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Your data matches 132 different statistics following compositions of up to 3 maps.
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Matching statistic: St000149
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
St000149: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0 = 1 - 1
[2]
=> 1 = 2 - 1
[1,1]
=> 0 = 1 - 1
[3]
=> 1 = 2 - 1
[2,1]
=> 0 = 1 - 1
[1,1,1]
=> 0 = 1 - 1
[3,1]
=> 1 = 2 - 1
[2,2]
=> 1 = 2 - 1
[2,1,1]
=> 0 = 1 - 1
[3,2]
=> 1 = 2 - 1
[3,1,1]
=> 1 = 2 - 1
[2,2,1]
=> 0 = 1 - 1
[3,2,1]
=> 0 = 1 - 1
[4,2,1]
=> 1 = 2 - 1
[3,3,1]
=> 1 = 2 - 1
[3,2,2]
=> 1 = 2 - 1
[3,2,1,1]
=> 0 = 1 - 1
[4,3,1]
=> 1 = 2 - 1
[4,2,2]
=> 2 = 3 - 1
[4,2,1,1]
=> 1 = 2 - 1
[3,3,2]
=> 1 = 2 - 1
[3,3,1,1]
=> 1 = 2 - 1
[3,2,2,1]
=> 0 = 1 - 1
[4,3,2]
=> 1 = 2 - 1
[4,3,1,1]
=> 1 = 2 - 1
[4,2,2,1]
=> 1 = 2 - 1
[3,3,2,1]
=> 0 = 1 - 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 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
St000256: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0 = 1 - 1
[2]
=> 1 = 2 - 1
[1,1]
=> 0 = 1 - 1
[3]
=> 1 = 2 - 1
[2,1]
=> 0 = 1 - 1
[1,1,1]
=> 0 = 1 - 1
[3,1]
=> 1 = 2 - 1
[2,2]
=> 1 = 2 - 1
[2,1,1]
=> 0 = 1 - 1
[3,2]
=> 1 = 2 - 1
[3,1,1]
=> 1 = 2 - 1
[2,2,1]
=> 0 = 1 - 1
[3,2,1]
=> 0 = 1 - 1
[4,2,1]
=> 1 = 2 - 1
[3,3,1]
=> 1 = 2 - 1
[3,2,2]
=> 1 = 2 - 1
[3,2,1,1]
=> 0 = 1 - 1
[4,3,1]
=> 1 = 2 - 1
[4,2,2]
=> 2 = 3 - 1
[4,2,1,1]
=> 1 = 2 - 1
[3,3,2]
=> 1 = 2 - 1
[3,3,1,1]
=> 1 = 2 - 1
[3,2,2,1]
=> 0 = 1 - 1
[4,3,2]
=> 1 = 2 - 1
[4,3,1,1]
=> 1 = 2 - 1
[4,2,2,1]
=> 1 = 2 - 1
[3,3,2,1]
=> 0 = 1 - 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: St000142
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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]
=> [2]
=> 1 = 2 - 1
[1,1]
=> [1,1]
=> 0 = 1 - 1
[3]
=> [2,1]
=> 1 = 2 - 1
[2,1]
=> [3]
=> 0 = 1 - 1
[1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[3,1]
=> [2,1,1]
=> 1 = 2 - 1
[2,2]
=> [4]
=> 1 = 2 - 1
[2,1,1]
=> [3,1]
=> 0 = 1 - 1
[3,2]
=> [4,1]
=> 1 = 2 - 1
[3,1,1]
=> [2,1,1,1]
=> 1 = 2 - 1
[2,2,1]
=> [5]
=> 0 = 1 - 1
[3,2,1]
=> [3,3]
=> 0 = 1 - 1
[4,2,1]
=> [5,2]
=> 1 = 2 - 1
[3,3,1]
=> [3,2,1,1]
=> 1 = 2 - 1
[3,2,2]
=> [6,1]
=> 1 = 2 - 1
[3,2,1,1]
=> [3,3,1]
=> 0 = 1 - 1
[4,3,1]
=> [4,3,1]
=> 1 = 2 - 1
[4,2,2]
=> [6,2]
=> 2 = 3 - 1
[4,2,1,1]
=> [6,1,1]
=> 1 = 2 - 1
[3,3,2]
=> [5,2,1]
=> 1 = 2 - 1
[3,3,1,1]
=> [3,2,1,1,1]
=> 1 = 2 - 1
[3,2,2,1]
=> [5,3]
=> 0 = 1 - 1
[4,3,2]
=> [6,3]
=> 1 = 2 - 1
[4,3,1,1]
=> [5,2,1,1]
=> 1 = 2 - 1
[4,2,2,1]
=> [7,2]
=> 1 = 2 - 1
[3,3,2,1]
=> [3,3,3]
=> 0 = 1 - 1
[4,3,2,1]
=> [5,5]
=> 0 = 1 - 1
Description
The number of even parts of a partition.
Matching statistic: St000150
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000150: 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]
=> 1 = 2 - 1
[1,1]
=> [2]
=> 0 = 1 - 1
[3]
=> [1,1,1]
=> 1 = 2 - 1
[2,1]
=> [2,1]
=> 0 = 1 - 1
[1,1,1]
=> [3]
=> 0 = 1 - 1
[3,1]
=> [2,1,1]
=> 1 = 2 - 1
[2,2]
=> [2,2]
=> 1 = 2 - 1
[2,1,1]
=> [3,1]
=> 0 = 1 - 1
[3,2]
=> [2,2,1]
=> 1 = 2 - 1
[3,1,1]
=> [3,1,1]
=> 1 = 2 - 1
[2,2,1]
=> [3,2]
=> 0 = 1 - 1
[3,2,1]
=> [3,2,1]
=> 0 = 1 - 1
[4,2,1]
=> [3,2,1,1]
=> 1 = 2 - 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]
=> 0 = 1 - 1
[4,3,1]
=> [3,2,2,1]
=> 1 = 2 - 1
[4,2,2]
=> [3,3,1,1]
=> 2 = 3 - 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]
=> 1 = 2 - 1
[3,2,2,1]
=> [4,3,1]
=> 0 = 1 - 1
[4,3,2]
=> [3,3,2,1]
=> 1 = 2 - 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]
=> 0 = 1 - 1
[4,3,2,1]
=> [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 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000257: 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]
=> 1 = 2 - 1
[1,1]
=> [2]
=> 0 = 1 - 1
[3]
=> [1,1,1]
=> 1 = 2 - 1
[2,1]
=> [2,1]
=> 0 = 1 - 1
[1,1,1]
=> [3]
=> 0 = 1 - 1
[3,1]
=> [2,1,1]
=> 1 = 2 - 1
[2,2]
=> [2,2]
=> 1 = 2 - 1
[2,1,1]
=> [3,1]
=> 0 = 1 - 1
[3,2]
=> [2,2,1]
=> 1 = 2 - 1
[3,1,1]
=> [3,1,1]
=> 1 = 2 - 1
[2,2,1]
=> [3,2]
=> 0 = 1 - 1
[3,2,1]
=> [3,2,1]
=> 0 = 1 - 1
[4,2,1]
=> [3,2,1,1]
=> 1 = 2 - 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]
=> 0 = 1 - 1
[4,3,1]
=> [3,2,2,1]
=> 1 = 2 - 1
[4,2,2]
=> [3,3,1,1]
=> 2 = 3 - 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]
=> 1 = 2 - 1
[3,2,2,1]
=> [4,3,1]
=> 0 = 1 - 1
[4,3,2]
=> [3,3,2,1]
=> 1 = 2 - 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]
=> 0 = 1 - 1
[4,3,2,1]
=> [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: St000386
(load all 39 compositions to match this statistic)
(load all 39 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000386: 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]
=> 1 = 2 - 1
[1,1]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,1]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 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]
=> 0 = 1 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 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]
=> 0 = 1 - 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]
=> 1 = 2 - 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]
=> 0 = 1 - 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2 = 3 - 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]
=> 1 = 2 - 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 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]
=> 0 = 1 - 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St001092
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
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]
=> [2]
=> 1 = 2 - 1
[1,1]
=> [1,1]
=> 0 = 1 - 1
[3]
=> [2,1]
=> 1 = 2 - 1
[2,1]
=> [3]
=> 0 = 1 - 1
[1,1,1]
=> [1,1,1]
=> 0 = 1 - 1
[3,1]
=> [2,1,1]
=> 1 = 2 - 1
[2,2]
=> [4]
=> 1 = 2 - 1
[2,1,1]
=> [3,1]
=> 0 = 1 - 1
[3,2]
=> [4,1]
=> 1 = 2 - 1
[3,1,1]
=> [2,1,1,1]
=> 1 = 2 - 1
[2,2,1]
=> [5]
=> 0 = 1 - 1
[3,2,1]
=> [3,3]
=> 0 = 1 - 1
[4,2,1]
=> [5,2]
=> 1 = 2 - 1
[3,3,1]
=> [3,2,1,1]
=> 1 = 2 - 1
[3,2,2]
=> [6,1]
=> 1 = 2 - 1
[3,2,1,1]
=> [3,3,1]
=> 0 = 1 - 1
[4,3,1]
=> [4,3,1]
=> 1 = 2 - 1
[4,2,2]
=> [6,2]
=> 2 = 3 - 1
[4,2,1,1]
=> [6,1,1]
=> 1 = 2 - 1
[3,3,2]
=> [5,2,1]
=> 1 = 2 - 1
[3,3,1,1]
=> [3,2,1,1,1]
=> 1 = 2 - 1
[3,2,2,1]
=> [5,3]
=> 0 = 1 - 1
[4,3,2]
=> [6,3]
=> 1 = 2 - 1
[4,3,1,1]
=> [5,2,1,1]
=> 1 = 2 - 1
[4,2,2,1]
=> [7,2]
=> 1 = 2 - 1
[3,3,2,1]
=> [3,3,3]
=> 0 = 1 - 1
[4,3,2,1]
=> [5,5]
=> 0 = 1 - 1
Description
The number of distinct even parts of a partition.
See Section 3.3.1 of [1].
Matching statistic: St000092
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,2] => 1
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[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] => 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 2
[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
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 2
[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
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 3
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 2
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
Description
The number of outer peaks of a permutation.
An 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 $w_1 > w_2$ or $n$ if $w_{n} > w_{n-1}$.
In other words, it is a peak in the word $[0,w_1,..., w_n,0]$.
Matching statistic: St000201
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ 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]
=> [[.,.],[.,.]]
=> 2
[1,1]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [[[.,.],.],[.,.]]
=> 2
[2,1]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [.,[[[.,.],.],.]]
=> 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[.,[.,.]],[.,.]]
=> 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [.,[[.,[.,.]],.]]
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,.]]]
=> 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 1
[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]
=> [[.,[.,[.,.]]],[.,.]]
=> 2
[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]
=> [.,[[.,[.,[.,.]]],.]]
=> 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[.,.],[[.,.],[.,.]]]
=> 3
[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]
=> [.,[[.,.],[[.,.],.]]]
=> 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [.,[.,[[.,[.,.]],.]]]
=> 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 2
[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]
=> [.,[.,[.,[[.,.],.]]]]
=> 1
[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].
Matching statistic: St001487
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Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [[1,1],[]]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> 2
[1,1]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> 2
[2,1]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> 3
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> 2
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> 2
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> 2
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> 2
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> 1
Description
The number of inner corners of a skew partition.
The following 122 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000353The number of inner valleys of a permutation. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001960The number of descents of a permutation minus one if its first entry is not one. St000021The number of descents of a permutation. St000035The number of left outer peaks of a permutation. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000099The number of valleys of a permutation, including the boundary. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000251The number of nonsingleton blocks of a set partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000390The number of runs of ones in a binary word. St000527The width of the poset. St000568The hook number of a binary tree. St000647The number of big descents of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000884The number of isolated descents 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. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. 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. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001732The number of peaks visible from the left. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000023The number of inner peaks of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000159The number of distinct parts of the integer partition. 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. St000325The width of the tree associated to a permutation. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000392The length of the longest run of ones in a binary word. St000470The number of runs in a permutation. St000523The number of 2-protected nodes of a rooted tree. St000632The jump number of the poset. St000646The number of big ascents of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000711The number of big exceedences of a permutation. St000753The Grundy value for the game of Kayles on a binary word. St000779The tier of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000992The alternating sum of the parts of an integer partition. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. 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. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001372The length of a longest cyclic run of ones of a binary word. St001394The genus of a permutation. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001712The number of natural descents of a standard Young tableau. 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. St000660The number of rises of length at least 3 of a Dyck path. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000665The number of rafts of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001569The maximal modular displacement of a permutation. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001948The number of augmented double ascents of a permutation. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000352The Elizalde-Pak rank of a permutation. St000054The first entry of the permutation. St000455The second largest eigenvalue of a graph if it is integral. St001568The smallest positive integer that does not appear twice in the partition. St000929The constant term of the character polynomial of an integer partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000492The rob statistic of a set partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000654The first descent of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001151The number of blocks with odd minimum. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St001896The number of right descents of a signed permutations. St001904The length of the initial strictly increasing segment of a parking function. St000075The orbit size of a standard tableau under promotion. St000089The absolute variation of a composition. St000091The descent variation of a composition. St000260The radius of a connected graph. St000542The number of left-to-right-minima of a permutation. St000562The number of internal points of a set partition. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000839The largest opener of a set partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001867The number of alignments of type EN of a signed permutation. St000230Sum of the minimal elements of the blocks of a set partition. St000264The girth of a graph, which is not a tree. St000090The variation of a composition. St000454The largest eigenvalue of a graph if it is integral. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St000259The diameter of a connected graph. St000307The number of rowmotion orbits of a poset. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001811The Castelnuovo-Mumford regularity of a permutation. St000717The number of ordinal summands of a poset.
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