,0]
=> ? = 3 + 1
[8,5,4,2,3,6,7,1] => [1,4,5,7,6,3,2,8] => ?
=> ?
=> ? = 3 + 1
[8,3,4,2,5,6,7,1] => [1,6,5,7,4,3,2,8] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 3 + 1
[6,5,4,7,3,2,8,1] => [3,4,5,2,6,7,1,8] => ?
=> ?
=> ? = 2 + 1
[7,6,3,4,5,2,8,1] => [2,3,6,5,4,7,1,8] => ?
=> ?
=> ? = 2 + 1
[7,3,4,5,6,2,8,1] => [2,6,5,4,3,7,1,8] => ?
=> ?
=> ? = 2 + 1
[7,4,5,6,2,3,8,1] => [2,5,4,3,7,6,1,8] => ?
=> ?
=> ? = 2 + 1
[5,6,7,2,3,4,8,1] => [4,3,2,7,6,5,1,8] => ?
=> ?
=> ? = 2 + 1
[5,4,6,3,2,7,8,1] => [4,5,3,6,7,2,1,8] => ?
=> ?
=> ? = 2 + 1
[7,6,5,4,3,8,1,2] => [2,3,4,5,6,1,8,7] => [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[7,5,6,4,3,8,1,2] => [2,4,3,5,6,1,8,7] => [1,1,0,1,1,0,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 2 + 1
[5,6,7,4,3,8,1,2] => [4,3,2,5,6,1,8,7] => [1,1,1,1,0,0,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2 + 1
[6,7,4,5,3,8,1,2] => [3,2,5,4,6,1,8,7] => [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
[6,5,4,7,3,8,1,2] => [3,4,5,2,6,1,8,7] => [1,1,1,0,1,0,1,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[5,4,6,7,3,8,1,2] => [4,5,3,2,6,1,8,7] => [1,1,1,1,0,1,0,0,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2 + 1
[5,6,7,3,4,8,1,2] => [4,3,2,6,5,1,8,7] => [1,1,1,1,0,0,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 2 + 1
[7,6,3,4,5,8,1,2] => [2,3,6,5,4,1,8,7] => [1,1,0,1,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 2 + 1
[7,5,4,3,6,8,1,2] => [2,4,5,6,3,1,8,7] => ?
=> ?
=> ? = 2 + 1
[7,3,4,5,6,8,1,2] => [2,6,5,4,3,1,8,7] => [1,1,0,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 2 + 1
[7,8,5,4,3,2,1,6] => [2,1,4,5,6,7,8,3] => [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 2 + 1
[7,8,4,5,3,2,1,6] => [2,1,5,4,6,7,8,3] => [1,1,0,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 2 + 1
[8,7,3,4,5,2,1,6] => [1,2,6,5,4,7,8,3] => [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,1,0,0,0]
=> ? = 3 + 1
[7,8,3,4,5,2,1,6] => [2,1,6,5,4,7,8,3] => ?
=> ?
=> ? = 2 + 1
[8,7,4,5,2,3,1,6] => [1,2,5,4,7,6,8,3] => [1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,1,0,0,0]
=> ? = 3 + 1
[7,8,4,5,2,3,1,6] => [2,1,5,4,7,6,8,3] => [1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
[7,8,5,2,3,4,1,6] => [2,1,4,7,6,5,8,3] => [1,1,0,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 2 + 1
[7,8,4,3,2,5,1,6] => [2,1,5,6,7,4,8,3] => [1,1,0,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> ? = 2 + 1
[8,7,2,3,4,5,1,6] => [1,2,7,6,5,4,8,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,1,0,0,0]
=> ? = 3 + 1
[7,8,2,3,4,5,1,6] => [2,1,7,6,5,4,8,3] => [1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 2 + 1
[7,8,5,4,3,1,2,6] => [2,1,4,5,6,8,7,3] => [1,1,0,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2 + 1
[8,7,5,3,4,1,2,6] => [1,2,4,6,5,8,7,3] => [1,0,1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 3 + 1
[8,7,3,4,5,1,2,6] => [1,2,6,5,4,8,7,3] => [1,0,1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,1,0,0,0]
=> ? = 3 + 1
[7,8,3,4,5,1,2,6] => [2,1,6,5,4,8,7,3] => [1,1,0,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 2 + 1
[7,8,5,4,1,2,3,6] => [2,1,4,5,8,7,6,3] => [1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> ? = 2 + 1
[8,7,4,5,1,2,3,6] => [1,2,5,4,8,7,6,3] => [1,0,1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 3 + 1
[7,8,4,5,1,2,3,6] => [2,1,5,4,8,7,6,3] => [1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2 + 1
[8,7,5,3,2,1,4,6] => [1,2,4,6,7,8,5,3] => [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 3 + 1
Description
The position of the first down step of a Dyck path.
Matching statistic: St000288
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Mp00114: Permutations —connectivity set⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => 0 => 0 = 1 - 1
[2,1] => [1,2] => 1 => 1 = 2 - 1
[1,2,3] => [3,2,1] => 00 => 0 = 1 - 1
[1,3,2] => [3,1,2] => 00 => 0 = 1 - 1
[2,1,3] => [2,3,1] => 00 => 0 = 1 - 1
[2,3,1] => [2,1,3] => 01 => 1 = 2 - 1
[3,1,2] => [1,3,2] => 10 => 1 = 2 - 1
[3,2,1] => [1,2,3] => 11 => 2 = 3 - 1
[1,2,3,4] => [4,3,2,1] => 000 => 0 = 1 - 1
[1,2,4,3] => [4,3,1,2] => 000 => 0 = 1 - 1
[1,3,2,4] => [4,2,3,1] => 000 => 0 = 1 - 1
[1,3,4,2] => [4,2,1,3] => 000 => 0 = 1 - 1
[1,4,2,3] => [4,1,3,2] => 000 => 0 = 1 - 1
[1,4,3,2] => [4,1,2,3] => 000 => 0 = 1 - 1
[2,1,3,4] => [3,4,2,1] => 000 => 0 = 1 - 1
[2,1,4,3] => [3,4,1,2] => 000 => 0 = 1 - 1
[2,3,1,4] => [3,2,4,1] => 000 => 0 = 1 - 1
[2,3,4,1] => [3,2,1,4] => 001 => 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => 000 => 0 = 1 - 1
[2,4,3,1] => [3,1,2,4] => 001 => 1 = 2 - 1
[3,1,2,4] => [2,4,3,1] => 000 => 0 = 1 - 1
[3,1,4,2] => [2,4,1,3] => 000 => 0 = 1 - 1
[3,2,1,4] => [2,3,4,1] => 000 => 0 = 1 - 1
[3,2,4,1] => [2,3,1,4] => 001 => 1 = 2 - 1
[3,4,1,2] => [2,1,4,3] => 010 => 1 = 2 - 1
[3,4,2,1] => [2,1,3,4] => 011 => 2 = 3 - 1
[4,1,2,3] => [1,4,3,2] => 100 => 1 = 2 - 1
[4,1,3,2] => [1,4,2,3] => 100 => 1 = 2 - 1
[4,2,1,3] => [1,3,4,2] => 100 => 1 = 2 - 1
[4,2,3,1] => [1,3,2,4] => 101 => 2 = 3 - 1
[4,3,1,2] => [1,2,4,3] => 110 => 2 = 3 - 1
[4,3,2,1] => [1,2,3,4] => 111 => 3 = 4 - 1
[1,2,3,4,5] => [5,4,3,2,1] => 0000 => 0 = 1 - 1
[1,2,3,5,4] => [5,4,3,1,2] => 0000 => 0 = 1 - 1
[1,2,4,3,5] => [5,4,2,3,1] => 0000 => 0 = 1 - 1
[1,2,4,5,3] => [5,4,2,1,3] => 0000 => 0 = 1 - 1
[1,2,5,3,4] => [5,4,1,3,2] => 0000 => 0 = 1 - 1
[1,2,5,4,3] => [5,4,1,2,3] => 0000 => 0 = 1 - 1
[1,3,2,4,5] => [5,3,4,2,1] => 0000 => 0 = 1 - 1
[1,3,2,5,4] => [5,3,4,1,2] => 0000 => 0 = 1 - 1
[1,3,4,2,5] => [5,3,2,4,1] => 0000 => 0 = 1 - 1
[1,3,4,5,2] => [5,3,2,1,4] => 0000 => 0 = 1 - 1
[1,3,5,2,4] => [5,3,1,4,2] => 0000 => 0 = 1 - 1
[1,3,5,4,2] => [5,3,1,2,4] => 0000 => 0 = 1 - 1
[1,4,2,3,5] => [5,2,4,3,1] => 0000 => 0 = 1 - 1
[1,4,2,5,3] => [5,2,4,1,3] => 0000 => 0 = 1 - 1
[1,4,3,2,5] => [5,2,3,4,1] => 0000 => 0 = 1 - 1
[1,4,3,5,2] => [5,2,3,1,4] => 0000 => 0 = 1 - 1
[1,4,5,2,3] => [5,2,1,4,3] => 0000 => 0 = 1 - 1
[1,4,5,3,2] => [5,2,1,3,4] => 0000 => 0 = 1 - 1
[8,5,6,4,3,2,7,1] => [1,4,3,5,6,7,2,8] => ? => ? = 3 - 1
[8,6,4,5,3,2,7,1] => [1,3,5,4,6,7,2,8] => ? => ? = 3 - 1
[8,6,5,3,4,2,7,1] => [1,3,4,6,5,7,2,8] => ? => ? = 3 - 1
[8,5,4,3,6,2,7,1] => [1,4,5,6,3,7,2,8] => ? => ? = 3 - 1
[8,3,4,5,6,2,7,1] => [1,6,5,4,3,7,2,8] => ? => ? = 3 - 1
[8,6,5,4,2,3,7,1] => [1,3,4,5,7,6,2,8] => ? => ? = 3 - 1
[8,6,2,3,4,5,7,1] => [1,3,7,6,5,4,2,8] => ? => ? = 3 - 1
[8,4,5,3,2,6,7,1] => [1,5,4,6,7,3,2,8] => ? => ? = 3 - 1
[8,5,4,2,3,6,7,1] => [1,4,5,7,6,3,2,8] => ? => ? = 3 - 1
[7,5,6,4,3,2,8,1] => [2,4,3,5,6,7,1,8] => ? => ? = 2 - 1
[6,5,4,7,3,2,8,1] => [3,4,5,2,6,7,1,8] => ? => ? = 2 - 1
[7,6,3,4,5,2,8,1] => [2,3,6,5,4,7,1,8] => ? => ? = 2 - 1
[7,5,4,3,6,2,8,1] => [2,4,5,6,3,7,1,8] => ? => ? = 2 - 1
[7,3,4,5,6,2,8,1] => [2,6,5,4,3,7,1,8] => ? => ? = 2 - 1
[5,6,7,4,2,3,8,1] => [4,3,2,5,7,6,1,8] => ? => ? = 2 - 1
[7,4,5,6,2,3,8,1] => [2,5,4,3,7,6,1,8] => ? => ? = 2 - 1
[5,6,7,2,3,4,8,1] => [4,3,2,7,6,5,1,8] => ? => ? = 2 - 1
[5,4,6,3,2,7,8,1] => [4,5,3,6,7,2,1,8] => ? => ? = 2 - 1
[6,5,4,7,3,8,1,2] => [3,4,5,2,6,1,8,7] => ? => ? = 2 - 1
[7,5,6,3,4,8,1,2] => [2,4,3,6,5,1,8,7] => ? => ? = 2 - 1
[7,6,3,4,5,8,1,2] => [2,3,6,5,4,1,8,7] => ? => ? = 2 - 1
[7,5,4,3,6,8,1,2] => [2,4,5,6,3,1,8,7] => ? => ? = 2 - 1
[5,6,4,3,7,8,1,2] => [4,3,5,6,2,1,8,7] => ? => ? = 2 - 1
[6,5,3,4,7,8,1,2] => [3,4,6,5,2,1,8,7] => ? => ? = 2 - 1
[7,8,3,4,5,2,1,6] => [2,1,6,5,4,7,8,3] => ? => ? = 2 - 1
[8,7,2,3,4,5,1,6] => [1,2,7,6,5,4,8,3] => ? => ? = 3 - 1
[7,8,3,4,5,1,2,6] => [2,1,6,5,4,8,7,3] => ? => ? = 2 - 1
[8,7,3,4,2,1,5,6] => [1,2,6,5,7,8,4,3] => ? => ? = 3 - 1
[7,8,3,4,2,1,5,6] => [2,1,6,5,7,8,4,3] => ? => ? = 2 - 1
[7,8,4,3,1,2,5,6] => [2,1,5,6,8,7,4,3] => ? => ? = 2 - 1
[8,7,2,3,1,4,5,6] => [1,2,7,6,8,5,4,3] => ? => ? = 3 - 1
[7,8,2,3,1,4,5,6] => [2,1,7,6,8,5,4,3] => ? => ? = 2 - 1
[7,8,3,1,2,4,5,6] => [2,1,6,8,7,5,4,3] => ? => ? = 2 - 1
[8,4,3,5,6,2,1,7] => [1,5,6,4,3,7,8,2] => ? => ? = 2 - 1
[8,3,4,5,6,2,1,7] => [1,6,5,4,3,7,8,2] => ? => ? = 2 - 1
[8,5,6,4,2,3,1,7] => [1,4,3,5,7,6,8,2] => ? => ? = 2 - 1
[8,6,4,5,2,3,1,7] => [1,3,5,4,7,6,8,2] => ? => ? = 2 - 1
[8,4,5,6,2,3,1,7] => [1,5,4,3,7,6,8,2] => ? => ? = 2 - 1
[8,5,6,2,3,4,1,7] => [1,4,3,7,6,5,8,2] => ? => ? = 2 - 1
[8,6,3,2,4,5,1,7] => [1,3,6,7,5,4,8,2] => ? => ? = 2 - 1
[8,5,4,2,3,6,1,7] => [1,4,5,7,6,3,8,2] => ? => ? = 2 - 1
[8,3,4,2,5,6,1,7] => [1,6,5,7,4,3,8,2] => ? => ? = 2 - 1
[8,6,4,5,3,1,2,7] => [1,3,5,4,6,8,7,2] => ? => ? = 2 - 1
[8,4,5,6,1,2,3,7] => [1,5,4,3,8,7,6,2] => ? => ? = 2 - 1
[8,6,5,2,1,3,4,7] => [1,3,4,7,8,6,5,2] => ? => ? = 2 - 1
[8,4,3,5,2,1,6,7] => [1,5,6,4,7,8,3,2] => ? => ? = 2 - 1
[8,5,4,2,1,3,6,7] => [1,4,5,7,8,6,3,2] => ? => ? = 2 - 1
[6,7,5,3,4,2,1,8] => [3,2,4,6,5,7,8,1] => ? => ? = 1 - 1
[6,5,7,3,4,2,1,8] => [3,4,2,6,5,7,8,1] => ? => ? = 1 - 1
[5,4,6,3,7,2,1,8] => [4,5,3,6,2,7,8,1] => ? => ? = 1 - 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000237
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 49% ●values known / values provided: 49%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => [1,1,0,0]
=> [1,2] => 0 = 1 - 1
[2,1] => [1,2] => [1,0,1,0]
=> [2,1] => 1 = 2 - 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[2,1,3] => [2,3,1] => [1,1,0,1,0,0]
=> [3,1,2] => 0 = 1 - 1
[2,3,1] => [2,1,3] => [1,1,0,0,1,0]
=> [1,3,2] => 1 = 2 - 1
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1 = 2 - 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [2,3,1] => 2 = 3 - 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 0 = 1 - 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 0 = 1 - 1
[2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 0 = 1 - 1
[2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 0 = 1 - 1
[2,4,3,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1 = 2 - 1
[3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 0 = 1 - 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 0 = 1 - 1
[3,2,1,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 0 = 1 - 1
[3,2,4,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 1 = 2 - 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1 = 2 - 1
[3,4,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 2 = 3 - 1
[4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1 = 2 - 1
[4,1,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1 = 2 - 1
[4,2,1,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1 = 2 - 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2 = 3 - 1
[4,3,1,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2 = 3 - 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 3 = 4 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,3,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,2,5,3,4] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,3,2,4,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,3,2,5,4] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,2,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,3,5,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,3,5,4,2] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,4,2,5,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,4,3,5,2] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,4,5,2,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,4,5,3,2] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[2,3,4,1,5,6,7] => [6,5,4,7,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 1 - 1
[2,3,4,1,5,7,6] => [6,5,4,7,3,1,2] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 1 - 1
[2,3,4,1,6,5,7] => [6,5,4,7,2,3,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 1 - 1
[2,3,4,1,6,7,5] => [6,5,4,7,2,1,3] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 1 - 1
[2,3,4,1,7,5,6] => [6,5,4,7,1,3,2] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 1 - 1
[2,3,4,1,7,6,5] => [6,5,4,7,1,2,3] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 1 - 1
[2,3,4,5,1,6,7] => [6,5,4,3,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,3,7,4,5,6] => ? = 1 - 1
[2,3,4,5,1,7,6] => [6,5,4,3,7,1,2] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,3,7,4,5,6] => ? = 1 - 1
[2,3,4,5,6,1,7] => [6,5,4,3,2,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,2,3,4,7,5,6] => ? = 1 - 1
[2,3,4,5,7,1,6] => [6,5,4,3,1,7,2] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,2,3,4,7,5,6] => ? = 1 - 1
[2,3,4,6,1,5,7] => [6,5,4,2,7,3,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,3,7,4,5,6] => ? = 1 - 1
[2,3,4,6,1,7,5] => [6,5,4,2,7,1,3] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,3,7,4,5,6] => ? = 1 - 1
[2,3,4,6,5,1,7] => [6,5,4,2,3,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,2,3,4,7,5,6] => ? = 1 - 1
[2,3,4,6,7,1,5] => [6,5,4,2,1,7,3] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,2,3,4,7,5,6] => ? = 1 - 1
[2,3,4,7,1,5,6] => [6,5,4,1,7,3,2] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,3,7,4,5,6] => ? = 1 - 1
[2,3,4,7,1,6,5] => [6,5,4,1,7,2,3] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,3,7,4,5,6] => ? = 1 - 1
[2,3,4,7,5,1,6] => [6,5,4,1,3,7,2] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,2,3,4,7,5,6] => ? = 1 - 1
[2,3,4,7,6,1,5] => [6,5,4,1,2,7,3] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,2,3,4,7,5,6] => ? = 1 - 1
[2,3,5,1,4,6,7] => [6,5,3,7,4,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 1 - 1
[2,3,5,1,4,7,6] => [6,5,3,7,4,1,2] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 1 - 1
[2,3,5,1,6,4,7] => [6,5,3,7,2,4,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 1 - 1
[2,3,5,1,6,7,4] => [6,5,3,7,2,1,4] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 1 - 1
[2,3,5,1,7,4,6] => [6,5,3,7,1,4,2] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 1 - 1
[2,3,5,1,7,6,4] => [6,5,3,7,1,2,4] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 1 - 1
[2,3,5,4,1,6,7] => [6,5,3,4,7,2,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,3,7,4,5,6] => ? = 1 - 1
[2,3,5,4,1,7,6] => [6,5,3,4,7,1,2] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,3,7,4,5,6] => ? = 1 - 1
[2,3,5,4,6,1,7] => [6,5,3,4,2,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,2,3,4,7,5,6] => ? = 1 - 1
[2,3,5,4,7,1,6] => [6,5,3,4,1,7,2] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,2,3,4,7,5,6] => ? = 1 - 1
[2,3,5,6,1,4,7] => [6,5,3,2,7,4,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,3,7,4,5,6] => ? = 1 - 1
[2,3,5,6,1,7,4] => [6,5,3,2,7,1,4] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,3,7,4,5,6] => ? = 1 - 1
[2,3,5,6,4,1,7] => [6,5,3,2,4,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,2,3,4,7,5,6] => ? = 1 - 1
[2,3,5,6,7,1,4] => [6,5,3,2,1,7,4] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,2,3,4,7,5,6] => ? = 1 - 1
[2,3,5,7,1,4,6] => [6,5,3,1,7,4,2] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,3,7,4,5,6] => ? = 1 - 1
[2,3,5,7,1,6,4] => [6,5,3,1,7,2,4] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,3,7,4,5,6] => ? = 1 - 1
[2,3,5,7,4,1,6] => [6,5,3,1,4,7,2] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,2,3,4,7,5,6] => ? = 1 - 1
[2,3,5,7,6,1,4] => [6,5,3,1,2,7,4] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,2,3,4,7,5,6] => ? = 1 - 1
[2,3,6,1,4,5,7] => [6,5,2,7,4,3,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 1 - 1
[2,3,6,1,4,7,5] => [6,5,2,7,4,1,3] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 1 - 1
[2,3,6,1,5,4,7] => [6,5,2,7,3,4,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 1 - 1
[2,3,6,1,5,7,4] => [6,5,2,7,3,1,4] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 1 - 1
[2,3,6,1,7,4,5] => [6,5,2,7,1,4,3] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 1 - 1
[2,3,6,1,7,5,4] => [6,5,2,7,1,3,4] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,2,7,3,4,5,6] => ? = 1 - 1
[2,3,6,4,1,5,7] => [6,5,2,4,7,3,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,3,7,4,5,6] => ? = 1 - 1
[2,3,6,4,1,7,5] => [6,5,2,4,7,1,3] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,3,7,4,5,6] => ? = 1 - 1
[2,3,6,4,5,1,7] => [6,5,2,4,3,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,2,3,4,7,5,6] => ? = 1 - 1
[2,3,6,4,7,1,5] => [6,5,2,4,1,7,3] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,2,3,4,7,5,6] => ? = 1 - 1
[2,3,6,5,1,4,7] => [6,5,2,3,7,4,1] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,3,7,4,5,6] => ? = 1 - 1
[2,3,6,5,1,7,4] => [6,5,2,3,7,1,4] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,2,3,7,4,5,6] => ? = 1 - 1
[2,3,6,5,4,1,7] => [6,5,2,3,4,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,2,3,4,7,5,6] => ? = 1 - 1
[2,3,6,5,7,1,4] => [6,5,2,3,1,7,4] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,2,3,4,7,5,6] => ? = 1 - 1
Description
The number of small exceedances.
This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St000007
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => [1,1,0,0]
=> [1,2] => 1
[2,1] => [1,2] => [1,0,1,0]
=> [2,1] => 2
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 1
[1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 1
[2,1,3] => [2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => 1
[2,3,1] => [2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 2
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => 2
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 3
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[2,4,3,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[3,2,1,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[3,2,4,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[3,4,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2
[4,1,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2
[4,2,1,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3
[4,3,1,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 3
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 4
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
[1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
[1,2,4,3,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
[1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
[1,2,5,3,4] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
[1,3,2,4,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
[1,3,2,5,4] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
[1,3,4,2,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
[1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
[1,3,5,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
[1,3,5,4,2] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
[1,4,2,5,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
[1,4,3,5,2] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
[1,4,5,2,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
[1,4,5,3,2] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
[3,2,4,1,5,6,7] => [5,6,4,7,3,2,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1
[3,2,4,1,5,7,6] => [5,6,4,7,3,1,2] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1
[3,2,4,1,6,5,7] => [5,6,4,7,2,3,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1
[3,2,4,1,6,7,5] => [5,6,4,7,2,1,3] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1
[3,2,4,1,7,5,6] => [5,6,4,7,1,3,2] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1
[3,2,4,1,7,6,5] => [5,6,4,7,1,2,3] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1
[3,2,4,5,1,6,7] => [5,6,4,3,7,2,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1
[3,2,4,5,1,7,6] => [5,6,4,3,7,1,2] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1
[3,2,4,5,6,1,7] => [5,6,4,3,2,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1
[3,2,4,5,7,1,6] => [5,6,4,3,1,7,2] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1
[3,2,4,6,1,5,7] => [5,6,4,2,7,3,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1
[3,2,4,6,1,7,5] => [5,6,4,2,7,1,3] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1
[3,2,4,6,5,1,7] => [5,6,4,2,3,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1
[3,2,4,6,7,1,5] => [5,6,4,2,1,7,3] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1
[3,2,4,7,1,5,6] => [5,6,4,1,7,3,2] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1
[3,2,4,7,1,6,5] => [5,6,4,1,7,2,3] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1
[3,2,4,7,5,1,6] => [5,6,4,1,3,7,2] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1
[3,2,4,7,6,1,5] => [5,6,4,1,2,7,3] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1
[3,2,5,1,4,6,7] => [5,6,3,7,4,2,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1
[3,2,5,1,4,7,6] => [5,6,3,7,4,1,2] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1
[3,2,5,1,6,4,7] => [5,6,3,7,2,4,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1
[3,2,5,1,6,7,4] => [5,6,3,7,2,1,4] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1
[3,2,5,1,7,4,6] => [5,6,3,7,1,4,2] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1
[3,2,5,1,7,6,4] => [5,6,3,7,1,2,4] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1
[3,2,5,4,1,6,7] => [5,6,3,4,7,2,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1
[3,2,5,4,1,7,6] => [5,6,3,4,7,1,2] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1
[3,2,5,4,6,1,7] => [5,6,3,4,2,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1
[3,2,5,4,7,1,6] => [5,6,3,4,1,7,2] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1
[3,2,5,6,1,4,7] => [5,6,3,2,7,4,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1
[3,2,5,6,1,7,4] => [5,6,3,2,7,1,4] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1
[3,2,5,6,4,1,7] => [5,6,3,2,4,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1
[3,2,5,6,7,1,4] => [5,6,3,2,1,7,4] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1
[3,2,5,7,1,4,6] => [5,6,3,1,7,4,2] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1
[3,2,5,7,1,6,4] => [5,6,3,1,7,2,4] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1
[3,2,5,7,4,1,6] => [5,6,3,1,4,7,2] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1
[3,2,5,7,6,1,4] => [5,6,3,1,2,7,4] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1
[3,2,6,1,4,5,7] => [5,6,2,7,4,3,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1
[3,2,6,1,4,7,5] => [5,6,2,7,4,1,3] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1
[3,2,6,1,5,4,7] => [5,6,2,7,3,4,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1
[3,2,6,1,5,7,4] => [5,6,2,7,3,1,4] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1
[3,2,6,1,7,4,5] => [5,6,2,7,1,4,3] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1
[3,2,6,1,7,5,4] => [5,6,2,7,1,3,4] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1
[3,2,6,4,1,5,7] => [5,6,2,4,7,3,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1
[3,2,6,4,1,7,5] => [5,6,2,4,7,1,3] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1
[3,2,6,4,5,1,7] => [5,6,2,4,3,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1
[3,2,6,4,7,1,5] => [5,6,2,4,1,7,3] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1
[3,2,6,5,1,4,7] => [5,6,2,3,7,4,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1
[3,2,6,5,1,7,4] => [5,6,2,3,7,1,4] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1
[3,2,6,5,4,1,7] => [5,6,2,3,4,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1
[3,2,6,5,7,1,4] => [5,6,2,3,1,7,4] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000546
(load all 71 compositions to match this statistic)
(load all 71 compositions to match this statistic)
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000546: Permutations ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000546: Permutations ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => [1,1,0,0]
=> [1,2] => 0 = 1 - 1
[2,1] => [1,2] => [1,0,1,0]
=> [2,1] => 1 = 2 - 1
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[2,1,3] => [2,3,1] => [1,1,0,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[2,3,1] => [2,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 1 = 2 - 1
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 2 = 3 - 1
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 0 = 1 - 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 0 = 1 - 1
[2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 0 = 1 - 1
[2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1 = 2 - 1
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 0 = 1 - 1
[2,4,3,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1 = 2 - 1
[3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 0 = 1 - 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 0 = 1 - 1
[3,2,1,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 0 = 1 - 1
[3,2,4,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1 = 2 - 1
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1 = 2 - 1
[3,4,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2 = 3 - 1
[4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 2 - 1
[4,1,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1 = 2 - 1
[4,2,1,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1 = 2 - 1
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2 = 3 - 1
[4,3,1,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2 = 3 - 1
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3 = 4 - 1
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,2,3,5,4] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,3,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,2,4,5,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,2,5,3,4] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,3,2,4,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,3,2,5,4] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,2,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,3,4,5,2] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,3,5,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,3,5,4,2] => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,4,2,3,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,4,2,5,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,4,3,5,2] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,4,5,2,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,4,5,3,2] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[3,2,4,1,5,6,7] => [5,6,4,7,3,2,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1 - 1
[3,2,4,1,5,7,6] => [5,6,4,7,3,1,2] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1 - 1
[3,2,4,1,6,5,7] => [5,6,4,7,2,3,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1 - 1
[3,2,4,1,6,7,5] => [5,6,4,7,2,1,3] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1 - 1
[3,2,4,1,7,5,6] => [5,6,4,7,1,3,2] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1 - 1
[3,2,4,1,7,6,5] => [5,6,4,7,1,2,3] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1 - 1
[3,2,4,5,1,6,7] => [5,6,4,3,7,2,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1 - 1
[3,2,4,5,1,7,6] => [5,6,4,3,7,1,2] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1 - 1
[3,2,4,5,6,1,7] => [5,6,4,3,2,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1 - 1
[3,2,4,5,7,1,6] => [5,6,4,3,1,7,2] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1 - 1
[3,2,4,6,1,5,7] => [5,6,4,2,7,3,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1 - 1
[3,2,4,6,1,7,5] => [5,6,4,2,7,1,3] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1 - 1
[3,2,4,6,5,1,7] => [5,6,4,2,3,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1 - 1
[3,2,4,6,7,1,5] => [5,6,4,2,1,7,3] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1 - 1
[3,2,4,7,1,5,6] => [5,6,4,1,7,3,2] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1 - 1
[3,2,4,7,1,6,5] => [5,6,4,1,7,2,3] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1 - 1
[3,2,4,7,5,1,6] => [5,6,4,1,3,7,2] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1 - 1
[3,2,4,7,6,1,5] => [5,6,4,1,2,7,3] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1 - 1
[3,2,5,1,4,6,7] => [5,6,3,7,4,2,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1 - 1
[3,2,5,1,4,7,6] => [5,6,3,7,4,1,2] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1 - 1
[3,2,5,1,6,4,7] => [5,6,3,7,2,4,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1 - 1
[3,2,5,1,6,7,4] => [5,6,3,7,2,1,4] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1 - 1
[3,2,5,1,7,4,6] => [5,6,3,7,1,4,2] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1 - 1
[3,2,5,1,7,6,4] => [5,6,3,7,1,2,4] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1 - 1
[3,2,5,4,1,6,7] => [5,6,3,4,7,2,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1 - 1
[3,2,5,4,1,7,6] => [5,6,3,4,7,1,2] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1 - 1
[3,2,5,4,6,1,7] => [5,6,3,4,2,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1 - 1
[3,2,5,4,7,1,6] => [5,6,3,4,1,7,2] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1 - 1
[3,2,5,6,1,4,7] => [5,6,3,2,7,4,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1 - 1
[3,2,5,6,1,7,4] => [5,6,3,2,7,1,4] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1 - 1
[3,2,5,6,4,1,7] => [5,6,3,2,4,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1 - 1
[3,2,5,6,7,1,4] => [5,6,3,2,1,7,4] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1 - 1
[3,2,5,7,1,4,6] => [5,6,3,1,7,4,2] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1 - 1
[3,2,5,7,1,6,4] => [5,6,3,1,7,2,4] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1 - 1
[3,2,5,7,4,1,6] => [5,6,3,1,4,7,2] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1 - 1
[3,2,5,7,6,1,4] => [5,6,3,1,2,7,4] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1 - 1
[3,2,6,1,4,5,7] => [5,6,2,7,4,3,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1 - 1
[3,2,6,1,4,7,5] => [5,6,2,7,4,1,3] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1 - 1
[3,2,6,1,5,4,7] => [5,6,2,7,3,4,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1 - 1
[3,2,6,1,5,7,4] => [5,6,2,7,3,1,4] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1 - 1
[3,2,6,1,7,4,5] => [5,6,2,7,1,4,3] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1 - 1
[3,2,6,1,7,5,4] => [5,6,2,7,1,3,4] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [4,2,1,3,5,6,7] => ? = 1 - 1
[3,2,6,4,1,5,7] => [5,6,2,4,7,3,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1 - 1
[3,2,6,4,1,7,5] => [5,6,2,4,7,1,3] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1 - 1
[3,2,6,4,5,1,7] => [5,6,2,4,3,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1 - 1
[3,2,6,4,7,1,5] => [5,6,2,4,1,7,3] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1 - 1
[3,2,6,5,1,4,7] => [5,6,2,3,7,4,1] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1 - 1
[3,2,6,5,1,7,4] => [5,6,2,3,7,1,4] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [5,2,1,3,4,6,7] => ? = 1 - 1
[3,2,6,5,4,1,7] => [5,6,2,3,4,7,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1 - 1
[3,2,6,5,7,1,4] => [5,6,2,3,1,7,4] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 1 - 1
Description
The number of global descents of a permutation.
The global descents are the integers in the set
$$C(\pi)=\{i\in [n-1] : \forall 1 \leq j \leq i < k \leq n :\quad \pi(j) > \pi(k)\}.$$
In particular, if $i\in C(\pi)$ then $i$ is a descent.
For the number of global ascents, see [[St000234]].
Matching statistic: St000678
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => [1,1,0,0]
=> [1,1,0,0]
=> 1
[2,1] => [1,2] => [1,0,1,0]
=> [1,0,1,0]
=> 2
[1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 1
[2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,3,5,4] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,2,4,5,3] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,2,5,3,4] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,2,5,4] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,3,4,5,2] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,3,5,2,4] => [4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,5,4,2] => [2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,4,2,5,3] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,4,3,2,5] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,4,3,5,2] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,4,5,3,2] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[8,6,5,4,3,2,7,1] => [1,7,2,3,4,5,6,8] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 3
[8,5,6,4,3,2,7,1] => [1,7,2,3,4,6,5,8] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 3
[8,6,4,5,3,2,7,1] => [1,7,2,3,5,4,6,8] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 3
[8,6,5,3,4,2,7,1] => [1,7,2,4,3,5,6,8] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 3
[8,5,6,3,4,2,7,1] => [1,7,2,4,3,6,5,8] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 3
[8,6,3,4,5,2,7,1] => [1,7,2,5,4,3,6,8] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 3
[8,5,4,3,6,2,7,1] => [1,7,2,6,3,4,5,8] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 3
[8,3,4,5,6,2,7,1] => [1,7,2,6,5,4,3,8] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 3
[8,6,5,4,2,3,7,1] => [1,7,3,2,4,5,6,8] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 3
[8,5,6,4,2,3,7,1] => [1,7,3,2,4,6,5,8] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 3
[8,6,4,5,2,3,7,1] => [1,7,3,2,5,4,6,8] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 3
[8,6,4,3,2,5,7,1] => [1,7,5,2,3,4,6,8] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 3
[8,6,2,3,4,5,7,1] => [1,7,5,4,3,2,6,8] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 3
[8,4,5,3,2,6,7,1] => [1,7,6,2,3,5,4,8] => ?
=> ?
=> ? = 3
[8,5,4,2,3,6,7,1] => [1,7,6,3,2,4,5,8] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 3
[8,3,4,2,5,6,7,1] => [1,7,6,5,2,4,3,8] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 3
[8,2,3,4,5,6,7,1] => [1,7,6,5,4,3,2,8] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 3
[7,6,5,4,3,2,8,1] => [1,8,2,3,4,5,6,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[6,7,5,4,3,2,8,1] => [1,8,2,3,4,5,7,6] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[7,5,6,4,3,2,8,1] => [1,8,2,3,4,6,5,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[5,6,7,4,3,2,8,1] => [1,8,2,3,4,7,6,5] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[6,5,4,7,3,2,8,1] => [1,8,2,3,7,4,5,6] => ?
=> ?
=> ? = 2
[4,5,6,7,3,2,8,1] => [1,8,2,3,7,6,5,4] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[7,6,5,3,4,2,8,1] => [1,8,2,4,3,5,6,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[7,5,6,3,4,2,8,1] => [1,8,2,4,3,6,5,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[5,6,7,3,4,2,8,1] => [1,8,2,4,3,7,6,5] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[7,6,3,4,5,2,8,1] => [1,8,2,5,4,3,6,7] => ?
=> ?
=> ? = 2
[7,5,4,3,6,2,8,1] => [1,8,2,6,3,4,5,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[7,3,4,5,6,2,8,1] => [1,8,2,6,5,4,3,7] => ?
=> ?
=> ? = 2
[6,5,3,4,7,2,8,1] => [1,8,2,7,4,3,5,6] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[4,5,3,6,7,2,8,1] => [1,8,2,7,6,3,5,4] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[3,4,5,6,7,2,8,1] => [1,8,2,7,6,5,4,3] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[7,6,5,4,2,3,8,1] => [1,8,3,2,4,5,6,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[5,6,7,4,2,3,8,1] => [1,8,3,2,4,7,6,5] => ?
=> ?
=> ? = 2
[6,7,4,5,2,3,8,1] => [1,8,3,2,5,4,7,6] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[7,4,5,6,2,3,8,1] => [1,8,3,2,6,5,4,7] => ?
=> ?
=> ? = 2
[5,4,6,7,2,3,8,1] => [1,8,3,2,7,6,4,5] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[7,6,5,2,3,4,8,1] => [1,8,4,3,2,5,6,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[5,6,7,2,3,4,8,1] => [1,8,4,3,2,7,6,5] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[6,7,3,2,4,5,8,1] => [1,8,5,4,2,3,7,6] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[7,6,2,3,4,5,8,1] => [1,8,5,4,3,2,6,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[6,7,2,3,4,5,8,1] => [1,8,5,4,3,2,7,6] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[7,2,3,4,5,6,8,1] => [1,8,6,5,4,3,2,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[6,5,4,3,2,7,8,1] => [1,8,7,2,3,4,5,6] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[5,4,6,3,2,7,8,1] => [1,8,7,2,3,6,4,5] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[4,5,6,3,2,7,8,1] => [1,8,7,2,3,6,5,4] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[5,4,3,2,6,7,8,1] => [1,8,7,6,2,3,4,5] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[5,3,4,2,6,7,8,1] => [1,8,7,6,2,4,3,5] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[3,4,5,2,6,7,8,1] => [1,8,7,6,2,5,4,3] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
[4,5,2,3,6,7,8,1] => [1,8,7,6,3,2,5,4] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St000383
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2,1] => ([(0,1)],2)
=> [1,1] => 1
[2,1] => [1,2] => ([],2)
=> [2] => 2
[1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> [2,1] => 1
[1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
=> [1,1,1] => 1
[2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
=> [1,1,1] => 1
[2,3,1] => [1,3,2] => ([(1,2)],3)
=> [1,2] => 2
[3,1,2] => [2,1,3] => ([(1,2)],3)
=> [1,2] => 2
[3,2,1] => [1,2,3] => ([],3)
=> [3] => 3
[1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1] => 1
[1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 1
[1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 1
[1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
[1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
[1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => 1
[2,1,3,4] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 1
[2,1,4,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> [1,2,1] => 1
[2,3,1,4] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
[2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> [2,2] => 2
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1
[2,4,3,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> [1,1,2] => 2
[3,1,2,4] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> [1,1,1,1] => 1
[3,2,1,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> [1,2,1] => 1
[3,2,4,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> [1,1,2] => 2
[3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> [2,2] => 2
[3,4,2,1] => [1,2,4,3] => ([(2,3)],4)
=> [1,3] => 3
[4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> [2,2] => 2
[4,1,3,2] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> [1,1,2] => 2
[4,2,1,3] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> [1,1,2] => 2
[4,2,3,1] => [1,3,2,4] => ([(2,3)],4)
=> [1,3] => 3
[4,3,1,2] => [2,1,3,4] => ([(2,3)],4)
=> [1,3] => 3
[4,3,2,1] => [1,2,3,4] => ([],4)
=> [4] => 4
[1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1] => 1
[1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 1
[1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 1
[1,2,4,5,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 1
[1,2,5,3,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 1
[1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 1
[1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1] => 1
[1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 1
[1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 1
[1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,2,1,1] => 1
[1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
[1,3,5,4,2] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
[1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 1
[1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
[1,4,3,2,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 1
[1,4,3,5,2] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
[1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [1,2,1,1] => 1
[1,4,5,3,2] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,2,1] => 1
[8,6,5,4,3,2,7,1] => [1,7,2,3,4,5,6,8] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? => ? = 3
[8,5,6,3,4,2,7,1] => [1,7,2,4,3,6,5,8] => ([(2,7),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? => ? = 3
[8,5,4,3,6,2,7,1] => [1,7,2,6,3,4,5,8] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,1,2,1,3] => ? = 3
[8,3,4,5,6,2,7,1] => [1,7,2,6,5,4,3,8] => ([(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
[8,5,6,4,2,3,7,1] => [1,7,3,2,4,6,5,8] => ([(2,7),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? => ? = 3
[8,6,4,5,2,3,7,1] => [1,7,3,2,5,4,6,8] => ([(2,7),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? => ? = 3
[8,6,4,3,2,5,7,1] => [1,7,5,2,3,4,6,8] => ([(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [1,1,2,1,3] => ? = 3
[8,6,2,3,4,5,7,1] => [1,7,5,4,3,2,6,8] => ([(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
[8,4,5,3,2,6,7,1] => [1,7,6,2,3,5,4,8] => ?
=> ? => ? = 3
[8,3,4,2,5,6,7,1] => [1,7,6,5,2,4,3,8] => ([(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
[8,2,3,4,5,6,7,1] => [1,7,6,5,4,3,2,8] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
[7,6,5,4,3,2,8,1] => [1,8,2,3,4,5,6,7] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? => ? = 2
[6,7,5,4,3,2,8,1] => [1,8,2,3,4,5,7,6] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
[7,5,6,4,3,2,8,1] => [1,8,2,3,4,6,5,7] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
[5,6,7,4,3,2,8,1] => [1,8,2,3,4,7,6,5] => ([(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
[6,5,4,7,3,2,8,1] => [1,8,2,3,7,4,5,6] => ?
=> ? => ? = 2
[4,5,6,7,3,2,8,1] => [1,8,2,3,7,6,5,4] => ?
=> ? => ? = 2
[7,6,5,3,4,2,8,1] => [1,8,2,4,3,5,6,7] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
[7,5,6,3,4,2,8,1] => [1,8,2,4,3,6,5,7] => ([(1,7),(2,7),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? => ? = 2
[5,6,7,3,4,2,8,1] => [1,8,2,4,3,7,6,5] => ?
=> ? => ? = 2
[7,6,3,4,5,2,8,1] => [1,8,2,5,4,3,6,7] => ?
=> ? => ? = 2
[7,3,4,5,6,2,8,1] => [1,8,2,6,5,4,3,7] => ?
=> ? => ? = 2
[4,5,3,6,7,2,8,1] => [1,8,2,7,6,3,5,4] => ?
=> ? => ? = 2
[7,6,5,4,2,3,8,1] => [1,8,3,2,4,5,6,7] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
[5,6,7,4,2,3,8,1] => [1,8,3,2,4,7,6,5] => ?
=> ? => ? = 2
[7,4,5,6,2,3,8,1] => [1,8,3,2,6,5,4,7] => ?
=> ? => ? = 2
[5,4,6,7,2,3,8,1] => [1,8,3,2,7,6,4,5] => ?
=> ? => ? = 2
[7,6,5,2,3,4,8,1] => [1,8,4,3,2,5,6,7] => ([(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
[5,6,7,2,3,4,8,1] => [1,8,4,3,2,7,6,5] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
[6,5,4,3,2,7,8,1] => [1,8,7,2,3,4,5,6] => ([(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
[4,5,6,3,2,7,8,1] => [1,8,7,2,3,6,5,4] => ?
=> ? => ? = 2
[4,3,2,5,6,7,8,1] => [1,8,7,6,5,2,3,4] => ([(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [4,2,2] => ? = 2
[3,2,4,5,6,7,8,1] => [1,8,7,6,5,4,2,3] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [5,1,2] => ? = 2
[2,3,4,5,6,7,8,1] => [1,8,7,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [6,2] => ? = 2
[7,6,5,4,3,8,1,2] => [2,1,8,3,4,5,6,7] => ([(0,1),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> [1,1,4,2] => ? = 2
[7,5,6,4,3,8,1,2] => [2,1,8,3,4,6,5,7] => ?
=> ? => ? = 2
[5,6,7,4,3,8,1,2] => [2,1,8,3,4,7,6,5] => ([(0,1),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
[6,5,4,7,3,8,1,2] => [2,1,8,3,7,4,5,6] => ([(0,1),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
[5,4,6,7,3,8,1,2] => [2,1,8,3,7,6,4,5] => ([(0,1),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
[5,6,7,3,4,8,1,2] => [2,1,8,4,3,7,6,5] => ([(0,1),(2,3),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
[7,6,3,4,5,8,1,2] => [2,1,8,5,4,3,6,7] => ([(0,1),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
[7,5,4,3,6,8,1,2] => [2,1,8,6,3,4,5,7] => ?
=> ? => ? = 2
[7,3,4,5,6,8,1,2] => [2,1,8,6,5,4,3,7] => ([(0,7),(1,2),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
[5,6,4,3,7,8,1,2] => [2,1,8,7,3,4,6,5] => ([(0,1),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
[6,5,3,4,7,8,1,2] => [2,1,8,7,4,3,5,6] => ([(0,1),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
[4,5,3,6,7,8,1,2] => [2,1,8,7,6,3,5,4] => ([(0,1),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
[5,3,4,6,7,8,1,2] => [2,1,8,7,6,4,3,5] => ([(0,1),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
[3,4,5,6,7,8,1,2] => [2,1,8,7,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> [5,1,2] => ? = 2
[8,7,5,4,3,2,1,6] => [6,1,2,3,4,5,7,8] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? => ? = 3
[7,8,5,4,3,2,1,6] => [6,1,2,3,4,5,8,7] => ([(0,1),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> [1,1,4,2] => ? = 2
Description
The last part of an integer composition.
The following 40 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000010The length of the partition. St000675The number of centered multitunnels of a Dyck path. St000025The number of initial rises of a Dyck path. St000544The cop number of a graph. St000234The number of global ascents of a permutation. St001363The Euler characteristic of a graph according to Knill. St000287The number of connected components of a graph. St001828The Euler characteristic of a graph. St000717The number of ordinal summands of a poset. St000553The number of blocks of a graph. St000286The number of connected components of the complement of a graph. St000260The radius of a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000382The first part of an integer composition. St000843The decomposition number of a perfect matching. St000259The diameter of a connected graph. St000456The monochromatic index of a connected graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000264The girth of a graph, which is not a tree. St001461The number of topologically connected components of the chord diagram of a permutation. St000056The decomposition (or block) number of a permutation. St000061The number of nodes on the left branch of a binary tree. St000084The number of subtrees. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000181The number of connected components of the Hasse diagram for the poset. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001889The size of the connectivity set of a signed permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!