Your data matches 4 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000257
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1]
 => [1]
 => [1]
 => [1]
 => 0
[2,1]
 => [1]
 => [1]
 => [1]
 => 0
[1,1,1]
 => [1,1]
 => [2]
 => [1,1]
 => 1
[3,1]
 => [1]
 => [1]
 => [1]
 => 0
[2,2]
 => [2]
 => [1,1]
 => [2]
 => 0
[2,1,1]
 => [1,1]
 => [2]
 => [1,1]
 => 1
[1,1,1,1]
 => [1,1,1]
 => [3]
 => [3]
 => 0
[4,1]
 => [1]
 => [1]
 => [1]
 => 0
[3,2]
 => [2]
 => [1,1]
 => [2]
 => 0
[3,1,1]
 => [1,1]
 => [2]
 => [1,1]
 => 1
[2,2,1]
 => [2,1]
 => [2,1]
 => [1,1,1]
 => 1
[2,1,1,1]
 => [1,1,1]
 => [3]
 => [3]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => [2,2]
 => 1
[5,1]
 => [1]
 => [1]
 => [1]
 => 0
[4,2]
 => [2]
 => [1,1]
 => [2]
 => 0
[4,1,1]
 => [1,1]
 => [2]
 => [1,1]
 => 1
[3,3]
 => [3]
 => [1,1,1]
 => [2,1]
 => 0
[3,2,1]
 => [2,1]
 => [2,1]
 => [1,1,1]
 => 1
[3,1,1,1]
 => [1,1,1]
 => [3]
 => [3]
 => 0
[2,2,2]
 => [2,2]
 => [2,2]
 => [1,1,1,1]
 => 1
[2,2,1,1]
 => [2,1,1]
 => [3,1]
 => [3,1]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => [2,2]
 => 1
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => [5]
 => 0
[6,1]
 => [1]
 => [1]
 => [1]
 => 0
[5,2]
 => [2]
 => [1,1]
 => [2]
 => 0
[5,1,1]
 => [1,1]
 => [2]
 => [1,1]
 => 1
[4,3]
 => [3]
 => [1,1,1]
 => [2,1]
 => 0
[4,2,1]
 => [2,1]
 => [2,1]
 => [1,1,1]
 => 1
[4,1,1,1]
 => [1,1,1]
 => [3]
 => [3]
 => 0
[3,3,1]
 => [3,1]
 => [2,1,1]
 => [2,1,1]
 => 1
[3,2,2]
 => [2,2]
 => [2,2]
 => [1,1,1,1]
 => 1
[3,2,1,1]
 => [2,1,1]
 => [3,1]
 => [3,1]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => [2,2]
 => 1
[2,2,2,1]
 => [2,2,1]
 => [3,2]
 => [3,1,1]
 => 1
[2,2,1,1,1]
 => [2,1,1,1]
 => [4,1]
 => [2,2,1]
 => 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => [5]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [6]
 => [3,3]
 => 1
[7,1]
 => [1]
 => [1]
 => [1]
 => 0
[6,2]
 => [2]
 => [1,1]
 => [2]
 => 0
[6,1,1]
 => [1,1]
 => [2]
 => [1,1]
 => 1
[5,3]
 => [3]
 => [1,1,1]
 => [2,1]
 => 0
[5,2,1]
 => [2,1]
 => [2,1]
 => [1,1,1]
 => 1
[5,1,1,1]
 => [1,1,1]
 => [3]
 => [3]
 => 0
[4,4]
 => [4]
 => [1,1,1,1]
 => [4]
 => 0
[4,3,1]
 => [3,1]
 => [2,1,1]
 => [2,1,1]
 => 1
[4,2,2]
 => [2,2]
 => [2,2]
 => [1,1,1,1]
 => 1
[4,2,1,1]
 => [2,1,1]
 => [3,1]
 => [3,1]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => [2,2]
 => 1
[3,3,2]
 => [3,2]
 => [2,2,1]
 => [1,1,1,1,1]
 => 1
[3,3,1,1]
 => [3,1,1]
 => [3,1,1]
 => [3,2]
 => 0
Description
The number of distinct parts of a partition that occur at least twice. See Section 3.3.1 of [2].
Matching statistic: St001092
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,1]
 => [1]
 => [1]
 => 0
[2,1]
 => [1]
 => [1]
 => 0
[1,1,1]
 => [1,1]
 => [2]
 => 1
[3,1]
 => [1]
 => [1]
 => 0
[2,2]
 => [2]
 => [1,1]
 => 0
[2,1,1]
 => [1,1]
 => [2]
 => 1
[1,1,1,1]
 => [1,1,1]
 => [3]
 => 0
[4,1]
 => [1]
 => [1]
 => 0
[3,2]
 => [2]
 => [1,1]
 => 0
[3,1,1]
 => [1,1]
 => [2]
 => 1
[2,2,1]
 => [2,1]
 => [2,1]
 => 1
[2,1,1,1]
 => [1,1,1]
 => [3]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 1
[5,1]
 => [1]
 => [1]
 => 0
[4,2]
 => [2]
 => [1,1]
 => 0
[4,1,1]
 => [1,1]
 => [2]
 => 1
[3,3]
 => [3]
 => [1,1,1]
 => 0
[3,2,1]
 => [2,1]
 => [2,1]
 => 1
[3,1,1,1]
 => [1,1,1]
 => [3]
 => 0
[2,2,2]
 => [2,2]
 => [2,2]
 => 1
[2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 1
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => 0
[6,1]
 => [1]
 => [1]
 => 0
[5,2]
 => [2]
 => [1,1]
 => 0
[5,1,1]
 => [1,1]
 => [2]
 => 1
[4,3]
 => [3]
 => [1,1,1]
 => 0
[4,2,1]
 => [2,1]
 => [2,1]
 => 1
[4,1,1,1]
 => [1,1,1]
 => [3]
 => 0
[3,3,1]
 => [3,1]
 => [2,1,1]
 => 1
[3,2,2]
 => [2,2]
 => [2,2]
 => 1
[3,2,1,1]
 => [2,1,1]
 => [3,1]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 1
[2,2,2,1]
 => [2,2,1]
 => [3,2]
 => 1
[2,2,1,1,1]
 => [2,1,1,1]
 => [4,1]
 => 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [6]
 => 1
[7,1]
 => [1]
 => [1]
 => 0
[6,2]
 => [2]
 => [1,1]
 => 0
[6,1,1]
 => [1,1]
 => [2]
 => 1
[5,3]
 => [3]
 => [1,1,1]
 => 0
[5,2,1]
 => [2,1]
 => [2,1]
 => 1
[5,1,1,1]
 => [1,1,1]
 => [3]
 => 0
[4,4]
 => [4]
 => [1,1,1,1]
 => 0
[4,3,1]
 => [3,1]
 => [2,1,1]
 => 1
[4,2,2]
 => [2,2]
 => [2,2]
 => 1
[4,2,1,1]
 => [2,1,1]
 => [3,1]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 1
[3,3,2]
 => [3,2]
 => [2,2,1]
 => 1
[3,3,1,1]
 => [3,1,1]
 => [3,1,1]
 => 0
[2,2,2,2,2,1,1,1,1,1]
 => [2,2,2,2,1,1,1,1,1]
 => [9,4]
 => ? = 1
[3,2,2,2,2,1,1,1,1,1]
 => [2,2,2,2,1,1,1,1,1]
 => [9,4]
 => ? = 1
[4,2,2,2,2,1,1,1,1,1]
 => [2,2,2,2,1,1,1,1,1]
 => [9,4]
 => ? = 1
[3,3,3,3,1,1,1,1,1]
 => [3,3,3,1,1,1,1,1]
 => [8,3,3]
 => ? = 1
[3,3,3,1,1,1,1,1,1,1,1]
 => [3,3,1,1,1,1,1,1,1,1]
 => [10,2,2]
 => ? = 2
[2,2,2,2,2,2,1,1,1,1,1]
 => [2,2,2,2,2,1,1,1,1,1]
 => [10,5]
 => ? = 1
[2,2,2,2,2,1,1,1,1,1,1,1]
 => [2,2,2,2,1,1,1,1,1,1,1]
 => [11,4]
 => ? = 1
[9,8,7]
 => [8,7]
 => [2,2,2,2,2,2,2,1]
 => ? = 1
[3,3,3,3,1,1,1,1,1,1]
 => [3,3,3,1,1,1,1,1,1]
 => [9,3,3]
 => ? = 0
[7,7,1,1,1,1,1,1]
 => [7,1,1,1,1,1,1]
 => [7,1,1,1,1,1,1]
 => ? = 0
Description
The number of distinct even parts of a partition. See Section 3.3.1 of [1].
Matching statistic: St001115
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001115: Permutations ⟶ ℤResult quality: 75% ●values known / values provided: 82%●distinct values known / distinct values provided: 75%
Values
[1,1]
 => [1]
 => [1,0,1,0]
 => [2,1] => 0
[2,1]
 => [1]
 => [1,0,1,0]
 => [2,1] => 0
[1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[3,1]
 => [1]
 => [1,0,1,0]
 => [2,1] => 0
[2,2]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 0
[2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 0
[4,1]
 => [1]
 => [1,0,1,0]
 => [2,1] => 0
[3,2]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 0
[3,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 1
[2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[5,1]
 => [1]
 => [1,0,1,0]
 => [2,1] => 0
[4,2]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 0
[4,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[3,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 0
[3,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 1
[3,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 0
[2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 1
[2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 0
[6,1]
 => [1]
 => [1,0,1,0]
 => [2,1] => 0
[5,2]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 0
[5,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[4,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 0
[4,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 1
[4,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 0
[3,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 1
[3,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 1
[3,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[2,2,2,1]
 => [2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 1
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => 1
[7,1]
 => [1]
 => [1,0,1,0]
 => [2,1] => 0
[6,2]
 => [2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 0
[6,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 1
[5,3]
 => [3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 0
[5,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 1
[5,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 0
[4,4]
 => [4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 0
[4,3,1]
 => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 1
[4,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 1
[4,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 1
[3,3,2]
 => [3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 1
[3,3,1,1]
 => [3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 0
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ? = 0
[2,2,2,2,2,2,1]
 => [2,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,2,1] => ? = 1
[2,2,2,2,2,1,1,1]
 => [2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,2,7,8,1] => ? = 1
[2,2,2,2,1,1,1,1,1]
 => [2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,9,1] => ? = 1
[2,2,2,1,1,1,1,1,1,1]
 => [2,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,10,1] => ? = 1
[2,2,1,1,1,1,1,1,1,1,1]
 => [2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,10,11,1] => ? = 1
[2,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ? = 0
[1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ? = 1
[3,3,3,2,1,1,1]
 => [3,3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [4,5,3,2,6,7,1] => ? = 2
[3,3,3,1,1,1,1,1]
 => [3,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [4,5,2,3,6,7,8,1] => ? = 1
[3,3,2,2,2,2]
 => [3,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,7,1,2] => ? = 0
[3,3,2,2,2,1,1]
 => [3,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,2,7,1] => ? = 2
[3,3,2,1,1,1,1,1,1]
 => [3,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [4,3,2,5,6,7,8,9,1] => ? = 2
[3,2,2,2,2,2,1]
 => [2,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,2,1] => ? = 1
[3,2,2,2,2,1,1,1]
 => [2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,2,7,8,1] => ? = 1
[3,2,2,2,1,1,1,1,1]
 => [2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,9,1] => ? = 1
[3,2,2,1,1,1,1,1,1,1]
 => [2,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,10,1] => ? = 1
[3,2,1,1,1,1,1,1,1,1,1]
 => [2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,10,11,1] => ? = 1
[3,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ? = 0
[2,2,2,2,2,2,1,1]
 => [2,2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,7,2,8,1] => ? = 0
[2,2,2,2,2,1,1,1,1]
 => [2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [3,4,5,6,2,7,8,9,1] => ? = 2
[2,2,2,2,1,1,1,1,1,1]
 => [2,2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,9,10,1] => ? = 0
[2,2,2,1,1,1,1,1,1,1,1]
 => [2,2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,10,11,1] => ? = 2
[2,2,1,1,1,1,1,1,1,1,1,1]
 => [2,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,10,11,12,1] => ? = 0
[2,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ? = 1
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ? = 0
[4,4,3,1,1,1,1]
 => [4,3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,1,0,0,0]
 => [5,4,2,3,6,7,1] => ? = 2
[4,4,2,2,1,1,1]
 => [4,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
 => [5,3,4,2,6,7,1] => ? = 1
[4,4,1,1,1,1,1,1,1]
 => [4,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
 => [5,2,3,4,6,7,8,9,1] => ? = 1
[4,3,3,2,1,1,1]
 => [3,3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [4,5,3,2,6,7,1] => ? = 2
[4,3,3,1,1,1,1,1]
 => [3,3,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
 => [4,5,2,3,6,7,8,1] => ? = 1
[4,3,2,2,2,2]
 => [3,2,2,2,2]
 => [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,7,1,2] => ? = 0
[4,3,2,2,2,1,1]
 => [3,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,2,7,1] => ? = 2
[4,3,2,1,1,1,1,1,1]
 => [3,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
 => [4,3,2,5,6,7,8,9,1] => ? = 2
[4,2,2,2,2,2,1]
 => [2,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,4,5,6,7,2,1] => ? = 1
[4,2,2,2,2,1,1,1]
 => [2,2,2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,2,7,8,1] => ? = 1
[4,2,2,2,1,1,1,1,1]
 => [2,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,9,1] => ? = 1
[4,2,2,1,1,1,1,1,1,1]
 => [2,2,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
 => [3,4,2,5,6,7,8,9,10,1] => ? = 1
[4,2,1,1,1,1,1,1,1,1,1]
 => [2,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
 => [3,2,4,5,6,7,8,9,10,11,1] => ? = 1
[4,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
 => ? => ? = 0
[3,3,3,3,3]
 => [3,3,3,3]
 => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [4,5,6,7,1,2,3] => ? = 1
[3,3,3,3,1,1,1]
 => [3,3,3,1,1,1]
 => [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
 => [4,5,6,2,3,7,1] => ? = 1
[3,3,3,2,2,2]
 => [3,3,2,2,2]
 => [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
 => [4,5,3,6,7,1,2] => ? = 1
[3,3,3,2,2,1,1]
 => [3,3,2,2,1,1]
 => [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [4,5,3,6,2,7,1] => ? = 3
[3,3,2,2,2,2,1]
 => [3,2,2,2,2,1]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [4,3,5,6,7,2,1] => ? = 1
[3,3,2,2,1,1,1,1,1]
 => [3,2,2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => [4,3,5,2,6,7,8,9,1] => ? = 1
[3,3,1,1,1,1,1,1,1,1,1]
 => [3,1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
 => [4,2,3,5,6,7,8,9,10,11,1] => ? = 1
[3,2,2,2,2,2,1,1]
 => [2,2,2,2,2,1,1]
 => [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [3,4,5,6,7,2,8,1] => ? = 0
[3,2,2,2,2,1,1,1,1]
 => [2,2,2,2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
 => [3,4,5,6,2,7,8,9,1] => ? = 2
[3,2,2,2,1,1,1,1,1,1]
 => [2,2,2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
 => [3,4,5,2,6,7,8,9,10,1] => ? = 0
Description
The number of even descents of a permutation.
Matching statistic: St000338
(load all 2 compositions to match this statistic)
Mapping
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St000338: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Values
[1,1]
 => [[1],[2]]
 => [2,1] => [2,1] => 0
[2,1]
 => [[1,3],[2]]
 => [2,1,3] => [2,1,3] => 0
[1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => [3,2,1] => 1
[3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => [2,1,3,4] => 0
[2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => [3,1,4,2] => 0
[2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => [3,2,1,4] => 1
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => [4,3,2,1] => 0
[4,1]
 => [[1,3,4,5],[2]]
 => [2,1,3,4,5] => [2,1,3,4,5] => 0
[3,2]
 => [[1,2,5],[3,4]]
 => [3,4,1,2,5] => [3,1,4,2,5] => 0
[3,1,1]
 => [[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [3,2,1,4,5] => 1
[2,2,1]
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [2,4,1,5,3] => 1
[2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [4,3,2,1,5] => 0
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [5,4,3,2,1] => 1
[5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [2,1,3,4,5,6] => 0
[4,2]
 => [[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => [3,1,4,2,5,6] => 0
[4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => [3,2,1,4,5,6] => 1
[3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => [4,1,5,2,6,3] => 0
[3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [4,2,5,1,3,6] => [2,4,1,5,3,6] => 1
[3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => [4,3,2,1,5,6] => 0
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => [5,3,1,6,4,2] => 1
[2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [5,3,2,6,1,4] => [3,2,5,1,6,4] => 0
[2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [5,4,3,2,1,6] => [5,4,3,2,1,6] => 1
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 0
[6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 0
[5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => [3,1,4,2,5,6,7] => ? = 0
[5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => ? = 1
[4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => [4,1,5,2,6,3,7] => ? = 0
[4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [4,2,5,1,3,6,7] => [2,4,1,5,3,6,7] => ? = 1
[4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => ? = 0
[3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [5,2,6,7,1,3,4] => [5,6,2,1,3,7,4] => ? = 1
[3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [5,6,3,4,1,2,7] => [5,3,1,6,4,2,7] => ? = 1
[3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => [5,3,2,6,1,4,7] => [3,2,5,1,6,4,7] => ? = 0
[3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => ? = 1
[2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3] => [4,2,6,1,7,5,3] => ? = 1
[2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => [6,4,3,2,7,1,5] => [4,3,2,6,1,7,5] => ? = 1
[2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ? = 0
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 1
[7,1]
 => [[1,3,4,5,6,7,8],[2]]
 => [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => ? = 0
[6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [3,4,1,2,5,6,7,8] => [3,1,4,2,5,6,7,8] => ? = 0
[6,1,1]
 => [[1,4,5,6,7,8],[2],[3]]
 => [3,2,1,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => ? = 1
[5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => [4,5,6,1,2,3,7,8] => [4,1,5,2,6,3,7,8] => ? = 0
[5,2,1]
 => [[1,3,6,7,8],[2,5],[4]]
 => [4,2,5,1,3,6,7,8] => [2,4,1,5,3,6,7,8] => ? = 1
[5,1,1,1]
 => [[1,5,6,7,8],[2],[3],[4]]
 => [4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => ? = 0
[4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,8,4] => ? = 0
[4,3,1]
 => [[1,3,4,8],[2,6,7],[5]]
 => [5,2,6,7,1,3,4,8] => [5,6,2,1,3,7,4,8] => ? = 1
[4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => [5,6,3,4,1,2,7,8] => [5,3,1,6,4,2,7,8] => ? = 1
[4,2,1,1]
 => [[1,4,7,8],[2,6],[3],[5]]
 => [5,3,2,6,1,4,7,8] => [3,2,5,1,6,4,7,8] => ? = 0
[4,1,1,1,1]
 => [[1,6,7,8],[2],[3],[4],[5]]
 => [5,4,3,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => ? = 1
[3,3,2]
 => [[1,2,5],[3,4,8],[6,7]]
 => [6,7,3,4,8,1,2,5] => [3,6,4,1,7,2,8,5] => ? = 1
[3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => [6,3,2,7,8,1,4,5] => [3,6,7,2,1,4,8,5] => ? = 0
[3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3,8] => [4,2,6,1,7,5,3,8] => ? = 1
[3,2,1,1,1]
 => [[1,5,8],[2,7],[3],[4],[6]]
 => [6,4,3,2,7,1,5,8] => [4,3,2,6,1,7,5,8] => ? = 1
[3,1,1,1,1,1]
 => [[1,7,8],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,7,8] => ? = 0
[2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2] => [7,5,3,1,8,6,4,2] => ? = 0
[2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5],[7]]
 => [7,5,3,8,2,6,1,4] => [3,5,2,7,1,8,6,4] => ? = 2
[2,2,1,1,1,1]
 => [[1,6],[2,8],[3],[4],[5],[7]]
 => [7,5,4,3,2,8,1,6] => [5,4,3,2,7,1,8,6] => ? = 0
[2,1,1,1,1,1,1]
 => [[1,8],[2],[3],[4],[5],[6],[7]]
 => [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => ? = 1
[1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => ? = 0
[8,1]
 => [[1,3,4,5,6,7,8,9],[2]]
 => [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => ? = 0
[7,2]
 => [[1,2,5,6,7,8,9],[3,4]]
 => [3,4,1,2,5,6,7,8,9] => [3,1,4,2,5,6,7,8,9] => ? = 0
[7,1,1]
 => [[1,4,5,6,7,8,9],[2],[3]]
 => [3,2,1,4,5,6,7,8,9] => [3,2,1,4,5,6,7,8,9] => ? = 1
[6,3]
 => [[1,2,3,7,8,9],[4,5,6]]
 => [4,5,6,1,2,3,7,8,9] => [4,1,5,2,6,3,7,8,9] => ? = 0
[6,2,1]
 => [[1,3,6,7,8,9],[2,5],[4]]
 => [4,2,5,1,3,6,7,8,9] => [2,4,1,5,3,6,7,8,9] => ? = 1
[6,1,1,1]
 => [[1,5,6,7,8,9],[2],[3],[4]]
 => [4,3,2,1,5,6,7,8,9] => [4,3,2,1,5,6,7,8,9] => ? = 0
[5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4,9] => [5,1,6,2,7,3,8,4,9] => ? = 0
[5,3,1]
 => [[1,3,4,8,9],[2,6,7],[5]]
 => [5,2,6,7,1,3,4,8,9] => [5,6,2,1,3,7,4,8,9] => ? = 1
[5,2,2]
 => [[1,2,7,8,9],[3,4],[5,6]]
 => [5,6,3,4,1,2,7,8,9] => [5,3,1,6,4,2,7,8,9] => ? = 1
[5,2,1,1]
 => [[1,4,7,8,9],[2,6],[3],[5]]
 => [5,3,2,6,1,4,7,8,9] => [3,2,5,1,6,4,7,8,9] => ? = 0
[5,1,1,1,1]
 => [[1,6,7,8,9],[2],[3],[4],[5]]
 => [5,4,3,2,1,6,7,8,9] => [5,4,3,2,1,6,7,8,9] => ? = 1
[4,4,1]
 => [[1,3,4,5],[2,7,8,9],[6]]
 => [6,2,7,8,9,1,3,4,5] => [6,7,2,1,8,3,4,9,5] => ? = 1
[4,3,2]
 => [[1,2,5,9],[3,4,8],[6,7]]
 => [6,7,3,4,8,1,2,5,9] => [3,6,4,1,7,2,8,5,9] => ? = 1
[4,3,1,1]
 => [[1,4,5,9],[2,7,8],[3],[6]]
 => [6,3,2,7,8,1,4,5,9] => [3,6,7,2,1,4,8,5,9] => ? = 0
[4,2,2,1]
 => [[1,3,8,9],[2,5],[4,7],[6]]
 => [6,4,7,2,5,1,3,8,9] => [4,2,6,1,7,5,3,8,9] => ? = 1
Description
The number of pixed points of a permutation. For a permutation $\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines $$\textrm{pix} \, \sigma = \textrm{length} (p)$$.