Processing math: 100%

Your data matches 16 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001657
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
 => 0
[1,2] => [2] => [2]
 => 1
[2,1] => [1,1] => [1,1]
 => 0
[1,2,3] => [3] => [3]
 => 0
[1,3,2] => [2,1] => [2,1]
 => 1
[2,1,3] => [1,2] => [2,1]
 => 1
[2,3,1] => [2,1] => [2,1]
 => 1
[3,1,2] => [1,2] => [2,1]
 => 1
[3,2,1] => [1,1,1] => [1,1,1]
 => 0
[1,2,3,4] => [4] => [4]
 => 0
[1,2,4,3] => [3,1] => [3,1]
 => 0
[1,3,2,4] => [2,2] => [2,2]
 => 2
[1,3,4,2] => [3,1] => [3,1]
 => 0
[1,4,2,3] => [2,2] => [2,2]
 => 2
[1,4,3,2] => [2,1,1] => [2,1,1]
 => 1
[2,1,3,4] => [1,3] => [3,1]
 => 0
[2,1,4,3] => [1,2,1] => [2,1,1]
 => 1
[2,3,1,4] => [2,2] => [2,2]
 => 2
[2,3,4,1] => [3,1] => [3,1]
 => 0
[2,4,1,3] => [2,2] => [2,2]
 => 2
[2,4,3,1] => [2,1,1] => [2,1,1]
 => 1
[3,1,2,4] => [1,3] => [3,1]
 => 0
[3,1,4,2] => [1,2,1] => [2,1,1]
 => 1
[3,2,1,4] => [1,1,2] => [2,1,1]
 => 1
[3,2,4,1] => [1,2,1] => [2,1,1]
 => 1
[3,4,1,2] => [2,2] => [2,2]
 => 2
[3,4,2,1] => [2,1,1] => [2,1,1]
 => 1
[4,1,2,3] => [1,3] => [3,1]
 => 0
[4,1,3,2] => [1,2,1] => [2,1,1]
 => 1
[4,2,1,3] => [1,1,2] => [2,1,1]
 => 1
[4,2,3,1] => [1,2,1] => [2,1,1]
 => 1
[4,3,1,2] => [1,1,2] => [2,1,1]
 => 1
[4,3,2,1] => [1,1,1,1] => [1,1,1,1]
 => 0
[1,2,3,4,5] => [5] => [5]
 => 0
[1,2,3,5,4] => [4,1] => [4,1]
 => 0
[1,2,4,3,5] => [3,2] => [3,2]
 => 1
[1,2,4,5,3] => [4,1] => [4,1]
 => 0
[1,2,5,3,4] => [3,2] => [3,2]
 => 1
[1,2,5,4,3] => [3,1,1] => [3,1,1]
 => 0
[1,3,2,4,5] => [2,3] => [3,2]
 => 1
[1,3,2,5,4] => [2,2,1] => [2,2,1]
 => 2
[1,3,4,2,5] => [3,2] => [3,2]
 => 1
[1,3,4,5,2] => [4,1] => [4,1]
 => 0
[1,3,5,2,4] => [3,2] => [3,2]
 => 1
[1,3,5,4,2] => [3,1,1] => [3,1,1]
 => 0
[1,4,2,3,5] => [2,3] => [3,2]
 => 1
[1,4,2,5,3] => [2,2,1] => [2,2,1]
 => 2
[1,4,3,2,5] => [2,1,2] => [2,2,1]
 => 2
[1,4,3,5,2] => [2,2,1] => [2,2,1]
 => 2
[1,4,5,2,3] => [3,2] => [3,2]
 => 1
Description
The number of twos in an integer partition. The total number of twos in all partitions of n is equal to the total number of singletons [[St001484]] in all partitions of n−1, see [1].
Matching statistic: St000658
(load all 11 compositions to match this statistic)
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000658: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 71%
Values
[1] => [1] => [1] => [1,0]
 => ? = 0
[1,2] => [2] => [2] => [1,1,0,0]
 => 1
[2,1] => [1,1] => [1,1] => [1,0,1,0]
 => 0
[1,2,3] => [3] => [3] => [1,1,1,0,0,0]
 => 0
[1,3,2] => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,3] => [1,2] => [2,1] => [1,1,0,0,1,0]
 => 1
[2,3,1] => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 1
[3,1,2] => [1,2] => [2,1] => [1,1,0,0,1,0]
 => 1
[3,2,1] => [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[1,2,3,4] => [4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 0
[1,3,2,4] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,3,4,2] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 0
[1,4,2,3] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,4,3,2] => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[2,1,3,4] => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[2,1,4,3] => [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[2,3,1,4] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[2,3,4,1] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 0
[2,4,1,3] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[2,4,3,1] => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[3,1,2,4] => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[3,1,4,2] => [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[3,2,1,4] => [1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[3,2,4,1] => [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[3,4,1,2] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[3,4,2,1] => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[4,1,2,3] => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[4,1,3,2] => [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[4,2,1,3] => [1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[4,2,3,1] => [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[4,3,1,2] => [1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[4,3,2,1] => [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[1,2,3,4,5] => [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,5,4] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3,5] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,2,4,5,3] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,2,5,3,4] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,2,5,4,3] => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,3,2,4,5] => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,3,2,5,4] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,3,4,2,5] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,3,4,5,2] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,3,5,2,4] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,3,5,4,2] => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,4,2,3,5] => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,4,2,5,3] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,4,3,2,5] => [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,4,3,5,2] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,4,5,2,3] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,4,5,3,2] => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 0
[8,7,6,5,4,1,2,3] => [1,1,1,1,1,3] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[7,8,6,5,4,1,2,3] => [2,1,1,1,3] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[7,6,8,5,4,1,2,3] => [1,2,1,1,3] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1
[6,7,8,5,4,1,2,3] => [3,1,1,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 0
[7,8,5,6,4,1,2,3] => [2,2,1,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2
[8,6,5,7,4,1,2,3] => [1,1,2,1,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1
[8,5,6,7,4,1,2,3] => [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 0
[7,6,5,8,4,1,2,3] => [1,1,2,1,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1
[6,7,5,8,4,1,2,3] => [2,2,1,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2
[7,5,6,8,4,1,2,3] => [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 0
[6,5,7,8,4,1,2,3] => [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 0
[5,6,7,8,4,1,2,3] => [4,1,3] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 0
[8,7,6,4,5,1,2,3] => [1,1,1,2,3] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[7,8,6,4,5,1,2,3] => [2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2
[8,6,7,4,5,1,2,3] => [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[6,7,8,4,5,1,2,3] => [3,2,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 1
[8,7,5,4,6,1,2,3] => [1,1,1,2,3] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[7,8,5,4,6,1,2,3] => [2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2
[8,7,4,5,6,1,2,3] => [1,1,3,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 0
[7,8,4,5,6,1,2,3] => [2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1
[8,6,5,4,7,1,2,3] => [1,1,1,2,3] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[8,5,6,4,7,1,2,3] => [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[8,6,4,5,7,1,2,3] => [1,1,3,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 0
[8,5,4,6,7,1,2,3] => [1,1,3,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 0
[8,4,5,6,7,1,2,3] => [1,4,3] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 0
[7,6,5,4,8,1,2,3] => [1,1,1,2,3] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[6,7,5,4,8,1,2,3] => [2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2
[7,5,6,4,8,1,2,3] => [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[6,5,7,4,8,1,2,3] => [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[5,6,7,4,8,1,2,3] => [3,2,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 1
[7,6,4,5,8,1,2,3] => [1,1,3,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 0
[6,7,4,5,8,1,2,3] => [2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1
[7,5,4,6,8,1,2,3] => [1,1,3,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 0
[7,4,5,6,8,1,2,3] => [1,4,3] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 0
[6,5,4,7,8,1,2,3] => [1,1,3,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 0
[5,6,4,7,8,1,2,3] => [2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1
[6,4,5,7,8,1,2,3] => [1,4,3] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 0
[5,4,6,7,8,1,2,3] => [1,4,3] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 0
[4,5,6,7,8,1,2,3] => [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[8,7,6,5,3,1,2,4] => [1,1,1,1,1,3] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[7,8,6,5,3,1,2,4] => [2,1,1,1,3] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[8,6,7,5,3,1,2,4] => [1,2,1,1,3] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1
[7,6,8,5,3,1,2,4] => [1,2,1,1,3] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1
[6,7,8,5,3,1,2,4] => [3,1,1,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 0
[8,7,5,6,3,1,2,4] => [1,1,2,1,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1
[7,8,5,6,3,1,2,4] => [2,2,1,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2
[8,6,5,7,3,1,2,4] => [1,1,2,1,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1
[8,5,6,7,3,1,2,4] => [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 0
[7,6,5,8,3,1,2,4] => [1,1,2,1,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1
Description
The number of rises of length 2 of a Dyck path. This is also the number of (1,1) steps of the associated Łukasiewicz path, see [1]. A related statistic is the number of double rises in a Dyck path, [[St000024]].
Matching statistic: St001139
(load all 6 compositions to match this statistic)
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001139: Dyck paths ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 71%
Values
[1] => [1] => [1] => [1,0]
 => ? = 0
[1,2] => [2] => [2] => [1,1,0,0]
 => 1
[2,1] => [1,1] => [1,1] => [1,0,1,0]
 => 0
[1,2,3] => [3] => [3] => [1,1,1,0,0,0]
 => 0
[1,3,2] => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,3] => [1,2] => [2,1] => [1,1,0,0,1,0]
 => 1
[2,3,1] => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 1
[3,1,2] => [1,2] => [2,1] => [1,1,0,0,1,0]
 => 1
[3,2,1] => [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[1,2,3,4] => [4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 0
[1,3,2,4] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,3,4,2] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 0
[1,4,2,3] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,4,3,2] => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[2,1,3,4] => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[2,1,4,3] => [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[2,3,1,4] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[2,3,4,1] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 0
[2,4,1,3] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[2,4,3,1] => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[3,1,2,4] => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[3,1,4,2] => [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[3,2,1,4] => [1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[3,2,4,1] => [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[3,4,1,2] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[3,4,2,1] => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[4,1,2,3] => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 0
[4,1,3,2] => [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[4,2,1,3] => [1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[4,2,3,1] => [1,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[4,3,1,2] => [1,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1
[4,3,2,1] => [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[1,2,3,4,5] => [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,5,4] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3,5] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,2,4,5,3] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,2,5,3,4] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,2,5,4,3] => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,3,2,4,5] => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,3,2,5,4] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,3,4,2,5] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,3,4,5,2] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,3,5,2,4] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,3,5,4,2] => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 0
[1,4,2,3,5] => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,4,2,5,3] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,4,3,2,5] => [2,1,2] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,4,3,5,2] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,4,5,2,3] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,4,5,3,2] => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 0
[8,7,6,5,4,1,2,3] => [1,1,1,1,1,3] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[7,8,6,5,4,1,2,3] => [2,1,1,1,3] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[7,6,8,5,4,1,2,3] => [1,2,1,1,3] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1
[6,7,8,5,4,1,2,3] => [3,1,1,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 0
[7,8,5,6,4,1,2,3] => [2,2,1,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2
[8,6,5,7,4,1,2,3] => [1,1,2,1,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1
[8,5,6,7,4,1,2,3] => [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 0
[7,6,5,8,4,1,2,3] => [1,1,2,1,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1
[6,7,5,8,4,1,2,3] => [2,2,1,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2
[7,5,6,8,4,1,2,3] => [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 0
[6,5,7,8,4,1,2,3] => [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 0
[5,6,7,8,4,1,2,3] => [4,1,3] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 0
[8,7,6,4,5,1,2,3] => [1,1,1,2,3] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[7,8,6,4,5,1,2,3] => [2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2
[8,6,7,4,5,1,2,3] => [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[6,7,8,4,5,1,2,3] => [3,2,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 1
[8,7,5,4,6,1,2,3] => [1,1,1,2,3] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[7,8,5,4,6,1,2,3] => [2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2
[8,7,4,5,6,1,2,3] => [1,1,3,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 0
[7,8,4,5,6,1,2,3] => [2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1
[8,6,5,4,7,1,2,3] => [1,1,1,2,3] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[8,5,6,4,7,1,2,3] => [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[8,6,4,5,7,1,2,3] => [1,1,3,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 0
[8,5,4,6,7,1,2,3] => [1,1,3,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 0
[8,4,5,6,7,1,2,3] => [1,4,3] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 0
[7,6,5,4,8,1,2,3] => [1,1,1,2,3] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 1
[6,7,5,4,8,1,2,3] => [2,1,2,3] => [3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
 => ? = 2
[7,5,6,4,8,1,2,3] => [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[6,5,7,4,8,1,2,3] => [1,2,2,3] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 2
[5,6,7,4,8,1,2,3] => [3,2,3] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 1
[7,6,4,5,8,1,2,3] => [1,1,3,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 0
[6,7,4,5,8,1,2,3] => [2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1
[7,5,4,6,8,1,2,3] => [1,1,3,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 0
[7,4,5,6,8,1,2,3] => [1,4,3] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 0
[6,5,4,7,8,1,2,3] => [1,1,3,3] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 0
[5,6,4,7,8,1,2,3] => [2,3,3] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 1
[6,4,5,7,8,1,2,3] => [1,4,3] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 0
[5,4,6,7,8,1,2,3] => [1,4,3] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 0
[4,5,6,7,8,1,2,3] => [5,3] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 0
[8,7,6,5,3,1,2,4] => [1,1,1,1,1,3] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 0
[7,8,6,5,3,1,2,4] => [2,1,1,1,3] => [3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
[8,6,7,5,3,1,2,4] => [1,2,1,1,3] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1
[7,6,8,5,3,1,2,4] => [1,2,1,1,3] => [3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 1
[6,7,8,5,3,1,2,4] => [3,1,1,3] => [3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 0
[8,7,5,6,3,1,2,4] => [1,1,2,1,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1
[7,8,5,6,3,1,2,4] => [2,2,1,3] => [3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2
[8,6,5,7,3,1,2,4] => [1,1,2,1,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1
[8,5,6,7,3,1,2,4] => [1,3,1,3] => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 0
[7,6,5,8,3,1,2,4] => [1,1,2,1,3] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 1
Description
The number of occurrences of hills of size 2 in a Dyck path. A hill of size two is a subpath beginning at height zero, consisting of two up steps followed by two down steps.
Matching statistic: St000011
(load all 3 compositions to match this statistic)
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1,0]
 => [1,0]
 => 1 = 0 + 1
[1,2] => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
[2,1] => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 0 + 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1 = 0 + 1
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1 = 0 + 1
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 0 + 1
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1 = 0 + 1
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1 = 0 + 1
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[6,7,8,4,5,3,2,1] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 1
[7,8,4,5,6,3,2,1] => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[5,6,7,4,8,3,2,1] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1 + 1
[6,7,4,5,8,3,2,1] => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[5,6,4,7,8,3,2,1] => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[7,8,6,5,3,4,2,1] => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[8,6,7,5,3,4,2,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[7,6,8,5,3,4,2,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[6,7,8,5,3,4,2,1] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[8,7,5,6,3,4,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[7,8,5,6,3,4,2,1] => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3 + 1
[8,6,5,7,3,4,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[8,5,6,7,3,4,2,1] => [1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[7,6,5,8,3,4,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[6,7,5,8,3,4,2,1] => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3 + 1
[6,5,7,8,3,4,2,1] => [1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[5,6,7,8,3,4,2,1] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[7,8,6,4,3,5,2,1] => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[8,6,7,4,3,5,2,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[7,6,8,4,3,5,2,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[6,7,8,4,3,5,2,1] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[7,8,6,3,4,5,2,1] => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
[8,6,7,3,4,5,2,1] => [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
[7,6,8,3,4,5,2,1] => [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
[7,8,5,4,3,6,2,1] => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[8,7,4,5,3,6,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[7,8,5,3,4,6,2,1] => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
[7,8,4,3,5,6,2,1] => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
[7,8,3,4,5,6,2,1] => [2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[8,5,6,4,3,7,2,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[8,6,4,5,3,7,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[8,5,4,6,3,7,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[8,5,6,3,4,7,2,1] => [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
[6,7,5,4,3,8,2,1] => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[7,5,6,4,3,8,2,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[6,5,7,4,3,8,2,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[5,6,7,4,3,8,2,1] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[7,6,4,5,3,8,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[6,7,4,5,3,8,2,1] => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3 + 1
[7,5,4,6,3,8,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[7,4,5,6,3,8,2,1] => [1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[6,5,4,7,3,8,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2 + 1
[5,6,4,7,3,8,2,1] => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3 + 1
[6,4,5,7,3,8,2,1] => [1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[5,4,6,7,3,8,2,1] => [1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[4,5,6,7,3,8,2,1] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 1 + 1
[6,7,5,3,4,8,2,1] => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
[6,5,7,3,4,8,2,1] => [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
[5,6,4,3,7,8,2,1] => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
[5,4,6,3,7,8,2,1] => [1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => ? = 1 + 1
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: St001465
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => [1] => 0
[1,2] => [2] => [1,1,0,0]
 => [2,1] => 1
[2,1] => [1,1] => [1,0,1,0]
 => [1,2] => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [3,1,2] => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [2,1,3] => 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [1,3,2] => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [2,1,3] => 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [1,3,2] => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,2,3] => 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 0
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 0
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 0
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 0
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 0
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 0
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => 0
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => 0
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => 0
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => 0
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => 0
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 2
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => 1
[1,2,3,5,4,6,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 0
[1,2,3,5,4,7,6] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,3,6,4,5,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 0
[1,2,3,6,4,7,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,3,6,5,7,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,3,7,4,5,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 0
[1,2,3,7,4,6,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,3,7,5,6,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,4,3,5,6,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,1,2,7,4,5,6] => ? = 0
[1,2,4,3,5,7,6] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 0
[1,2,4,3,6,5,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,4,3,6,7,5] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 0
[1,2,4,3,7,5,6] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,4,3,7,6,5] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [3,1,2,5,4,6,7] => ? = 1
[1,2,4,5,3,6,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 0
[1,2,4,5,3,7,6] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,4,6,3,5,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 0
[1,2,4,6,3,7,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,4,6,5,7,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,4,7,3,5,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 0
[1,2,4,7,3,6,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,4,7,5,6,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,5,3,4,6,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,1,2,7,4,5,6] => ? = 0
[1,2,5,3,4,7,6] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 0
[1,2,5,3,6,4,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,5,3,6,7,4] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 0
[1,2,5,3,7,4,6] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,5,3,7,6,4] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [3,1,2,5,4,6,7] => ? = 1
[1,2,5,4,3,6,7] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [3,1,2,4,7,5,6] => ? = 0
[1,2,5,4,3,7,6] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [3,1,2,4,6,5,7] => ? = 1
[1,2,5,4,6,3,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,5,4,6,7,3] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 0
[1,2,5,4,7,3,6] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,5,4,7,6,3] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [3,1,2,5,4,6,7] => ? = 1
[1,2,5,6,3,4,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 0
[1,2,5,6,3,7,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,5,6,4,7,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,5,7,3,4,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 0
[1,2,5,7,3,6,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,5,7,4,6,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,6,3,4,5,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,1,2,7,4,5,6] => ? = 0
[1,2,6,3,4,7,5] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 0
[1,2,6,3,5,4,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,6,3,5,7,4] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 0
[1,2,6,3,7,4,5] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,6,3,7,5,4] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [3,1,2,5,4,6,7] => ? = 1
[1,2,6,4,3,5,7] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [3,1,2,4,7,5,6] => ? = 0
[1,2,6,4,3,7,5] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [3,1,2,4,6,5,7] => ? = 1
[1,2,6,4,5,3,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,6,4,5,7,3] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 0
Description
The number of adjacent transpositions in the cycle decomposition of a permutation.
Matching statistic: St000214
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => [1] => 0
[1,2] => [2] => [1,1,0,0]
 => [2,1] => 1
[2,1] => [1,1] => [1,0,1,0]
 => [1,2] => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [3,1,2] => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [2,1,3] => 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [1,3,2] => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [2,1,3] => 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [1,3,2] => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,2,3] => 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 0
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 0
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 0
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 0
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 0
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 0
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => 0
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => 0
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => 0
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => 0
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => 0
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 2
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => 1
[1,2,3,5,4,6,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 0
[1,2,3,5,4,7,6] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,3,6,4,5,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 0
[1,2,3,6,4,7,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,3,6,5,7,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,3,7,4,5,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 0
[1,2,3,7,4,6,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,3,7,5,6,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,4,3,5,6,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,1,2,7,4,5,6] => ? = 0
[1,2,4,3,5,7,6] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 0
[1,2,4,3,6,5,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,4,3,6,7,5] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 0
[1,2,4,3,7,5,6] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,4,3,7,6,5] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [3,1,2,5,4,6,7] => ? = 1
[1,2,4,5,3,6,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 0
[1,2,4,5,3,7,6] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,4,6,3,5,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 0
[1,2,4,6,3,7,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,4,6,5,7,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,4,7,3,5,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 0
[1,2,4,7,3,6,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,4,7,5,6,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,5,3,4,6,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,1,2,7,4,5,6] => ? = 0
[1,2,5,3,4,7,6] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 0
[1,2,5,3,6,4,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,5,3,6,7,4] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 0
[1,2,5,3,7,4,6] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,5,3,7,6,4] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [3,1,2,5,4,6,7] => ? = 1
[1,2,5,4,3,6,7] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [3,1,2,4,7,5,6] => ? = 0
[1,2,5,4,3,7,6] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [3,1,2,4,6,5,7] => ? = 1
[1,2,5,4,6,3,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,5,4,6,7,3] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 0
[1,2,5,4,7,3,6] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,5,4,7,6,3] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [3,1,2,5,4,6,7] => ? = 1
[1,2,5,6,3,4,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 0
[1,2,5,6,3,7,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,5,6,4,7,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,5,7,3,4,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 0
[1,2,5,7,3,6,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,5,7,4,6,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,6,3,4,5,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,1,2,7,4,5,6] => ? = 0
[1,2,6,3,4,7,5] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 0
[1,2,6,3,5,4,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,6,3,5,7,4] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 0
[1,2,6,3,7,4,5] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,6,3,7,5,4] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [3,1,2,5,4,6,7] => ? = 1
[1,2,6,4,3,5,7] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [3,1,2,4,7,5,6] => ? = 0
[1,2,6,4,3,7,5] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [3,1,2,4,6,5,7] => ? = 1
[1,2,6,4,5,3,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,6,4,5,7,3] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 0
Description
The number of adjacencies of a permutation. An adjacency of a permutation π is an index i such that π(i)−1=π(i+1). Adjacencies are also known as ''small descents''. This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].
Matching statistic: St000237
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
 => [1] => 0
[1,2] => [2] => [1,1,0,0]
 => [2,1] => 1
[2,1] => [1,1] => [1,0,1,0]
 => [1,2] => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [3,1,2] => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [2,1,3] => 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [1,3,2] => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [2,1,3] => 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [1,3,2] => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,2,3] => 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [4,1,2,3] => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 0
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 0
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 0
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 0
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 0
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,2,3] => 0
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => 0
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => 0
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => 0
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => 0
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => 0
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 2
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => 1
[1,2,3,5,4,6,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 0
[1,2,3,5,4,7,6] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,3,6,4,5,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 0
[1,2,3,6,4,7,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,3,6,5,7,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,3,7,4,5,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 0
[1,2,3,7,4,6,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,3,7,5,6,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,4,3,5,6,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,1,2,7,4,5,6] => ? = 0
[1,2,4,3,5,7,6] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 0
[1,2,4,3,6,5,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,4,3,6,7,5] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 0
[1,2,4,3,7,5,6] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,4,3,7,6,5] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [3,1,2,5,4,6,7] => ? = 1
[1,2,4,5,3,6,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 0
[1,2,4,5,3,7,6] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,4,6,3,5,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 0
[1,2,4,6,3,7,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,4,6,5,7,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,4,7,3,5,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 0
[1,2,4,7,3,6,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,4,7,5,6,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,5,3,4,6,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,1,2,7,4,5,6] => ? = 0
[1,2,5,3,4,7,6] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 0
[1,2,5,3,6,4,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,5,3,6,7,4] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 0
[1,2,5,3,7,4,6] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,5,3,7,6,4] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [3,1,2,5,4,6,7] => ? = 1
[1,2,5,4,3,6,7] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [3,1,2,4,7,5,6] => ? = 0
[1,2,5,4,3,7,6] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [3,1,2,4,6,5,7] => ? = 1
[1,2,5,4,6,3,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,5,4,6,7,3] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 0
[1,2,5,4,7,3,6] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,5,4,7,6,3] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [3,1,2,5,4,6,7] => ? = 1
[1,2,5,6,3,4,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 0
[1,2,5,6,3,7,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,5,6,4,7,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,5,7,3,4,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,1,2,3,7,5,6] => ? = 0
[1,2,5,7,3,6,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,5,7,4,6,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,1,2,3,6,5,7] => ? = 1
[1,2,6,3,4,5,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,1,2,7,4,5,6] => ? = 0
[1,2,6,3,4,7,5] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 0
[1,2,6,3,5,4,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,6,3,5,7,4] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 0
[1,2,6,3,7,4,5] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,6,3,7,5,4] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [3,1,2,5,4,6,7] => ? = 1
[1,2,6,4,3,5,7] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [3,1,2,4,7,5,6] => ? = 0
[1,2,6,4,3,7,5] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [3,1,2,4,6,5,7] => ? = 1
[1,2,6,4,5,3,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,1,2,5,4,7,6] => ? = 2
[1,2,6,4,5,7,3] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,1,2,6,4,5,7] => ? = 0
Description
The number of small exceedances. This is the number of indices i such that πi=i+1.
Matching statistic: St000884
(load all 29 compositions to match this statistic)
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1,0]
 => [1] => 0
[1,2] => [2] => [1,1,0,0]
 => [2,1] => 1
[2,1] => [1,1] => [1,0,1,0]
 => [1,2] => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [3,2,1] => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [2,1,3] => 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [1,3,2] => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [2,1,3] => 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [1,3,2] => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,2,3] => 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 0
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 0
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 0
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 0
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 0
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 0
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 0
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 0
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 0
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 0
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 0
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 2
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 1
[1,2,3,5,4,6,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,3,2,1,7,6,5] => ? = 0
[1,2,3,5,4,7,6] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? = 1
[1,2,3,6,4,5,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,3,2,1,7,6,5] => ? = 0
[1,2,3,6,4,7,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? = 1
[1,2,3,6,5,4,7] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [4,3,2,1,5,7,6] => ? = 1
[1,2,3,6,5,7,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? = 1
[1,2,3,7,4,5,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,3,2,1,7,6,5] => ? = 0
[1,2,3,7,4,6,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? = 1
[1,2,3,7,5,4,6] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [4,3,2,1,5,7,6] => ? = 1
[1,2,3,7,5,6,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? = 1
[1,2,3,7,6,4,5] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [4,3,2,1,5,7,6] => ? = 1
[1,2,4,3,5,6,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,2,1,7,6,5,4] => ? = 0
[1,2,4,3,5,7,6] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,2,1,6,5,4,7] => ? = 0
[1,2,4,3,6,5,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,2,1,5,4,7,6] => ? = 2
[1,2,4,3,6,7,5] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,2,1,6,5,4,7] => ? = 0
[1,2,4,3,7,5,6] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,2,1,5,4,7,6] => ? = 2
[1,2,4,3,7,6,5] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [3,2,1,5,4,6,7] => ? = 1
[1,2,4,5,3,6,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,3,2,1,7,6,5] => ? = 0
[1,2,4,5,3,7,6] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? = 1
[1,2,4,6,3,5,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,3,2,1,7,6,5] => ? = 0
[1,2,4,6,3,7,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? = 1
[1,2,4,6,5,3,7] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [4,3,2,1,5,7,6] => ? = 1
[1,2,4,6,5,7,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? = 1
[1,2,4,7,3,5,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,3,2,1,7,6,5] => ? = 0
[1,2,4,7,3,6,5] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? = 1
[1,2,4,7,5,3,6] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [4,3,2,1,5,7,6] => ? = 1
[1,2,4,7,5,6,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? = 1
[1,2,4,7,6,3,5] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [4,3,2,1,5,7,6] => ? = 1
[1,2,5,3,4,6,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,2,1,7,6,5,4] => ? = 0
[1,2,5,3,4,7,6] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,2,1,6,5,4,7] => ? = 0
[1,2,5,3,6,4,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,2,1,5,4,7,6] => ? = 2
[1,2,5,3,6,7,4] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,2,1,6,5,4,7] => ? = 0
[1,2,5,3,7,4,6] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,2,1,5,4,7,6] => ? = 2
[1,2,5,3,7,6,4] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [3,2,1,5,4,6,7] => ? = 1
[1,2,5,4,3,6,7] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [3,2,1,4,7,6,5] => ? = 0
[1,2,5,4,3,7,6] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [3,2,1,4,6,5,7] => ? = 1
[1,2,5,4,6,3,7] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,2,1,5,4,7,6] => ? = 2
[1,2,5,4,6,7,3] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [3,2,1,6,5,4,7] => ? = 0
[1,2,5,4,7,3,6] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [3,2,1,5,4,7,6] => ? = 2
[1,2,5,4,7,6,3] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [3,2,1,5,4,6,7] => ? = 1
[1,2,5,6,3,4,7] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,3,2,1,7,6,5] => ? = 0
[1,2,5,6,3,7,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? = 1
[1,2,5,6,4,3,7] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [4,3,2,1,5,7,6] => ? = 1
[1,2,5,6,4,7,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? = 1
[1,2,5,7,3,4,6] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [4,3,2,1,7,6,5] => ? = 0
[1,2,5,7,3,6,4] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? = 1
[1,2,5,7,4,3,6] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [4,3,2,1,5,7,6] => ? = 1
[1,2,5,7,4,6,3] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [4,3,2,1,6,5,7] => ? = 1
[1,2,5,7,6,3,4] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [4,3,2,1,5,7,6] => ? = 1
[1,2,6,3,4,5,7] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [3,2,1,7,6,5,4] => ? = 0
Description
The number of isolated descents of a permutation. A descent i is isolated if neither i+1 nor i−1 are descents. If a permutation has only isolated descents, then it is called primitive in [1].
Matching statistic: St001125
(load all 4 compositions to match this statistic)
Mapping
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001125: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [1,0]
 => [1,0]
 => 0
[1,2] => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 1
[2,1] => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 0
[1,3,2] => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[2,1,3] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[2,3,1] => [2,1] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[3,1,2] => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 0
[1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 0
[1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 0
[1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 0
[2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 0
[2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 0
[3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 1
[4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 0
[4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 0
[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 0
[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1
[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 0
[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1
[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 0
[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1
[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 0
[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1
[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 0
[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 1
[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 1
[8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[7,8,6,5,4,3,2,1] => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
[8,6,7,5,4,3,2,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[7,6,8,5,4,3,2,1] => [1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
[6,7,8,5,4,3,2,1] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[8,7,5,6,4,3,2,1] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 1
[7,8,5,6,4,3,2,1] => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[8,6,5,7,4,3,2,1] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 1
[8,5,6,7,4,3,2,1] => [1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 0
[7,6,5,8,4,3,2,1] => [1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 1
[6,7,5,8,4,3,2,1] => [2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 2
[7,5,6,8,4,3,2,1] => [1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 0
[6,5,7,8,4,3,2,1] => [1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 0
[5,6,7,8,4,3,2,1] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[8,7,6,4,5,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1
[7,8,6,4,5,3,2,1] => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[8,6,7,4,5,3,2,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[7,6,8,4,5,3,2,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[6,7,8,4,5,3,2,1] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1
[8,7,5,4,6,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1
[7,8,5,4,6,3,2,1] => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[8,7,4,5,6,3,2,1] => [1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 0
[7,8,4,5,6,3,2,1] => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 1
[8,6,5,4,7,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1
[8,5,6,4,7,3,2,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[8,6,4,5,7,3,2,1] => [1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 0
[8,5,4,6,7,3,2,1] => [1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 0
[8,4,5,6,7,3,2,1] => [1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 0
[7,6,5,4,8,3,2,1] => [1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1
[6,7,5,4,8,3,2,1] => [2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[7,5,6,4,8,3,2,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[6,5,7,4,8,3,2,1] => [1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 2
[5,6,7,4,8,3,2,1] => [3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1
[7,6,4,5,8,3,2,1] => [1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 0
[6,7,4,5,8,3,2,1] => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 1
[7,5,4,6,8,3,2,1] => [1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 0
[7,4,5,6,8,3,2,1] => [1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 0
[6,5,4,7,8,3,2,1] => [1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 0
[5,6,4,7,8,3,2,1] => [2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => ? = 1
[6,4,5,7,8,3,2,1] => [1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 0
[5,4,6,7,8,3,2,1] => [1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 0
[4,5,6,7,8,3,2,1] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 0
[8,7,6,5,3,4,2,1] => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 1
[7,8,6,5,3,4,2,1] => [2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 2
[8,6,7,5,3,4,2,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 2
[7,6,8,5,3,4,2,1] => [1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 2
[6,7,8,5,3,4,2,1] => [3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 1
[8,7,5,6,3,4,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2
[7,8,5,6,3,4,2,1] => [2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 3
[8,6,5,7,3,4,2,1] => [1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2
Description
The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra.
Matching statistic: St001276
(load all 2 compositions to match this statistic)
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001276: Dyck paths ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [1] => [1,0]
 => 0
[1,2] => [2,1] => [1,1] => [1,0,1,0]
 => 1
[2,1] => [1,2] => [2] => [1,1,0,0]
 => 0
[1,2,3] => [3,2,1] => [1,1,1] => [1,0,1,0,1,0]
 => 0
[1,3,2] => [2,3,1] => [2,1] => [1,1,0,0,1,0]
 => 1
[2,1,3] => [3,1,2] => [1,2] => [1,0,1,1,0,0]
 => 1
[2,3,1] => [1,3,2] => [2,1] => [1,1,0,0,1,0]
 => 1
[3,1,2] => [2,1,3] => [1,2] => [1,0,1,1,0,0]
 => 1
[3,2,1] => [1,2,3] => [3] => [1,1,1,0,0,0]
 => 0
[1,2,3,4] => [4,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0
[1,2,4,3] => [3,4,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 0
[1,3,2,4] => [4,2,3,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[1,3,4,2] => [2,4,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 0
[1,4,2,3] => [3,2,4,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[1,4,3,2] => [2,3,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[2,1,3,4] => [4,3,1,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 0
[2,1,4,3] => [3,4,1,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,3,1,4] => [4,1,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[2,3,4,1] => [1,4,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 0
[2,4,1,3] => [3,1,4,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[2,4,3,1] => [1,3,4,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[3,1,2,4] => [4,2,1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 0
[3,1,4,2] => [2,4,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,2,1,4] => [4,1,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[3,2,4,1] => [1,4,2,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,4,1,2] => [2,1,4,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[3,4,2,1] => [1,2,4,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[4,1,2,3] => [3,2,1,4] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 0
[4,1,3,2] => [2,3,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[4,2,1,3] => [3,1,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[4,2,3,1] => [1,3,2,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[4,3,1,2] => [2,1,3,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[4,3,2,1] => [1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,3,4,5] => [5,4,3,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,2,3,5,4] => [4,5,3,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 0
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,2,4,5,3] => [3,5,4,2,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 0
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 0
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,3,2,5,4] => [4,5,2,3,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,3,4,5,2] => [2,5,4,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 0
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,3,5,4,2] => [2,4,5,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 0
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,4,2,5,3] => [3,5,2,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,4,3,2,5] => [5,2,3,4,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,4,3,5,2] => [2,5,3,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
[8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 0
[7,8,6,5,4,3,2,1] => [1,2,3,4,5,6,8,7] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 1
[8,6,7,5,4,3,2,1] => [1,2,3,4,5,7,6,8] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 1
[7,6,8,5,4,3,2,1] => [1,2,3,4,5,8,6,7] => [6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
 => ? = 1
[6,7,8,5,4,3,2,1] => [1,2,3,4,5,8,7,6] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
 => ? = 0
[8,7,5,6,4,3,2,1] => [1,2,3,4,6,5,7,8] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 1
[7,8,5,6,4,3,2,1] => [1,2,3,4,6,5,8,7] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 2
[8,6,5,7,4,3,2,1] => [1,2,3,4,7,5,6,8] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 1
[8,5,6,7,4,3,2,1] => [1,2,3,4,7,6,5,8] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 0
[7,6,5,8,4,3,2,1] => [1,2,3,4,8,5,6,7] => [5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
 => ? = 1
[6,7,5,8,4,3,2,1] => [1,2,3,4,8,5,7,6] => [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
 => ? = 2
[7,5,6,8,4,3,2,1] => [1,2,3,4,8,6,5,7] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 0
[6,5,7,8,4,3,2,1] => [1,2,3,4,8,7,5,6] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
 => ? = 0
[5,6,7,8,4,3,2,1] => [1,2,3,4,8,7,6,5] => [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
 => ? = 0
[8,7,6,4,5,3,2,1] => [1,2,3,5,4,6,7,8] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1
[7,8,6,4,5,3,2,1] => [1,2,3,5,4,6,8,7] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 2
[8,6,7,4,5,3,2,1] => [1,2,3,5,4,7,6,8] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
[7,6,8,4,5,3,2,1] => [1,2,3,5,4,8,6,7] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
[6,7,8,4,5,3,2,1] => [1,2,3,5,4,8,7,6] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 1
[8,7,5,4,6,3,2,1] => [1,2,3,6,4,5,7,8] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1
[7,8,5,4,6,3,2,1] => [1,2,3,6,4,5,8,7] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 2
[8,7,4,5,6,3,2,1] => [1,2,3,6,5,4,7,8] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 0
[7,8,4,5,6,3,2,1] => [1,2,3,6,5,4,8,7] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 1
[8,6,5,4,7,3,2,1] => [1,2,3,7,4,5,6,8] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1
[8,5,6,4,7,3,2,1] => [1,2,3,7,4,6,5,8] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
[8,6,4,5,7,3,2,1] => [1,2,3,7,5,4,6,8] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 0
[8,5,4,6,7,3,2,1] => [1,2,3,7,6,4,5,8] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 0
[8,4,5,6,7,3,2,1] => [1,2,3,7,6,5,4,8] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 0
[7,6,5,4,8,3,2,1] => [1,2,3,8,4,5,6,7] => [4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 1
[6,7,5,4,8,3,2,1] => [1,2,3,8,4,5,7,6] => [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
 => ? = 2
[7,5,6,4,8,3,2,1] => [1,2,3,8,4,6,5,7] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
[6,5,7,4,8,3,2,1] => [1,2,3,8,4,7,5,6] => [4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 2
[5,6,7,4,8,3,2,1] => [1,2,3,8,4,7,6,5] => [4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
 => ? = 1
[7,6,4,5,8,3,2,1] => [1,2,3,8,5,4,6,7] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 0
[6,7,4,5,8,3,2,1] => [1,2,3,8,5,4,7,6] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 1
[7,5,4,6,8,3,2,1] => [1,2,3,8,6,4,5,7] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 0
[7,4,5,6,8,3,2,1] => [1,2,3,8,6,5,4,7] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 0
[6,5,4,7,8,3,2,1] => [1,2,3,8,7,4,5,6] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 0
[5,6,4,7,8,3,2,1] => [1,2,3,8,7,4,6,5] => [4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
 => ? = 1
[6,4,5,7,8,3,2,1] => [1,2,3,8,7,5,4,6] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 0
[5,4,6,7,8,3,2,1] => [1,2,3,8,7,6,4,5] => [4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
 => ? = 0
[4,5,6,7,8,3,2,1] => [1,2,3,8,7,6,5,4] => [4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => ? = 0
[8,7,6,5,3,4,2,1] => [1,2,4,3,5,6,7,8] => [3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 1
[7,8,6,5,3,4,2,1] => [1,2,4,3,5,6,8,7] => [3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 2
[8,6,7,5,3,4,2,1] => [1,2,4,3,5,7,6,8] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2
[7,6,8,5,3,4,2,1] => [1,2,4,3,5,8,6,7] => [3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2
[6,7,8,5,3,4,2,1] => [1,2,4,3,5,8,7,6] => [3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 1
[8,7,5,6,3,4,2,1] => [1,2,4,3,6,5,7,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2
[7,8,5,6,3,4,2,1] => [1,2,4,3,6,5,8,7] => [3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 3
[8,6,5,7,3,4,2,1] => [1,2,4,3,7,5,6,8] => [3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2
Description
The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. Generalising the notion of k-regular modules from simple to arbitrary indecomposable modules, we call an indecomposable module M over an algebra A k-regular in case it has projective dimension k and ExtiA(M,A)=0 for i≠k and ExtkA(M,A) is 1-dimensional. The number of Dyck paths where the statistic returns 0 might be given by [[OEIS:A035929]] .
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001274The number of indecomposable injective modules with projective dimension equal to two.