Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001685
(load all 6 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001685: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [1,2] => 0
[2]
 => [1,1,0,0,1,0]
 => [2,1,3] => 0
[1,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 0
[2,1]
 => [1,0,1,0,1,0]
 => [1,2,3] => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 0
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 0
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [5,4,3,2,1,6] => 0
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [4,2,3,1,5] => 0
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 0
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,5,3,4,2] => 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,6,5,4,3,2] => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [5,3,4,2,1,6] => 0
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [3,2,4,1,5] => 0
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [2,4,3,1,5] => 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 3
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 0
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,4,3,5,2] => 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 2
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,4,2] => 2
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,6,4,5,3,2] => 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [5,3,2,4,1,6] => 0
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [5,2,4,3,1,6] => 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 0
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => 0
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => 3
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => 2
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,6,4,3,5,2] => 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 2
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,6,3,5,4,2] => 2
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [4,3,2,5,1,6] => 0
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [5,2,3,4,1,6] => 0
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [2,5,4,3,1,6] => 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [4,3,2,1,6,5] => 4
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => 0
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,5,4,3,6,2] => 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 3
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 3
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => 2
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,6,3,4,5,2] => 1
Description
The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation.
Matching statistic: St001587
Mapping
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => 0
[2]
 => [1,1]
 => 0
[1,1]
 => [2]
 => 1
[3]
 => [3]
 => 0
[2,1]
 => [1,1,1]
 => 0
[1,1,1]
 => [2,1]
 => 1
[4]
 => [1,1,1,1]
 => 0
[3,1]
 => [3,1]
 => 0
[2,2]
 => [4]
 => 2
[2,1,1]
 => [2,1,1]
 => 1
[1,1,1,1]
 => [2,2]
 => 1
[5]
 => [5]
 => 0
[4,1]
 => [1,1,1,1,1]
 => 0
[3,2]
 => [3,1,1]
 => 0
[3,1,1]
 => [3,2]
 => 1
[2,2,1]
 => [4,1]
 => 2
[2,1,1,1]
 => [2,1,1,1]
 => 1
[1,1,1,1,1]
 => [2,2,1]
 => 1
[5,1]
 => [5,1]
 => 0
[4,2]
 => [1,1,1,1,1,1]
 => 0
[4,1,1]
 => [2,1,1,1,1]
 => 1
[3,3]
 => [6]
 => 3
[3,2,1]
 => [3,1,1,1]
 => 0
[3,1,1,1]
 => [3,2,1]
 => 1
[2,2,2]
 => [4,1,1]
 => 2
[2,2,1,1]
 => [4,2]
 => 2
[2,1,1,1,1]
 => [2,2,1,1]
 => 1
[5,2]
 => [5,1,1]
 => 0
[5,1,1]
 => [5,2]
 => 1
[4,3]
 => [3,1,1,1,1]
 => 0
[4,2,1]
 => [1,1,1,1,1,1,1]
 => 0
[4,1,1,1]
 => [2,1,1,1,1,1]
 => 1
[3,3,1]
 => [6,1]
 => 3
[3,2,2]
 => [4,3]
 => 2
[3,2,1,1]
 => [3,2,1,1]
 => 1
[3,1,1,1,1]
 => [3,2,2]
 => 1
[2,2,2,1]
 => [4,1,1,1]
 => 2
[2,2,1,1,1]
 => [4,2,1]
 => 2
[5,3]
 => [5,3]
 => 0
[5,2,1]
 => [5,1,1,1]
 => 0
[5,1,1,1]
 => [5,2,1]
 => 1
[4,4]
 => [8]
 => 4
[4,3,1]
 => [3,1,1,1,1,1]
 => 0
[4,2,2]
 => [4,1,1,1,1]
 => 2
[4,2,1,1]
 => [2,1,1,1,1,1,1]
 => 1
[4,1,1,1,1]
 => [2,2,1,1,1,1]
 => 1
[3,3,2]
 => [6,1,1]
 => 3
[3,3,1,1]
 => [6,2]
 => 3
[3,2,2,1]
 => [4,3,1]
 => 2
[3,2,1,1,1]
 => [3,2,1,1,1]
 => 1
[6,5,4,3,2,1]
 => ?
 => ? = 0
[5,5,4,3,2,1]
 => ?
 => ? = 5
[6,4,4,3,2,1]
 => ?
 => ? = 4
[5,4,4,3,2,1]
 => ?
 => ? = 4
[4,4,4,3,2,1]
 => ?
 => ? = 4
[6,5,3,3,2,1]
 => ?
 => ? = 3
[5,5,3,3,2,1]
 => ?
 => ? = 5
[6,4,3,3,2,1]
 => ?
 => ? = 3
[5,4,3,3,2,1]
 => ?
 => ? = 3
[6,3,3,3,2,1]
 => ?
 => ? = 3
[6,5,4,2,2,1]
 => ?
 => ? = 2
[5,5,4,2,2,1]
 => ?
 => ? = 5
[6,4,4,2,2,1]
 => ?
 => ? = 4
[5,4,4,2,2,1]
 => ?
 => ? = 4
[6,5,3,2,2,1]
 => ?
 => ? = 2
[5,5,3,2,2,1]
 => ?
 => ? = 5
[6,4,3,2,2,1]
 => ?
 => ? = 2
[6,5,2,2,2,1]
 => ?
 => ? = 2
[6,5,4,3,1,1]
 => ?
 => ? = 1
[5,5,4,3,1,1]
 => ?
 => ? = 5
[6,4,4,3,1,1]
 => ?
 => ? = 4
[5,4,4,3,1,1]
 => ?
 => ? = 4
[6,5,3,3,1,1]
 => ?
 => ? = 3
[5,5,3,3,1,1]
 => ?
 => ? = 5
[6,4,3,3,1,1]
 => ?
 => ? = 3
[6,5,4,2,1,1]
 => ?
 => ? = 1
[5,5,4,2,1,1]
 => ?
 => ? = 5
[6,4,4,2,1,1]
 => ?
 => ? = 4
[6,5,3,2,1,1]
 => ?
 => ? = 1
[6,5,4,1,1,1]
 => ?
 => ? = 1
Description
Half of the largest even part of an integer partition. The largest even part is recorded by [[St000995]].
Matching statistic: St000143
Mapping
St000143: Integer partitions ⟶ ℤResult quality: 58% ●values known / values provided: 58%●distinct values known / distinct values provided: 83%
Values
[1]
 => 0
[2]
 => 0
[1,1]
 => 1
[3]
 => 0
[2,1]
 => 0
[1,1,1]
 => 1
[4]
 => 0
[3,1]
 => 0
[2,2]
 => 2
[2,1,1]
 => 1
[1,1,1,1]
 => 1
[5]
 => 0
[4,1]
 => 0
[3,2]
 => 0
[3,1,1]
 => 1
[2,2,1]
 => 2
[2,1,1,1]
 => 1
[1,1,1,1,1]
 => 1
[5,1]
 => 0
[4,2]
 => 0
[4,1,1]
 => 1
[3,3]
 => 3
[3,2,1]
 => 0
[3,1,1,1]
 => 1
[2,2,2]
 => 2
[2,2,1,1]
 => 2
[2,1,1,1,1]
 => 1
[5,2]
 => 0
[5,1,1]
 => 1
[4,3]
 => 0
[4,2,1]
 => 0
[4,1,1,1]
 => 1
[3,3,1]
 => 3
[3,2,2]
 => 2
[3,2,1,1]
 => 1
[3,1,1,1,1]
 => 1
[2,2,2,1]
 => 2
[2,2,1,1,1]
 => 2
[5,3]
 => 0
[5,2,1]
 => 0
[5,1,1,1]
 => 1
[4,4]
 => 4
[4,3,1]
 => 0
[4,2,2]
 => 2
[4,2,1,1]
 => 1
[4,1,1,1,1]
 => 1
[3,3,2]
 => 3
[3,3,1,1]
 => 3
[3,2,2,1]
 => 2
[3,2,1,1,1]
 => 1
[3,2,2,2,1,1]
 => ? = 2
[2,2,2,2,2,1]
 => ? = 2
[4,2,2,2,1,1]
 => ? = 2
[3,3,3,1,1,1]
 => ? = 3
[3,2,2,2,2,1]
 => ? = 2
[5,2,2,2,1,1]
 => ? = 2
[4,3,3,1,1,1]
 => ? = 3
[4,3,2,2,1,1]
 => ? = 2
[4,2,2,2,2,1]
 => ? = 2
[3,3,3,2,1,1]
 => ? = 3
[3,3,2,2,2,1]
 => ? = 3
[6,2,2,2,1,1]
 => ? = 2
[5,3,3,1,1,1]
 => ? = 3
[5,3,2,2,1,1]
 => ? = 2
[5,2,2,2,2,1]
 => ? = 2
[4,4,3,1,1,1]
 => ? = 4
[4,4,2,2,1,1]
 => ? = 4
[4,3,3,2,1,1]
 => ? = 3
[4,3,2,2,2,1]
 => ? = 2
[3,3,3,3,1,1]
 => ? = 3
[3,3,3,2,2,1]
 => ? = 3
[6,3,3,1,1,1]
 => ? = 3
[6,3,2,2,1,1]
 => ? = 2
[6,2,2,2,2,1]
 => ? = 2
[5,4,3,1,1,1]
 => ? = 1
[5,4,2,2,1,1]
 => ? = 2
[5,3,3,2,1,1]
 => ? = 3
[5,3,2,2,2,1]
 => ? = 2
[4,4,4,1,1,1]
 => ? = 4
[4,4,3,2,1,1]
 => ? = 4
[4,4,2,2,2,1]
 => ? = 4
[4,3,3,3,1,1]
 => ? = 3
[4,3,3,2,2,1]
 => ? = 3
[3,3,3,3,2,1]
 => ? = 3
[6,4,3,1,1,1]
 => ? = 1
[6,4,2,2,1,1]
 => ? = 2
[6,3,3,2,1,1]
 => ? = 3
[6,3,2,2,2,1]
 => ? = 2
[5,5,3,1,1,1]
 => ? = 5
[5,5,2,2,1,1]
 => ? = 5
[5,4,4,1,1,1]
 => ? = 4
[5,4,3,2,1,1]
 => ? = 1
[5,4,2,2,2,1]
 => ? = 2
[5,3,3,3,1,1]
 => ? = 3
[5,3,3,2,2,1]
 => ? = 3
[4,4,4,2,1,1]
 => ? = 4
[4,4,3,3,1,1]
 => ? = 4
[4,4,3,2,2,1]
 => ? = 4
[4,3,3,3,2,1]
 => ? = 3
[6,5,3,1,1,1]
 => ? = 1
Description
The largest repeated part of a partition. If the parts of the partition are all distinct, the value of the statistic is defined to be zero.