Your data matches 123 different statistics following compositions of up to 3 maps.
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Matching statistic: St000150
(load all 3 compositions to match this statistic)
Mapping
St000150: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0 = 1 - 1
[2]
 => 0 = 1 - 1
[1,1]
 => 1 = 2 - 1
[3]
 => 0 = 1 - 1
[2,1]
 => 0 = 1 - 1
[1,1,1]
 => 1 = 2 - 1
[3,1]
 => 0 = 1 - 1
[2,2]
 => 1 = 2 - 1
[2,1,1]
 => 1 = 2 - 1
[3,2]
 => 0 = 1 - 1
[3,1,1]
 => 1 = 2 - 1
[2,2,1]
 => 1 = 2 - 1
[3,2,1]
 => 0 = 1 - 1
[4,2,1]
 => 0 = 1 - 1
[3,3,1]
 => 1 = 2 - 1
[3,2,2]
 => 1 = 2 - 1
[3,2,1,1]
 => 1 = 2 - 1
[4,3,1]
 => 0 = 1 - 1
[4,2,2]
 => 1 = 2 - 1
[4,2,1,1]
 => 1 = 2 - 1
[3,3,2]
 => 1 = 2 - 1
[3,3,1,1]
 => 2 = 3 - 1
[3,2,2,1]
 => 1 = 2 - 1
[4,3,2]
 => 0 = 1 - 1
[4,3,1,1]
 => 1 = 2 - 1
[4,2,2,1]
 => 1 = 2 - 1
[3,3,2,1]
 => 1 = 2 - 1
[4,3,2,1]
 => 0 = 1 - 1
Description
The floored half-sum of the multiplicities of a partition. This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].
Matching statistic: St000257
(load all 4 compositions to match this statistic)
Mapping
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0 = 1 - 1
[2]
 => 0 = 1 - 1
[1,1]
 => 1 = 2 - 1
[3]
 => 0 = 1 - 1
[2,1]
 => 0 = 1 - 1
[1,1,1]
 => 1 = 2 - 1
[3,1]
 => 0 = 1 - 1
[2,2]
 => 1 = 2 - 1
[2,1,1]
 => 1 = 2 - 1
[3,2]
 => 0 = 1 - 1
[3,1,1]
 => 1 = 2 - 1
[2,2,1]
 => 1 = 2 - 1
[3,2,1]
 => 0 = 1 - 1
[4,2,1]
 => 0 = 1 - 1
[3,3,1]
 => 1 = 2 - 1
[3,2,2]
 => 1 = 2 - 1
[3,2,1,1]
 => 1 = 2 - 1
[4,3,1]
 => 0 = 1 - 1
[4,2,2]
 => 1 = 2 - 1
[4,2,1,1]
 => 1 = 2 - 1
[3,3,2]
 => 1 = 2 - 1
[3,3,1,1]
 => 2 = 3 - 1
[3,2,2,1]
 => 1 = 2 - 1
[4,3,2]
 => 0 = 1 - 1
[4,3,1,1]
 => 1 = 2 - 1
[4,2,2,1]
 => 1 = 2 - 1
[3,3,2,1]
 => 1 = 2 - 1
[4,3,2,1]
 => 0 = 1 - 1
Description
The number of distinct parts of a partition that occur at least twice. See Section 3.3.1 of [2].
Matching statistic: St000142
(load all 2 compositions to match this statistic)
Mapping
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => 0 = 1 - 1
[2]
 => [1,1]
 => 0 = 1 - 1
[1,1]
 => [2]
 => 1 = 2 - 1
[3]
 => [3]
 => 0 = 1 - 1
[2,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,1,1]
 => [2,1]
 => 1 = 2 - 1
[3,1]
 => [3,1]
 => 0 = 1 - 1
[2,2]
 => [4]
 => 1 = 2 - 1
[2,1,1]
 => [2,1,1]
 => 1 = 2 - 1
[3,2]
 => [3,1,1]
 => 0 = 1 - 1
[3,1,1]
 => [3,2]
 => 1 = 2 - 1
[2,2,1]
 => [4,1]
 => 1 = 2 - 1
[3,2,1]
 => [3,1,1,1]
 => 0 = 1 - 1
[4,2,1]
 => [1,1,1,1,1,1,1]
 => 0 = 1 - 1
[3,3,1]
 => [6,1]
 => 1 = 2 - 1
[3,2,2]
 => [4,3]
 => 1 = 2 - 1
[3,2,1,1]
 => [3,2,1,1]
 => 1 = 2 - 1
[4,3,1]
 => [3,1,1,1,1,1]
 => 0 = 1 - 1
[4,2,2]
 => [4,1,1,1,1]
 => 1 = 2 - 1
[4,2,1,1]
 => [2,1,1,1,1,1,1]
 => 1 = 2 - 1
[3,3,2]
 => [6,1,1]
 => 1 = 2 - 1
[3,3,1,1]
 => [6,2]
 => 2 = 3 - 1
[3,2,2,1]
 => [4,3,1]
 => 1 = 2 - 1
[4,3,2]
 => [3,1,1,1,1,1,1]
 => 0 = 1 - 1
[4,3,1,1]
 => [3,2,1,1,1,1]
 => 1 = 2 - 1
[4,2,2,1]
 => [4,1,1,1,1,1]
 => 1 = 2 - 1
[3,3,2,1]
 => [6,1,1,1]
 => 1 = 2 - 1
[4,3,2,1]
 => [3,1,1,1,1,1,1,1]
 => 0 = 1 - 1
Description
The number of even parts of a partition.
Matching statistic: St000149
(load all 2 compositions to match this statistic)
Mapping
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => 0 = 1 - 1
[2]
 => [1,1]
 => 0 = 1 - 1
[1,1]
 => [2]
 => 1 = 2 - 1
[3]
 => [1,1,1]
 => 0 = 1 - 1
[2,1]
 => [2,1]
 => 0 = 1 - 1
[1,1,1]
 => [3]
 => 1 = 2 - 1
[3,1]
 => [2,1,1]
 => 0 = 1 - 1
[2,2]
 => [2,2]
 => 1 = 2 - 1
[2,1,1]
 => [3,1]
 => 1 = 2 - 1
[3,2]
 => [2,2,1]
 => 0 = 1 - 1
[3,1,1]
 => [3,1,1]
 => 1 = 2 - 1
[2,2,1]
 => [3,2]
 => 1 = 2 - 1
[3,2,1]
 => [3,2,1]
 => 0 = 1 - 1
[4,2,1]
 => [3,2,1,1]
 => 0 = 1 - 1
[3,3,1]
 => [3,2,2]
 => 1 = 2 - 1
[3,2,2]
 => [3,3,1]
 => 1 = 2 - 1
[3,2,1,1]
 => [4,2,1]
 => 1 = 2 - 1
[4,3,1]
 => [3,2,2,1]
 => 0 = 1 - 1
[4,2,2]
 => [3,3,1,1]
 => 1 = 2 - 1
[4,2,1,1]
 => [4,2,1,1]
 => 1 = 2 - 1
[3,3,2]
 => [3,3,2]
 => 1 = 2 - 1
[3,3,1,1]
 => [4,2,2]
 => 2 = 3 - 1
[3,2,2,1]
 => [4,3,1]
 => 1 = 2 - 1
[4,3,2]
 => [3,3,2,1]
 => 0 = 1 - 1
[4,3,1,1]
 => [4,2,2,1]
 => 1 = 2 - 1
[4,2,2,1]
 => [4,3,1,1]
 => 1 = 2 - 1
[3,3,2,1]
 => [4,3,2]
 => 1 = 2 - 1
[4,3,2,1]
 => [4,3,2,1]
 => 0 = 1 - 1
Description
The number of cells of the partition whose leg is zero and arm is odd. This statistic is equidistributed with [[St000143]], see [1].
Matching statistic: St000256
(load all 4 compositions to match this statistic)
Mapping
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => 0 = 1 - 1
[2]
 => [1,1]
 => 0 = 1 - 1
[1,1]
 => [2]
 => 1 = 2 - 1
[3]
 => [1,1,1]
 => 0 = 1 - 1
[2,1]
 => [2,1]
 => 0 = 1 - 1
[1,1,1]
 => [3]
 => 1 = 2 - 1
[3,1]
 => [2,1,1]
 => 0 = 1 - 1
[2,2]
 => [2,2]
 => 1 = 2 - 1
[2,1,1]
 => [3,1]
 => 1 = 2 - 1
[3,2]
 => [2,2,1]
 => 0 = 1 - 1
[3,1,1]
 => [3,1,1]
 => 1 = 2 - 1
[2,2,1]
 => [3,2]
 => 1 = 2 - 1
[3,2,1]
 => [3,2,1]
 => 0 = 1 - 1
[4,2,1]
 => [3,2,1,1]
 => 0 = 1 - 1
[3,3,1]
 => [3,2,2]
 => 1 = 2 - 1
[3,2,2]
 => [3,3,1]
 => 1 = 2 - 1
[3,2,1,1]
 => [4,2,1]
 => 1 = 2 - 1
[4,3,1]
 => [3,2,2,1]
 => 0 = 1 - 1
[4,2,2]
 => [3,3,1,1]
 => 1 = 2 - 1
[4,2,1,1]
 => [4,2,1,1]
 => 1 = 2 - 1
[3,3,2]
 => [3,3,2]
 => 1 = 2 - 1
[3,3,1,1]
 => [4,2,2]
 => 2 = 3 - 1
[3,2,2,1]
 => [4,3,1]
 => 1 = 2 - 1
[4,3,2]
 => [3,3,2,1]
 => 0 = 1 - 1
[4,3,1,1]
 => [4,2,2,1]
 => 1 = 2 - 1
[4,2,2,1]
 => [4,3,1,1]
 => 1 = 2 - 1
[3,3,2,1]
 => [4,3,2]
 => 1 = 2 - 1
[4,3,2,1]
 => [4,3,2,1]
 => 0 = 1 - 1
Description
The number of parts from which one can substract 2 and still get an integer partition.
Matching statistic: St001037
(load all 20 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => 0 = 1 - 1
[2]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[1,1]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[2,1]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St001092
(load all 2 compositions to match this statistic)
Mapping
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => 0 = 1 - 1
[2]
 => [1,1]
 => 0 = 1 - 1
[1,1]
 => [2]
 => 1 = 2 - 1
[3]
 => [3]
 => 0 = 1 - 1
[2,1]
 => [1,1,1]
 => 0 = 1 - 1
[1,1,1]
 => [2,1]
 => 1 = 2 - 1
[3,1]
 => [3,1]
 => 0 = 1 - 1
[2,2]
 => [4]
 => 1 = 2 - 1
[2,1,1]
 => [2,1,1]
 => 1 = 2 - 1
[3,2]
 => [3,1,1]
 => 0 = 1 - 1
[3,1,1]
 => [3,2]
 => 1 = 2 - 1
[2,2,1]
 => [4,1]
 => 1 = 2 - 1
[3,2,1]
 => [3,1,1,1]
 => 0 = 1 - 1
[4,2,1]
 => [1,1,1,1,1,1,1]
 => 0 = 1 - 1
[3,3,1]
 => [6,1]
 => 1 = 2 - 1
[3,2,2]
 => [4,3]
 => 1 = 2 - 1
[3,2,1,1]
 => [3,2,1,1]
 => 1 = 2 - 1
[4,3,1]
 => [3,1,1,1,1,1]
 => 0 = 1 - 1
[4,2,2]
 => [4,1,1,1,1]
 => 1 = 2 - 1
[4,2,1,1]
 => [2,1,1,1,1,1,1]
 => 1 = 2 - 1
[3,3,2]
 => [6,1,1]
 => 1 = 2 - 1
[3,3,1,1]
 => [6,2]
 => 2 = 3 - 1
[3,2,2,1]
 => [4,3,1]
 => 1 = 2 - 1
[4,3,2]
 => [3,1,1,1,1,1,1]
 => 0 = 1 - 1
[4,3,1,1]
 => [3,2,1,1,1,1]
 => 1 = 2 - 1
[4,2,2,1]
 => [4,1,1,1,1,1]
 => 1 = 2 - 1
[3,3,2,1]
 => [6,1,1,1]
 => 1 = 2 - 1
[4,3,2,1]
 => [3,1,1,1,1,1,1,1]
 => 0 = 1 - 1
Description
The number of distinct even parts of a partition. See Section 3.3.1 of [1].
Matching statistic: St000035
(load all 3 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [3,1,2] => 1
[2]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => 1
[1,1]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 1
[2,1]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 2
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => 2
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => 2
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 2
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => 2
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => 2
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => 3
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => 2
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => 2
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => 2
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => 2
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 1
Description
The number of left outer peaks of a permutation. A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$. In other words, it is a peak in the word $[0,w_1,..., w_n]$. This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.
Matching statistic: St000099
(load all 6 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [2,1] => 1
[2]
 => [1,1,0,0,1,0]
 => [3,1,2] => 1
[1,1]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 1
[2,1]
 => [1,0,1,0,1,0]
 => [3,2,1] => 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 2
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [3,4,2,1] => 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => 2
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => 2
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => 2
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => 2
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => 2
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => 3
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => 2
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => 2
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => 2
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => 2
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => 1
Description
The number of valleys of a permutation, including the boundary. The number of valleys excluding the boundary is [[St000353]].
Matching statistic: St000201
(load all 6 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1,0,1,0]
 => [[.,.],.]
 => 1
[2]
 => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [[.,.],[.,.]]
 => 2
[3]
 => [1,1,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],.]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[.,.],[.,[.,.]]]
 => 2
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [[.,[.,.]],[.,.]]
 => 2
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [[[.,.],[.,.]],.]
 => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [[[.,.],.],[.,.]]
 => 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [[[[.,.],.],.],.]
 => 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[.,[[[.,.],.],.]],.]
 => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[.,[[.,.],.]],[.,.]]
 => 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => 2
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[.,.],[[[.,.],.],.]]
 => 2
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[[.,[[.,.],.]],.],.]
 => 1
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[[.,[.,.]],[.,.]],.]
 => 2
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [[[.,.],[[.,.],.]],.]
 => 2
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[[.,[.,.]],.],[.,.]]
 => 2
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[[.,.],[.,.]],[.,.]]
 => 3
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => 2
[4,3,2]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[[[.,[.,.]],.],.],.]
 => 1
[4,3,1,1]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[[[.,.],[.,.]],.],.]
 => 2
[4,2,2,1]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[[[.,.],.],[.,.]],.]
 => 2
[3,3,2,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[[[.,.],.],.],[.,.]]
 => 2
[4,3,2,1]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[[[[.,.],.],.],.],.]
 => 1
Description
The number of leaf nodes in a binary tree. Equivalently, the number of cherries [1] in the complete binary tree. The number of binary trees of size $n$, at least $1$, with exactly one leaf node for is $2^{n-1}$, see [2]. The number of binary tree of size $n$, at least $3$, with exactly two leaf nodes is $n(n+1)2^{n-2}$, see [3].
The following 113 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000659The number of rises of length at least 2 of a Dyck path. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000023The number of inner peaks of a permutation. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000386The number of factors DDU in a Dyck path. St000779The tier of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000071The number of maximal chains in a poset. St000092The number of outer peaks of a permutation. St000159The number of distinct parts of the integer partition. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000354The number of recoils of a permutation. St000390The number of runs of ones in a binary word. St000527The width of the poset. St000568The hook number of a binary tree. St000742The number of big ascents of a permutation after prepending zero. St000834The number of right outer peaks of a permutation. St000919The number of maximal left branches of a binary tree. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001487The number of inner corners of a skew partition. St001732The number of peaks visible from the left. St000252The number of nodes of degree 3 of a binary tree. St000288The number of ones in a binary word. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000353The number of inner valleys of a permutation. St000360The number of occurrences of the pattern 32-1. St000392The length of the longest run of ones in a binary word. St000523The number of 2-protected nodes of a rooted tree. St000632The jump number of the poset. St000646The number of big ascents of a permutation. St000647The number of big descents of a permutation. St000711The number of big exceedences of a permutation. St000753The Grundy value for the game of Kayles on a binary word. St000992The alternating sum of the parts of an integer partition. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001549The number of restricted non-inversions between exceedances. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001840The number of descents of a set partition. St001597The Frobenius rank of a skew partition. St000665The number of rafts of a permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000764The number of strong records in an integer composition. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St000805The number of peaks of the associated bargraph. St000807The sum of the heights of the valleys of the associated bargraph. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001520The number of strict 3-descents. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St000007The number of saliances of the permutation. St000455The second largest eigenvalue of a graph if it is integral. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000939The number of characters of the symmetric group whose value on the partition is positive. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000941The number of characters of the symmetric group whose value on the partition is even. St000090The variation of a composition. St000091The descent variation of a composition. St000498The lcs statistic of a set partition. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000668The least common multiple of the parts of the partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St001946The number of descents in a parking function. St000365The number of double ascents of a permutation. St000383The last part of an integer composition. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000709The number of occurrences of 14-2-3 or 14-3-2. St001568The smallest positive integer that does not appear twice in the partition. St001868The number of alignments of type NE of a signed permutation. St000264The girth of a graph, which is not a tree. St000284The Plancherel distribution on integer partitions. St000509The diagonal index (content) of a partition. St000929The constant term of the character polynomial of an integer partition. St001516The number of cyclic bonds of a permutation. St000735The last entry on the main diagonal of a standard tableau. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000454The largest eigenvalue of a graph if it is integral. St000736The last entry in the first row of a semistandard tableau. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001569The maximal modular displacement of a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000075The orbit size of a standard tableau under promotion. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001095The number of non-isomorphic posets with precisely one further covering relation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001948The number of augmented double ascents of a permutation.