Your data matches 8 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001640
(load all 10 compositions to match this statistic)
Mapping
St001640: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 1
[2,1] => 0
[1,2,3] => 2
[1,3,2] => 0
[2,1,3] => 1
[2,3,1] => 0
[3,1,2] => 1
[3,2,1] => 0
[1,2,3,4] => 3
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 0
[1,4,2,3] => 1
[1,4,3,2] => 0
[2,1,3,4] => 2
[2,1,4,3] => 0
[2,3,1,4] => 1
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,3,1] => 0
[3,1,2,4] => 2
[3,1,4,2] => 0
[3,2,1,4] => 1
[3,2,4,1] => 0
[3,4,1,2] => 1
[3,4,2,1] => 0
[4,1,2,3] => 2
[4,1,3,2] => 0
[4,2,1,3] => 1
[4,2,3,1] => 0
[4,3,1,2] => 1
[4,3,2,1] => 0
[1,2,3,4,5] => 4
[1,2,3,5,4] => 2
[1,2,4,3,5] => 2
[1,2,4,5,3] => 1
[1,2,5,3,4] => 2
[1,2,5,4,3] => 1
[1,3,2,4,5] => 2
[1,3,2,5,4] => 0
[1,3,4,2,5] => 1
[1,3,4,5,2] => 0
[1,3,5,2,4] => 1
[1,3,5,4,2] => 0
[1,4,2,3,5] => 2
[1,4,2,5,3] => 0
[1,4,3,2,5] => 1
[1,4,3,5,2] => 0
[1,4,5,2,3] => 1
Description
The number of ascent tops in the permutation such that all smaller elements appear before.
Matching statistic: St001067
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
 => 0
[1,2] => [2,1] => [1,2] => [1,0,1,0]
 => 1
[2,1] => [1,2] => [2,1] => [1,1,0,0]
 => 0
[1,2,3] => [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[1,3,2] => [2,3,1] => [2,1,3] => [1,1,0,0,1,0]
 => 0
[2,1,3] => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => 1
[2,3,1] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => 0
[3,1,2] => [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
 => 1
[3,2,1] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 0
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 3
[1,2,4,3] => [3,4,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 1
[1,3,2,4] => [4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 1
[1,3,4,2] => [2,4,3,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 0
[1,4,2,3] => [3,2,4,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 1
[1,4,3,2] => [2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 0
[2,1,3,4] => [4,3,1,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 2
[2,1,4,3] => [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 0
[2,3,1,4] => [4,1,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,3,4,1] => [1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 0
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 1
[2,4,3,1] => [1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 0
[3,1,2,4] => [4,2,1,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 2
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 0
[3,2,1,4] => [4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1
[3,2,4,1] => [1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 0
[3,4,1,2] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 1
[3,4,2,1] => [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 0
[4,1,2,3] => [3,2,1,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 2
[4,1,3,2] => [2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 0
[4,2,1,3] => [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 1
[4,2,3,1] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 0
[4,3,1,2] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 1
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 4
[1,2,3,5,4] => [4,5,3,2,1] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 2
[1,2,4,3,5] => [5,3,4,2,1] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[1,2,4,5,3] => [3,5,4,2,1] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,2,5,3,4] => [4,3,5,2,1] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => 2
[1,2,5,4,3] => [3,4,5,2,1] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[1,3,2,5,4] => [4,5,2,3,1] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,3,4,2,5] => [5,2,4,3,1] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,3,4,5,2] => [2,5,4,3,1] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,3,5,2,4] => [4,2,5,3,1] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => 1
[1,3,5,4,2] => [2,4,5,3,1] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,4,2,3,5] => [5,3,2,4,1] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => 2
[1,4,2,5,3] => [3,5,2,4,1] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
 => 0
[1,4,3,2,5] => [5,2,3,4,1] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,4,3,5,2] => [2,5,3,4,1] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,4,5,2,3] => [3,2,5,4,1] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => 1
Description
The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St000932
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1,0]
 => ? = 0
[1,2] => [2,1] => [1,2] => [1,0,1,0]
 => 1
[2,1] => [1,2] => [2,1] => [1,1,0,0]
 => 0
[1,2,3] => [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[1,3,2] => [2,3,1] => [2,1,3] => [1,1,0,0,1,0]
 => 0
[2,1,3] => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => 1
[2,3,1] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => 0
[3,1,2] => [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
 => 1
[3,2,1] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 0
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 3
[1,2,4,3] => [3,4,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 1
[1,3,2,4] => [4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 1
[1,3,4,2] => [2,4,3,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 0
[1,4,2,3] => [3,2,4,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 1
[1,4,3,2] => [2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 0
[2,1,3,4] => [4,3,1,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 2
[2,1,4,3] => [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 0
[2,3,1,4] => [4,1,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,3,4,1] => [1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 0
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 1
[2,4,3,1] => [1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 0
[3,1,2,4] => [4,2,1,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 2
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 0
[3,2,1,4] => [4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1
[3,2,4,1] => [1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 0
[3,4,1,2] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 1
[3,4,2,1] => [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 0
[4,1,2,3] => [3,2,1,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 2
[4,1,3,2] => [2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 0
[4,2,1,3] => [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 1
[4,2,3,1] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 0
[4,3,1,2] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 1
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 4
[1,2,3,5,4] => [4,5,3,2,1] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 2
[1,2,4,3,5] => [5,3,4,2,1] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[1,2,4,5,3] => [3,5,4,2,1] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,2,5,3,4] => [4,3,5,2,1] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => 2
[1,2,5,4,3] => [3,4,5,2,1] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[1,3,2,5,4] => [4,5,2,3,1] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,3,4,2,5] => [5,2,4,3,1] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,3,4,5,2] => [2,5,4,3,1] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,3,5,2,4] => [4,2,5,3,1] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => 1
[1,3,5,4,2] => [2,4,5,3,1] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,4,2,3,5] => [5,3,2,4,1] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => 2
[1,4,2,5,3] => [3,5,2,4,1] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
 => 0
[1,4,3,2,5] => [5,2,3,4,1] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,4,3,5,2] => [2,5,3,4,1] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,4,5,2,3] => [3,2,5,4,1] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => 1
[1,4,5,3,2] => [2,3,5,4,1] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => 0
Description
The number of occurrences of the pattern UDU in a Dyck path. The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St001484
Mapping
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => []
 => 0
[1,2] => [1,2] => [1,0,1,0]
 => [1]
 => 1
[2,1] => [2,1] => [1,1,0,0]
 => []
 => 0
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => [2,1]
 => 2
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1]
 => 0
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => [2]
 => 1
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
 => []
 => 0
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
 => [1]
 => 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => []
 => 0
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [3,2,1]
 => 3
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [2,2,1]
 => 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [3,1,1]
 => 1
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [2,1,1]
 => 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1]
 => 0
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [3,2]
 => 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [2,2]
 => 0
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => 1
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => []
 => 0
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [2]
 => 1
[2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => []
 => 0
[3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [3,1]
 => 2
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 0
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [3]
 => 1
[3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => []
 => 0
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1]
 => 1
[3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => []
 => 0
[4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [2,1]
 => 2
[4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1]
 => 0
[4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [2]
 => 1
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => []
 => 0
[4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1]
 => 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => []
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [4,3,2,1]
 => 4
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [3,3,2,1]
 => 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [4,2,2,1]
 => 2
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [3,2,2,1]
 => 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,2,2,1]
 => 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [4,3,1,1]
 => 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [3,3,1,1]
 => 0
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,3,5,2,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,1,1]
 => 1
[1,3,5,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [4,2,1,1]
 => 2
[1,4,2,5,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [2,2,1,1]
 => 0
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [4,1,1,1]
 => 1
[1,4,3,5,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,1]
 => 0
[1,4,5,2,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [2,1,1,1]
 => 1
[1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [6,4,3,3,2,1]
 => ? = 4
[1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [5,4,4,2,2,1]
 => ? = 2
[1,2,5,3,4,6,7] => [1,2,4,5,3,6,7] => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [6,5,3,2,2,1]
 => ? = 4
[1,2,5,3,4,7,6] => [1,2,4,5,3,7,6] => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [5,5,3,2,2,1]
 => ? = 2
[1,2,6,3,4,5,7] => [1,2,4,5,6,3,7] => [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [6,4,3,2,2,1]
 => ? = 4
[1,3,2,4,7,5,6] => [1,3,2,4,6,7,5] => [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [5,4,4,3,1,1]
 => ? = 2
[1,3,2,6,4,5,7] => [1,3,2,5,6,4,7] => [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [6,4,3,3,1,1]
 => ? = 2
[1,4,2,3,5,6,7] => [1,3,4,2,5,6,7] => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => [6,5,4,2,1,1]
 => ? = 4
[1,4,2,3,5,7,6] => [1,3,4,2,5,7,6] => [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
 => [5,5,4,2,1,1]
 => ? = 2
[1,4,2,3,6,5,7] => [1,3,4,2,6,5,7] => [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
 => [6,4,4,2,1,1]
 => ? = 2
Description
The number of singletons of an integer partition. A singleton in an integer partition is a part that appear precisely once.
Matching statistic: St000445
Mapping
Mp00066: Permutations —inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => [1,1,0,0]
 => 0
[1,2] => [1,2] => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[2,1] => [2,1] => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 0
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 2
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 0
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[2,3,1] => [3,1,2] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[3,1,2] => [2,3,1] => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 3
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1
[1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 0
[1,4,2,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 0
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 2
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 0
[2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[3,1,2,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 0
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[3,2,4,1] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[3,4,2,1] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[4,1,2,3] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
[4,1,3,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 0
[4,2,1,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[4,3,1,2] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 4
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 2
[1,2,4,5,3] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => 0
[1,3,4,2,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[1,3,5,2,4] => [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 1
[1,3,5,4,2] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[1,4,2,3,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 2
[1,4,2,5,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 0
[1,4,3,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 1
[1,4,3,5,2] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0
[1,4,5,2,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 1
[1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
 => ? = 4
[1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
 => ? = 2
[1,2,4,6,3,5,7] => [1,2,5,3,6,4,7] => [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 3
[1,2,4,6,3,7,5] => [1,2,5,3,7,4,6] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => ? = 1
[1,2,4,6,7,3,5] => [1,2,6,3,7,4,5] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 2
[1,2,4,7,3,6,5] => [1,2,5,3,7,6,4] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => ? = 1
[1,2,4,7,6,3,5] => [1,2,6,3,7,5,4] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 2
[1,2,5,3,4,7,6] => [1,2,4,5,3,7,6] => [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
 => ? = 2
[1,2,5,3,6,4,7] => [1,2,4,6,3,5,7] => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 2
[1,2,5,3,6,7,4] => [1,2,4,7,3,5,6] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 1
[1,2,5,3,7,4,6] => [1,2,4,6,3,7,5] => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => ? = 2
[1,2,5,3,7,6,4] => [1,2,4,7,3,6,5] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 1
[1,2,5,6,3,4,7] => [1,2,5,6,3,4,7] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 3
[1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 2
[1,2,6,3,5,4,7] => [1,2,4,6,5,3,7] => [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
 => ? = 2
[1,2,6,3,5,7,4] => [1,2,4,7,5,3,6] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 1
[1,2,6,3,7,5,4] => [1,2,4,7,6,3,5] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 1
[1,2,6,4,3,5,7] => [1,2,5,4,6,3,7] => [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
 => ? = 3
[1,2,6,4,3,7,5] => [1,2,5,4,7,3,6] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => ? = 1
[1,2,6,4,7,3,5] => [1,2,6,4,7,3,5] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 2
[1,2,6,5,3,4,7] => [1,2,5,6,4,3,7] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
 => ? = 3
[1,2,6,7,4,3,5] => [1,2,6,5,7,3,4] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 2
[1,2,7,3,4,6,5] => [1,2,4,5,7,6,3] => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => ? = 2
[1,2,7,3,5,4,6] => [1,2,4,6,5,7,3] => [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
 => ? = 2
[1,2,7,3,5,6,4] => [1,2,4,7,5,6,3] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 1
[1,2,7,3,6,5,4] => [1,2,4,7,6,5,3] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => ? = 1
[1,2,7,4,3,6,5] => [1,2,5,4,7,6,3] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
 => ? = 1
[1,2,7,4,6,3,5] => [1,2,6,4,7,5,3] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 2
[1,2,7,6,4,3,5] => [1,2,6,5,7,4,3] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => ? = 2
[1,3,2,4,7,5,6] => [1,3,2,4,6,7,5] => [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
 => ? = 2
[1,3,2,6,4,5,7] => [1,3,2,5,6,4,7] => [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,1,0,1,0,0,1,0,0]
 => ? = 2
[1,3,2,6,4,7,5] => [1,3,2,5,7,4,6] => [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => ? = 0
[1,3,2,7,4,5,6] => [1,3,2,5,6,7,4] => [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0]
 => ? = 2
[1,3,2,7,4,6,5] => [1,3,2,5,7,6,4] => [1,0,1,1,0,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
 => ? = 0
[1,3,4,2,7,5,6] => [1,4,2,3,6,7,5] => [1,0,1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,1,1,0,1,0,0,0]
 => ? = 1
[1,3,4,6,2,5,7] => [1,5,2,3,6,4,7] => [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,1,0,0,1,0,0]
 => ? = 2
[1,3,4,6,2,7,5] => [1,5,2,3,7,4,6] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => ? = 0
[1,3,4,7,2,6,5] => [1,5,2,3,7,6,4] => [1,0,1,1,1,1,0,0,0,1,1,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,1,1,0,0,0,0]
 => ? = 0
[1,3,5,2,4,6,7] => [1,4,2,5,3,6,7] => [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,1,0,0,1,0,1,0,0]
 => ? = 3
[1,3,5,2,4,7,6] => [1,4,2,5,3,7,6] => [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,1,1,0,0,0]
 => ? = 1
[1,3,5,2,6,4,7] => [1,4,2,6,3,5,7] => [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
 => ? = 1
[1,3,5,2,6,7,4] => [1,4,2,7,3,5,6] => [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => ? = 0
[1,3,5,2,7,6,4] => [1,4,2,7,3,6,5] => [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => ? = 0
[1,3,5,6,2,4,7] => [1,5,2,6,3,4,7] => [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,0,1,0,0]
 => ? = 2
[1,3,5,7,2,4,6] => [1,5,2,6,3,7,4] => [1,0,1,1,1,1,0,0,1,0,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,1,0,0,0]
 => ? = 2
[1,3,6,2,4,5,7] => [1,4,2,5,6,3,7] => [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,1,0,1,0,0,1,0,0]
 => ? = 3
[1,3,6,2,4,7,5] => [1,4,2,5,7,3,6] => [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
 => ? = 1
[1,3,6,2,5,4,7] => [1,4,2,6,5,3,7] => [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
 => ? = 1
[1,3,6,2,5,7,4] => [1,4,2,7,5,3,6] => [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => ? = 0
[1,3,6,2,7,4,5] => [1,4,2,6,7,3,5] => [1,0,1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,0,1,1,0,1,0,0,0,0]
 => ? = 1
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St000237
(load all 4 compositions to match this statistic)
Mapping
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => [2,1] => 1
[2,1] => [2,1] => [2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => [1,2,3] => [2,3,1] => 2
[1,3,2] => [1,3,2] => [1,3,2] => [3,2,1] => 0
[2,1,3] => [2,1,3] => [2,1,3] => [1,3,2] => 1
[2,3,1] => [3,1,2] => [3,1,2] => [3,1,2] => 0
[3,1,2] => [2,3,1] => [3,2,1] => [2,1,3] => 1
[3,2,1] => [3,2,1] => [2,3,1] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 3
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [2,4,3,1] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [3,2,4,1] => 1
[1,3,4,2] => [1,4,2,3] => [1,4,2,3] => [3,4,2,1] => 0
[1,4,2,3] => [1,3,4,2] => [1,4,3,2] => [4,3,2,1] => 1
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => [4,2,3,1] => 0
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [1,3,4,2] => 2
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [1,4,3,2] => 0
[2,3,1,4] => [3,1,2,4] => [3,1,2,4] => [3,1,4,2] => 1
[2,3,4,1] => [4,1,2,3] => [4,1,2,3] => [3,4,1,2] => 0
[2,4,1,3] => [3,4,1,2] => [4,1,3,2] => [4,3,1,2] => 1
[2,4,3,1] => [4,3,1,2] => [3,1,4,2] => [4,1,3,2] => 0
[3,1,2,4] => [2,3,1,4] => [3,2,1,4] => [2,1,4,3] => 2
[3,1,4,2] => [4,2,3,1] => [3,4,2,1] => [3,1,2,4] => 0
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => [1,2,4,3] => 1
[3,2,4,1] => [4,1,3,2] => [3,4,1,2] => [4,1,2,3] => 0
[3,4,1,2] => [2,4,1,3] => [4,2,1,3] => [2,4,1,3] => 1
[3,4,2,1] => [4,2,1,3] => [2,4,1,3] => [1,4,2,3] => 0
[4,1,2,3] => [2,3,4,1] => [4,2,3,1] => [2,3,1,4] => 2
[4,1,3,2] => [3,2,4,1] => [4,3,2,1] => [3,2,1,4] => 0
[4,2,1,3] => [3,4,2,1] => [2,4,3,1] => [1,3,2,4] => 1
[4,2,3,1] => [3,1,4,2] => [4,3,1,2] => [4,2,1,3] => 0
[4,3,1,2] => [2,4,3,1] => [3,2,4,1] => [2,1,3,4] => 1
[4,3,2,1] => [4,3,2,1] => [2,3,4,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 4
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => 2
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => 2
[1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,1] => 1
[1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,4,3] => [2,5,4,3,1] => 2
[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => [2,5,3,4,1] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,1] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,1] => 0
[1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,1] => 1
[1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => [3,4,5,2,1] => 0
[1,3,5,2,4] => [1,4,5,2,3] => [1,5,2,4,3] => [3,5,4,2,1] => 1
[1,3,5,4,2] => [1,5,4,2,3] => [1,4,2,5,3] => [3,5,2,4,1] => 0
[1,4,2,3,5] => [1,3,4,2,5] => [1,4,3,2,5] => [4,3,2,5,1] => 2
[1,4,2,5,3] => [1,5,3,4,2] => [1,4,5,3,2] => [5,4,2,3,1] => 0
[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [4,2,3,5,1] => 1
[1,4,3,5,2] => [1,5,2,4,3] => [1,4,5,2,3] => [4,5,2,3,1] => 0
[1,4,5,2,3] => [1,3,5,2,4] => [1,5,3,2,4] => [4,3,5,2,1] => 1
[1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [2,3,4,5,7,6,1] => ? = 4
[1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [2,3,4,6,5,7,1] => ? = 4
[1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [2,3,4,6,7,5,1] => ? = 3
[1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => [2,3,4,7,6,5,1] => ? = 4
[1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => [2,3,4,7,5,6,1] => ? = 3
[1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [2,3,5,4,6,7,1] => ? = 4
[1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [2,3,5,4,7,6,1] => ? = 2
[1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [2,3,5,6,4,7,1] => ? = 3
[1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [2,3,5,6,7,4,1] => ? = 2
[1,2,3,5,7,4,6] => [1,2,3,6,7,4,5] => [1,2,3,7,4,6,5] => [2,3,5,7,6,4,1] => ? = 3
[1,2,3,5,7,6,4] => [1,2,3,7,6,4,5] => [1,2,3,6,4,7,5] => [2,3,5,7,4,6,1] => ? = 2
[1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => [2,3,6,5,4,7,1] => ? = 4
[1,2,3,6,4,7,5] => [1,2,3,7,5,6,4] => [1,2,3,6,7,5,4] => [2,3,7,6,4,5,1] => ? = 2
[1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => [1,2,3,5,6,4,7] => [2,3,6,4,5,7,1] => ? = 3
[1,2,3,6,5,7,4] => [1,2,3,7,4,6,5] => [1,2,3,6,7,4,5] => [2,3,6,7,4,5,1] => ? = 2
[1,2,3,6,7,4,5] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [2,3,6,5,7,4,1] => ? = 3
[1,2,3,6,7,5,4] => [1,2,3,7,5,4,6] => [1,2,3,5,7,4,6] => [2,3,6,4,7,5,1] => ? = 2
[1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [1,2,3,7,5,6,4] => [2,3,7,5,6,4,1] => ? = 4
[1,2,3,7,4,6,5] => [1,2,3,6,5,7,4] => [1,2,3,7,6,5,4] => [2,3,7,6,5,4,1] => ? = 2
[1,2,3,7,5,4,6] => [1,2,3,6,7,5,4] => [1,2,3,5,7,6,4] => [2,3,7,4,6,5,1] => ? = 3
[1,2,3,7,5,6,4] => [1,2,3,6,4,7,5] => [1,2,3,7,6,4,5] => [2,3,6,7,5,4,1] => ? = 2
[1,2,3,7,6,4,5] => [1,2,3,5,7,6,4] => [1,2,3,6,5,7,4] => [2,3,7,5,4,6,1] => ? = 3
[1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => [1,2,3,5,6,7,4] => [2,3,7,4,5,6,1] => ? = 2
[1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [2,4,3,5,6,7,1] => ? = 4
[1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [2,4,3,5,7,6,1] => ? = 2
[1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [2,4,3,6,5,7,1] => ? = 2
[1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [2,4,3,6,7,5,1] => ? = 1
[1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => [1,2,4,3,7,6,5] => [2,4,3,7,6,5,1] => ? = 2
[1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => [1,2,4,3,6,7,5] => [2,4,3,7,5,6,1] => ? = 1
[1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [2,4,5,3,6,7,1] => ? = 3
[1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [2,4,5,3,7,6,1] => ? = 1
[1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => [1,2,6,3,4,5,7] => [2,4,5,6,3,7,1] => ? = 2
[1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => [2,4,5,6,7,3,1] => ? = 1
[1,2,4,5,7,3,6] => [1,2,6,7,3,4,5] => [1,2,7,3,4,6,5] => [2,4,5,7,6,3,1] => ? = 2
[1,2,4,5,7,6,3] => [1,2,7,6,3,4,5] => [1,2,6,3,4,7,5] => [2,4,5,7,3,6,1] => ? = 1
[1,2,4,6,3,5,7] => [1,2,5,6,3,4,7] => [1,2,6,3,5,4,7] => [2,4,6,5,3,7,1] => ? = 3
[1,2,4,6,3,7,5] => [1,2,7,5,6,3,4] => [1,2,6,3,7,5,4] => [2,4,7,6,3,5,1] => ? = 1
[1,2,4,6,5,3,7] => [1,2,6,5,3,4,7] => [1,2,5,3,6,4,7] => [2,4,6,3,5,7,1] => ? = 2
[1,2,4,6,5,7,3] => [1,2,7,3,4,6,5] => [1,2,6,3,7,4,5] => [2,4,6,7,3,5,1] => ? = 1
[1,2,4,6,7,3,5] => [1,2,5,7,3,4,6] => [1,2,7,3,5,4,6] => [2,4,6,5,7,3,1] => ? = 2
[1,2,4,6,7,5,3] => [1,2,7,5,3,4,6] => [1,2,5,3,7,4,6] => [2,4,6,3,7,5,1] => ? = 1
[1,2,4,7,3,5,6] => [1,2,5,6,7,3,4] => [1,2,7,3,5,6,4] => [2,4,7,5,6,3,1] => ? = 3
[1,2,4,7,3,6,5] => [1,2,6,5,7,3,4] => [1,2,7,3,6,5,4] => [2,4,7,6,5,3,1] => ? = 1
[1,2,4,7,5,3,6] => [1,2,6,7,5,3,4] => [1,2,5,3,7,6,4] => [2,4,7,3,6,5,1] => ? = 2
[1,2,4,7,5,6,3] => [1,2,6,3,4,7,5] => [1,2,7,3,6,4,5] => [2,4,6,7,5,3,1] => ? = 1
[1,2,4,7,6,3,5] => [1,2,5,7,6,3,4] => [1,2,6,3,5,7,4] => [2,4,7,5,3,6,1] => ? = 2
[1,2,4,7,6,5,3] => [1,2,7,6,5,3,4] => [1,2,5,3,6,7,4] => [2,4,7,3,5,6,1] => ? = 1
[1,2,5,3,4,6,7] => [1,2,4,5,3,6,7] => [1,2,5,4,3,6,7] => [2,5,4,3,6,7,1] => ? = 4
[1,2,5,3,4,7,6] => [1,2,4,5,3,7,6] => [1,2,5,4,3,7,6] => [2,5,4,3,7,6,1] => ? = 2
[1,2,5,3,6,4,7] => [1,2,6,4,5,3,7] => [1,2,5,6,4,3,7] => [2,6,5,3,4,7,1] => ? = 2
Description
The number of small exceedances. This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St001223
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001223: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1] => [1,0]
 => 0
[1,2] => [2,1] => [1,2] => [1,0,1,0]
 => 1
[2,1] => [1,2] => [2,1] => [1,1,0,0]
 => 0
[1,2,3] => [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
 => 2
[1,3,2] => [2,3,1] => [2,1,3] => [1,1,0,0,1,0]
 => 0
[2,1,3] => [3,1,2] => [1,3,2] => [1,0,1,1,0,0]
 => 1
[2,3,1] => [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
 => 0
[3,1,2] => [2,1,3] => [2,3,1] => [1,1,0,1,0,0]
 => 1
[3,2,1] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
 => 0
[1,2,3,4] => [4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 3
[1,2,4,3] => [3,4,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 1
[1,3,2,4] => [4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 1
[1,3,4,2] => [2,4,3,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 0
[1,4,2,3] => [3,2,4,1] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 1
[1,4,3,2] => [2,3,4,1] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 0
[2,1,3,4] => [4,3,1,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 2
[2,1,4,3] => [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 0
[2,3,1,4] => [4,1,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
 => 1
[2,3,4,1] => [1,4,3,2] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 0
[2,4,1,3] => [3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 1
[2,4,3,1] => [1,3,4,2] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 0
[3,1,2,4] => [4,2,1,3] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 2
[3,1,4,2] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
 => 0
[3,2,1,4] => [4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 1
[3,2,4,1] => [1,4,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
 => 0
[3,4,1,2] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 1
[3,4,2,1] => [1,2,4,3] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 0
[4,1,2,3] => [3,2,1,4] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 2
[4,1,3,2] => [2,3,1,4] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 0
[4,2,1,3] => [3,1,2,4] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 1
[4,2,3,1] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 0
[4,3,1,2] => [2,1,3,4] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => 1
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,3,4,5] => [5,4,3,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 4
[1,2,3,5,4] => [4,5,3,2,1] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 2
[1,2,4,3,5] => [5,3,4,2,1] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[1,2,4,5,3] => [3,5,4,2,1] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,2,5,3,4] => [4,3,5,2,1] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => 2
[1,2,5,4,3] => [3,4,5,2,1] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 1
[1,3,2,4,5] => [5,4,2,3,1] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => 2
[1,3,2,5,4] => [4,5,2,3,1] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => 0
[1,3,4,2,5] => [5,2,4,3,1] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,3,4,5,2] => [2,5,4,3,1] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,3,5,2,4] => [4,2,5,3,1] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => 1
[1,3,5,4,2] => [2,4,5,3,1] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,4,2,3,5] => [5,3,2,4,1] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => 2
[1,4,2,5,3] => [3,5,2,4,1] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
 => 0
[1,4,3,2,5] => [5,2,3,4,1] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
 => 1
[1,4,3,5,2] => [2,5,3,4,1] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => 0
[1,4,5,2,3] => [3,2,5,4,1] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => 1
[1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 6
[1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => [2,1,3,4,5,6,7] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 4
[1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => [1,3,2,4,5,6,7] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 4
[1,2,3,4,6,7,5] => [5,7,6,4,3,2,1] => [3,1,2,4,5,6,7] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 3
[1,2,3,4,7,5,6] => [6,5,7,4,3,2,1] => [2,3,1,4,5,6,7] => [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 4
[1,2,3,4,7,6,5] => [5,6,7,4,3,2,1] => [3,2,1,4,5,6,7] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 3
[1,2,3,5,4,6,7] => [7,6,4,5,3,2,1] => [1,2,4,3,5,6,7] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 4
[1,2,3,5,4,7,6] => [6,7,4,5,3,2,1] => [2,1,4,3,5,6,7] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 2
[1,2,3,5,6,4,7] => [7,4,6,5,3,2,1] => [1,4,2,3,5,6,7] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 3
[1,2,3,5,6,7,4] => [4,7,6,5,3,2,1] => [4,1,2,3,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 2
[1,2,3,5,7,4,6] => [6,4,7,5,3,2,1] => [2,4,1,3,5,6,7] => [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 3
[1,2,3,5,7,6,4] => [4,6,7,5,3,2,1] => [4,2,1,3,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 2
[1,2,3,6,4,5,7] => [7,5,4,6,3,2,1] => [1,3,4,2,5,6,7] => [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 4
[1,2,3,6,4,7,5] => [5,7,4,6,3,2,1] => [3,1,4,2,5,6,7] => [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => ? = 2
[1,2,3,6,5,4,7] => [7,4,5,6,3,2,1] => [1,4,3,2,5,6,7] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 3
[1,2,3,6,5,7,4] => [4,7,5,6,3,2,1] => [4,1,3,2,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 2
[1,2,3,6,7,4,5] => [5,4,7,6,3,2,1] => [3,4,1,2,5,6,7] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => ? = 3
[1,2,3,6,7,5,4] => [4,5,7,6,3,2,1] => [4,3,1,2,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 2
[1,2,3,7,4,5,6] => [6,5,4,7,3,2,1] => [2,3,4,1,5,6,7] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
 => ? = 4
[1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => [3,2,4,1,5,6,7] => [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => ? = 2
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => [2,4,3,1,5,6,7] => [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 3
[1,2,3,7,5,6,4] => [4,6,5,7,3,2,1] => [4,2,3,1,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 2
[1,2,3,7,6,4,5] => [5,4,6,7,3,2,1] => [3,4,2,1,5,6,7] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
 => ? = 3
[1,2,3,7,6,5,4] => [4,5,6,7,3,2,1] => [4,3,2,1,5,6,7] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => ? = 2
[1,2,4,3,5,6,7] => [7,6,5,3,4,2,1] => [1,2,3,5,4,6,7] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 4
[1,2,4,3,5,7,6] => [6,7,5,3,4,2,1] => [2,1,3,5,4,6,7] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 2
[1,2,4,3,6,5,7] => [7,5,6,3,4,2,1] => [1,3,2,5,4,6,7] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 2
[1,2,4,3,6,7,5] => [5,7,6,3,4,2,1] => [3,1,2,5,4,6,7] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 1
[1,2,4,3,7,5,6] => [6,5,7,3,4,2,1] => [2,3,1,5,4,6,7] => [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 2
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => [3,2,1,5,4,6,7] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 1
[1,2,4,5,3,6,7] => [7,6,3,5,4,2,1] => [1,2,5,3,4,6,7] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3
[1,2,4,5,3,7,6] => [6,7,3,5,4,2,1] => [2,1,5,3,4,6,7] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[1,2,4,5,6,3,7] => [7,3,6,5,4,2,1] => [1,5,2,3,4,6,7] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
[1,2,4,5,6,7,3] => [3,7,6,5,4,2,1] => [5,1,2,3,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 1
[1,2,4,5,7,3,6] => [6,3,7,5,4,2,1] => [2,5,1,3,4,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
[1,2,4,5,7,6,3] => [3,6,7,5,4,2,1] => [5,2,1,3,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 1
[1,2,4,6,3,5,7] => [7,5,3,6,4,2,1] => [1,3,5,2,4,6,7] => [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
 => ? = 3
[1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => [3,1,5,2,4,6,7] => [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
 => ? = 1
[1,2,4,6,5,3,7] => [7,3,5,6,4,2,1] => [1,5,3,2,4,6,7] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
[1,2,4,6,5,7,3] => [3,7,5,6,4,2,1] => [5,1,3,2,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 1
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => [3,5,1,2,4,6,7] => [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
[1,2,4,6,7,5,3] => [3,5,7,6,4,2,1] => [5,3,1,2,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 1
[1,2,4,7,3,5,6] => [6,5,3,7,4,2,1] => [2,3,5,1,4,6,7] => [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
 => ? = 3
[1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => [3,2,5,1,4,6,7] => [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
 => ? = 1
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => [2,5,3,1,4,6,7] => [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
[1,2,4,7,5,6,3] => [3,6,5,7,4,2,1] => [5,2,3,1,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 1
[1,2,4,7,6,3,5] => [5,3,6,7,4,2,1] => [3,5,2,1,4,6,7] => [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
[1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => [5,3,2,1,4,6,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => ? = 1
[1,2,5,3,4,6,7] => [7,6,4,3,5,2,1] => [1,2,4,5,3,6,7] => [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => ? = 4
[1,2,5,3,4,7,6] => [6,7,4,3,5,2,1] => [2,1,4,5,3,6,7] => [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
 => ? = 2
Description
Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless.
Matching statistic: St001491
(load all 2 compositions to match this statistic)
Mapping
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 14%
Values
[1] => [1] => [1] => => ? = 0 + 1
[1,2] => [2,1] => [2,1] => 0 => ? = 1 + 1
[2,1] => [1,2] => [1,2] => 1 => 1 = 0 + 1
[1,2,3] => [2,3,1] => [2,3,1] => 00 => ? = 2 + 1
[1,3,2] => [2,1,3] => [2,1,3] => 01 => 1 = 0 + 1
[2,1,3] => [3,2,1] => [3,2,1] => 00 => ? = 1 + 1
[2,3,1] => [1,2,3] => [1,3,2] => 10 => 1 = 0 + 1
[3,1,2] => [3,1,2] => [3,1,2] => 00 => ? = 1 + 1
[3,2,1] => [1,3,2] => [1,3,2] => 10 => 1 = 0 + 1
[1,2,3,4] => [2,3,4,1] => [2,4,3,1] => 000 => ? = 3 + 1
[1,2,4,3] => [2,3,1,4] => [2,4,1,3] => 000 => ? = 1 + 1
[1,3,2,4] => [2,4,3,1] => [2,4,3,1] => 000 => ? = 1 + 1
[1,3,4,2] => [2,1,3,4] => [2,1,4,3] => 010 => 1 = 0 + 1
[1,4,2,3] => [2,4,1,3] => [2,4,1,3] => 000 => ? = 1 + 1
[1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 010 => 1 = 0 + 1
[2,1,3,4] => [3,2,4,1] => [3,2,4,1] => 000 => ? = 2 + 1
[2,1,4,3] => [3,2,1,4] => [3,2,1,4] => 001 => 1 = 0 + 1
[2,3,1,4] => [4,2,3,1] => [4,2,3,1] => 000 => ? = 1 + 1
[2,3,4,1] => [1,2,3,4] => [1,4,3,2] => 100 => 1 = 0 + 1
[2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 000 => ? = 1 + 1
[2,4,3,1] => [1,2,4,3] => [1,4,3,2] => 100 => 1 = 0 + 1
[3,1,2,4] => [3,4,2,1] => [3,4,2,1] => 000 => ? = 2 + 1
[3,1,4,2] => [3,1,2,4] => [3,1,4,2] => 000 => ? = 0 + 1
[3,2,1,4] => [4,3,2,1] => [4,3,2,1] => 000 => ? = 1 + 1
[3,2,4,1] => [1,3,2,4] => [1,4,3,2] => 100 => 1 = 0 + 1
[3,4,1,2] => [4,1,2,3] => [4,1,3,2] => 000 => ? = 1 + 1
[3,4,2,1] => [1,4,2,3] => [1,4,3,2] => 100 => 1 = 0 + 1
[4,1,2,3] => [3,4,1,2] => [3,4,1,2] => 000 => ? = 2 + 1
[4,1,3,2] => [3,1,4,2] => [3,1,4,2] => 000 => ? = 0 + 1
[4,2,1,3] => [4,3,1,2] => [4,3,1,2] => 000 => ? = 1 + 1
[4,2,3,1] => [1,3,4,2] => [1,4,3,2] => 100 => 1 = 0 + 1
[4,3,1,2] => [4,1,3,2] => [4,1,3,2] => 000 => ? = 1 + 1
[4,3,2,1] => [1,4,3,2] => [1,4,3,2] => 100 => 1 = 0 + 1
[1,2,3,4,5] => [2,3,4,5,1] => [2,5,4,3,1] => 0000 => ? = 4 + 1
[1,2,3,5,4] => [2,3,4,1,5] => [2,5,4,1,3] => 0000 => ? = 2 + 1
[1,2,4,3,5] => [2,3,5,4,1] => [2,5,4,3,1] => 0000 => ? = 2 + 1
[1,2,4,5,3] => [2,3,1,4,5] => [2,5,1,4,3] => 0000 => ? = 1 + 1
[1,2,5,3,4] => [2,3,5,1,4] => [2,5,4,1,3] => 0000 => ? = 2 + 1
[1,2,5,4,3] => [2,3,1,5,4] => [2,5,1,4,3] => 0000 => ? = 1 + 1
[1,3,2,4,5] => [2,4,3,5,1] => [2,5,4,3,1] => 0000 => ? = 2 + 1
[1,3,2,5,4] => [2,4,3,1,5] => [2,5,4,1,3] => 0000 => ? = 0 + 1
[1,3,4,2,5] => [2,5,3,4,1] => [2,5,4,3,1] => 0000 => ? = 1 + 1
[1,3,4,5,2] => [2,1,3,4,5] => [2,1,5,4,3] => 0100 => 1 = 0 + 1
[1,3,5,2,4] => [2,5,3,1,4] => [2,5,4,1,3] => 0000 => ? = 1 + 1
[1,3,5,4,2] => [2,1,3,5,4] => [2,1,5,4,3] => 0100 => 1 = 0 + 1
[1,4,2,3,5] => [2,4,5,3,1] => [2,5,4,3,1] => 0000 => ? = 2 + 1
[1,4,2,5,3] => [2,4,1,3,5] => [2,5,1,4,3] => 0000 => ? = 0 + 1
[1,4,3,2,5] => [2,5,4,3,1] => [2,5,4,3,1] => 0000 => ? = 1 + 1
[1,4,3,5,2] => [2,1,4,3,5] => [2,1,5,4,3] => 0100 => 1 = 0 + 1
[1,4,5,2,3] => [2,5,1,3,4] => [2,5,1,4,3] => 0000 => ? = 1 + 1
[1,4,5,3,2] => [2,1,5,3,4] => [2,1,5,4,3] => 0100 => 1 = 0 + 1
[1,5,2,3,4] => [2,4,5,1,3] => [2,5,4,1,3] => 0000 => ? = 2 + 1
[1,5,2,4,3] => [2,4,1,5,3] => [2,5,1,4,3] => 0000 => ? = 0 + 1
[1,5,3,2,4] => [2,5,4,1,3] => [2,5,4,1,3] => 0000 => ? = 1 + 1
[1,5,3,4,2] => [2,1,4,5,3] => [2,1,5,4,3] => 0100 => 1 = 0 + 1
[1,5,4,2,3] => [2,5,1,4,3] => [2,5,1,4,3] => 0000 => ? = 1 + 1
[1,5,4,3,2] => [2,1,5,4,3] => [2,1,5,4,3] => 0100 => 1 = 0 + 1
[2,1,3,4,5] => [3,2,4,5,1] => [3,2,5,4,1] => 0000 => ? = 3 + 1
[2,1,3,5,4] => [3,2,4,1,5] => [3,2,5,1,4] => 0000 => ? = 1 + 1
[2,1,4,3,5] => [3,2,5,4,1] => [3,2,5,4,1] => 0000 => ? = 1 + 1
[2,1,4,5,3] => [3,2,1,4,5] => [3,2,1,5,4] => 0010 => 1 = 0 + 1
[2,1,5,3,4] => [3,2,5,1,4] => [3,2,5,1,4] => 0000 => ? = 1 + 1
[2,1,5,4,3] => [3,2,1,5,4] => [3,2,1,5,4] => 0010 => 1 = 0 + 1
[2,3,1,4,5] => [4,2,3,5,1] => [4,2,5,3,1] => 0000 => ? = 2 + 1
[2,3,1,5,4] => [4,2,3,1,5] => [4,2,5,1,3] => 0000 => ? = 0 + 1
[2,3,4,1,5] => [5,2,3,4,1] => [5,2,4,3,1] => 0000 => ? = 1 + 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[2,3,5,1,4] => [5,2,3,1,4] => [5,2,4,1,3] => 0000 => ? = 1 + 1
[2,3,5,4,1] => [1,2,3,5,4] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[2,4,1,3,5] => [4,2,5,3,1] => [4,2,5,3,1] => 0000 => ? = 2 + 1
[2,4,1,5,3] => [4,2,1,3,5] => [4,2,1,5,3] => 0000 => ? = 0 + 1
[2,4,3,1,5] => [5,2,4,3,1] => [5,2,4,3,1] => 0000 => ? = 1 + 1
[2,4,3,5,1] => [1,2,4,3,5] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[2,4,5,1,3] => [5,2,1,3,4] => [5,2,1,4,3] => 0000 => ? = 1 + 1
[2,4,5,3,1] => [1,2,5,3,4] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[2,5,3,4,1] => [1,2,4,5,3] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[2,5,4,3,1] => [1,2,5,4,3] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[3,2,1,5,4] => [4,3,2,1,5] => [4,3,2,1,5] => 0001 => 1 = 0 + 1
[3,2,4,5,1] => [1,3,2,4,5] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[3,2,5,4,1] => [1,3,2,5,4] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[3,4,2,5,1] => [1,4,2,3,5] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[3,4,5,2,1] => [1,5,2,3,4] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[3,5,2,4,1] => [1,4,2,5,3] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[3,5,4,2,1] => [1,5,2,4,3] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[4,2,3,5,1] => [1,3,4,2,5] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[4,2,5,3,1] => [1,3,5,2,4] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[4,3,2,5,1] => [1,4,3,2,5] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[4,3,5,2,1] => [1,5,3,2,4] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[4,5,2,3,1] => [1,4,5,2,3] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[4,5,3,2,1] => [1,5,4,2,3] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[5,2,3,4,1] => [1,3,4,5,2] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[5,2,4,3,1] => [1,3,5,4,2] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[5,3,2,4,1] => [1,4,3,5,2] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[5,3,4,2,1] => [1,5,3,4,2] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[5,4,2,3,1] => [1,4,5,3,2] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
[5,4,3,2,1] => [1,5,4,3,2] => [1,5,4,3,2] => 1000 => 1 = 0 + 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let $A_n=K[x]/(x^n)$. We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.