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Your data matches 15 different statistics following compositions of up to 3 maps.
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Matching statistic: St001092
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 1
[1,1]
=> 0
[3]
=> 0
[2,1]
=> 1
[1,1,1]
=> 0
[4]
=> 1
[3,1]
=> 0
[2,2]
=> 1
[2,1,1]
=> 1
[1,1,1,1]
=> 0
[5]
=> 0
[4,1]
=> 1
[3,2]
=> 1
[3,1,1]
=> 0
[2,2,1]
=> 1
[2,1,1,1]
=> 1
[1,1,1,1,1]
=> 0
[6]
=> 1
[5,1]
=> 0
[4,2]
=> 2
[4,1,1]
=> 1
[3,3]
=> 0
[3,2,1]
=> 1
[3,1,1,1]
=> 0
[2,2,2]
=> 1
[2,2,1,1]
=> 1
[2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 0
[7]
=> 0
[6,1]
=> 1
[5,2]
=> 1
[5,1,1]
=> 0
[4,3]
=> 1
[4,2,1]
=> 2
[4,1,1,1]
=> 1
[3,3,1]
=> 0
[3,2,2]
=> 1
[3,2,1,1]
=> 1
[3,1,1,1,1]
=> 0
[2,2,2,1]
=> 1
[2,2,1,1,1]
=> 1
[2,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> 0
[8]
=> 1
[7,1]
=> 0
[6,2]
=> 2
[6,1,1]
=> 1
[5,3]
=> 0
[5,2,1]
=> 1
Description
The number of distinct even parts of a partition.
See Section 3.3.1 of [1].
Matching statistic: St000257
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 44% ●values known / values provided: 48%●distinct values known / distinct values provided: 44%
St000257: Integer partitions ⟶ ℤResult quality: 44% ●values known / values provided: 48%●distinct values known / distinct values provided: 44%
Values
[1]
=> [1]
=> 0
[2]
=> [1,1]
=> 1
[1,1]
=> [2]
=> 0
[3]
=> [3]
=> 0
[2,1]
=> [1,1,1]
=> 1
[1,1,1]
=> [2,1]
=> 0
[4]
=> [2,2]
=> 1
[3,1]
=> [3,1]
=> 0
[2,2]
=> [1,1,1,1]
=> 1
[2,1,1]
=> [2,1,1]
=> 1
[1,1,1,1]
=> [4]
=> 0
[5]
=> [5]
=> 0
[4,1]
=> [2,2,1]
=> 1
[3,2]
=> [3,1,1]
=> 1
[3,1,1]
=> [3,2]
=> 0
[2,2,1]
=> [1,1,1,1,1]
=> 1
[2,1,1,1]
=> [2,1,1,1]
=> 1
[1,1,1,1,1]
=> [4,1]
=> 0
[6]
=> [3,3]
=> 1
[5,1]
=> [5,1]
=> 0
[4,2]
=> [2,2,1,1]
=> 2
[4,1,1]
=> [2,2,2]
=> 1
[3,3]
=> [6]
=> 0
[3,2,1]
=> [3,1,1,1]
=> 1
[3,1,1,1]
=> [3,2,1]
=> 0
[2,2,2]
=> [1,1,1,1,1,1]
=> 1
[2,2,1,1]
=> [2,1,1,1,1]
=> 1
[2,1,1,1,1]
=> [4,1,1]
=> 1
[1,1,1,1,1,1]
=> [4,2]
=> 0
[7]
=> [7]
=> 0
[6,1]
=> [3,3,1]
=> 1
[5,2]
=> [5,1,1]
=> 1
[5,1,1]
=> [5,2]
=> 0
[4,3]
=> [3,2,2]
=> 1
[4,2,1]
=> [2,2,1,1,1]
=> 2
[4,1,1,1]
=> [2,2,2,1]
=> 1
[3,3,1]
=> [6,1]
=> 0
[3,2,2]
=> [3,1,1,1,1]
=> 1
[3,2,1,1]
=> [3,2,1,1]
=> 1
[3,1,1,1,1]
=> [4,3]
=> 0
[2,2,2,1]
=> [1,1,1,1,1,1,1]
=> 1
[2,2,1,1,1]
=> [2,1,1,1,1,1]
=> 1
[2,1,1,1,1,1]
=> [4,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> [4,2,1]
=> 0
[8]
=> [4,4]
=> 1
[7,1]
=> [7,1]
=> 0
[6,2]
=> [3,3,1,1]
=> 2
[6,1,1]
=> [3,3,2]
=> 1
[5,3]
=> [5,3]
=> 0
[5,2,1]
=> [5,1,1,1]
=> 1
[9,1,1]
=> [9,2]
=> ? = 0
[7,3,1]
=> [7,3,1]
=> ? = 0
[7,2,1,1]
=> [7,2,1,1]
=> ? = 1
[6,2,1,1,1]
=> [3,3,2,1,1,1]
=> ? = 2
[4,2,1,1,1,1,1]
=> [4,2,2,1,1,1]
=> ? = 2
[3,3,2,1,1,1]
=> [6,2,1,1,1]
=> ? = 1
[2,2,1,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> ? = 1
[9,2,1]
=> [9,1,1,1]
=> ? = 1
[9,1,1,1]
=> [9,2,1]
=> ? = 0
[8,2,2]
=> [4,4,1,1,1,1]
=> ? = 2
[7,4,1]
=> [7,2,2,1]
=> ? = 1
[7,3,2]
=> [7,3,1,1]
=> ? = 1
[7,3,1,1]
=> [7,3,2]
=> ? = 0
[7,2,2,1]
=> [7,1,1,1,1,1]
=> ? = 1
[7,2,1,1,1]
=> [7,2,1,1,1]
=> ? = 1
[6,3,2,1]
=> [3,3,3,1,1,1]
=> ? = 2
[6,2,2,1,1]
=> [3,3,2,1,1,1,1]
=> ? = 2
[4,4,1,1,1,1]
=> [4,2,2,2,2]
=> ? = 1
[4,3,2,1,1,1]
=> [3,2,2,2,1,1,1]
=> ? = 2
[4,2,2,2,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> ? = 2
[4,2,2,1,1,1,1]
=> [4,2,2,1,1,1,1]
=> ? = 2
[4,2,1,1,1,1,1,1]
=> [4,2,2,2,1,1]
=> ? = 2
[3,3,3,1,1,1]
=> [6,3,2,1]
=> ? = 0
[3,2,2,2,2,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> ? = 1
[3,2,2,2,1,1,1]
=> [3,2,1,1,1,1,1,1,1]
=> ? = 1
[3,2,1,1,1,1,1,1,1]
=> [4,3,2,1,1,1]
=> ? = 1
[2,2,2,2,2,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[2,2,2,2,1,1,1,1]
=> [4,1,1,1,1,1,1,1,1]
=> ? = 1
[2,2,2,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1,1]
=> ? = 1
[9,2,2]
=> [9,1,1,1,1]
=> ? = 1
[7,4,2]
=> [7,2,2,1,1]
=> ? = 2
[6,4,2,1]
=> [3,3,2,2,1,1,1]
=> ? = 3
[6,3,2,2]
=> [3,3,3,1,1,1,1]
=> ? = 2
[5,4,4]
=> [5,2,2,2,2]
=> ? = 1
[3,3,2,1,1,1,1,1]
=> [6,4,1,1,1]
=> ? = 1
[3,2,2,2,2,1,1]
=> [3,2,1,1,1,1,1,1,1,1]
=> ? = 1
[2,2,2,2,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1]
=> ? = 1
[8,5,1]
=> [5,4,4,1]
=> ? = 1
[8,4,2]
=> [4,4,2,2,1,1]
=> ? = 3
[7,5,2]
=> [7,5,1,1]
=> ? = 1
[7,4,3]
=> [7,3,2,2]
=> ? = 1
[6,4,2,2]
=> [3,3,2,2,1,1,1,1]
=> ? = 3
[6,1,1,1,1,1,1,1,1]
=> [8,3,3]
=> ? = 1
[5,5,4]
=> [10,2,2]
=> ? = 1
[5,5,1,1,1,1]
=> [10,4]
=> ? = 0
[5,4,2,1,1,1]
=> [5,2,2,2,1,1,1]
=> ? = 2
[5,2,2,2,2,1]
=> [5,1,1,1,1,1,1,1,1,1]
=> ? = 1
[4,4,4,2]
=> [2,2,2,2,2,2,1,1]
=> ? = 2
[4,3,2,2,2,1]
=> [3,2,2,1,1,1,1,1,1,1]
=> ? = 2
[3,3,3,2,2,1]
=> [6,3,1,1,1,1,1]
=> ? = 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 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 44%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 44%
Values
[1]
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[2]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1]
=> [2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 0
[3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[2,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[4]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,2]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,1]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[4,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,2]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,1,1]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[2,2,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,1,1,1,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
[6]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[5,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 0
[4,2]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[4,1,1]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[3,3]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[3,2,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[3,1,1,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,2,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
[2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 1
[2,1,1,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,1,1,1,1]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0
[7]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 0
[6,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[5,2]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 1
[5,1,1]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 0
[4,3]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[4,2,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 2
[4,1,1,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[3,3,1]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 0
[3,2,2]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 1
[3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[3,1,1,1,1]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0
[2,2,2,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> 1
[2,2,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> 1
[2,1,1,1,1,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,1,1,1,1,1]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[8]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
[7,1]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 0
[6,2]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[6,1,1]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[5,3]
=> [5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 0
[5,2,1]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 1
[11]
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [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
[9,1,1]
=> [9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[5,5,1]
=> [10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[4,4,2,1]
=> [2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 2
[4,3,2,2]
=> [3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> ? = 2
[4,2,2,2,1]
=> [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]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 2
[4,2,2,1,1,1]
=> [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]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> ? = 2
[3,2,2,2,1,1]
=> [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]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 1
[3,1,1,1,1,1,1,1,1]
=> [8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[2,2,2,2,2,1]
=> [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]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 1
[2,2,2,2,1,1,1]
=> [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]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 1
[2,2,1,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 1
[2,1,1,1,1,1,1,1,1,1]
=> [8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 0
[11,1]
=> [11,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ? = 0
[9,3]
=> [9,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[9,2,1]
=> [9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> ? = 1
[9,1,1,1]
=> [9,2,1]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? = 0
[6,2,2,1,1]
=> [3,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> ? = 2
[5,5,2]
=> [10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ? = 1
[5,5,1,1]
=> [10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[5,2,2,2,1]
=> [5,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[5,2,2,1,1,1]
=> [5,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 1
[4,4,2,2]
=> [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]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> ? = 2
[4,4,2,1,1]
=> [2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 2
[4,3,2,2,1]
=> [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]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 2
[4,3,2,1,1,1]
=> [3,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> ? = 2
[4,2,2,2,2]
=> [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]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 2
[4,2,2,2,1,1]
=> [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]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> ? = 2
[4,2,2,1,1,1,1]
=> [4,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 2
[4,1,1,1,1,1,1,1,1]
=> [8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> ? = 1
[3,3,3,3]
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [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]
=> ? = 0
[3,3,2,2,2]
=> [6,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[3,2,2,2,2,1]
=> [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]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 1
[3,2,2,2,1,1,1]
=> [3,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[3,1,1,1,1,1,1,1,1,1]
=> [8,3,1]
=> [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> ? = 0
[2,2,2,2,2,2]
=> [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
[2,2,2,2,2,1,1]
=> [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]
=> ?
=> ? = 1
[2,2,2,2,1,1,1,1]
=> [4,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 1
[2,2,2,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 1
[2,2,1,1,1,1,1,1,1,1]
=> [8,1,1,1,1]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? = 1
[2,1,1,1,1,1,1,1,1,1,1]
=> [8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [8,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[13]
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [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
[9,2,2]
=> [9,1,1,1,1]
=> [1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> ? = 1
[6,4,2,1]
=> [3,3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? = 3
[6,2,2,1,1,1]
=> [3,3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 2
[5,5,3]
=> [10,3]
=> [1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 0
[5,4,2,2]
=> [5,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 2
[5,3,2,2,1]
=> [5,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 1
Description
The number of factors DDU in a Dyck path.
Matching statistic: St000201
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
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: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 44%
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: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 44%
Values
[1]
=> [1]
=> [1,0,1,0]
=> [[.,.],.]
=> 1 = 0 + 1
[2]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 2 = 1 + 1
[1,1]
=> [2]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 1 = 0 + 1
[3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> 1 = 0 + 1
[2,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> 2 = 1 + 1
[1,1,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 1 = 0 + 1
[4]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> 2 = 1 + 1
[3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[.,[[.,.],.]],.]
=> 1 = 0 + 1
[2,2]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 2 = 1 + 1
[2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> 2 = 1 + 1
[1,1,1,1]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> 1 = 0 + 1
[5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> 1 = 0 + 1
[4,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> 2 = 1 + 1
[3,2]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> 2 = 1 + 1
[3,1,1]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> 1 = 0 + 1
[2,2,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> 2 = 1 + 1
[2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 2 = 1 + 1
[1,1,1,1,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[.,[.,[[.,.],.]]],.]
=> 1 = 0 + 1
[6]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> 2 = 1 + 1
[5,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[.,[.,[.,[[.,.],.]]]],.]
=> 1 = 0 + 1
[4,2]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> 3 = 2 + 1
[4,1,1]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 2 = 1 + 1
[3,3]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> 1 = 0 + 1
[3,2,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 2 = 1 + 1
[3,1,1,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> 1 = 0 + 1
[2,2,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> 2 = 1 + 1
[2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[.,.],[.,[.,[[.,.],.]]]]
=> 2 = 1 + 1
[2,1,1,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],[.,.]]],.]
=> 2 = 1 + 1
[1,1,1,1,1,1]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> 1 = 0 + 1
[7]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> 1 = 0 + 1
[6,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> 2 = 1 + 1
[5,2]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[.,[.,[[.,.],[.,.]]]],.]
=> 2 = 1 + 1
[5,1,1]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[.,[.,[[.,[.,.]],.]]],.]
=> 1 = 0 + 1
[4,3]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 2 = 1 + 1
[4,2,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> 3 = 2 + 1
[4,1,1,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> 2 = 1 + 1
[3,3,1]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[[.,.],.]]]]],.]
=> 1 = 0 + 1
[3,2,2]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[.,.],[.,[[.,[.,.]],.]]]
=> 2 = 1 + 1
[3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 2 = 1 + 1
[3,1,1,1,1]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> 1 = 0 + 1
[2,2,2,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> 2 = 1 + 1
[2,2,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> 2 = 1 + 1
[2,1,1,1,1,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> 2 = 1 + 1
[1,1,1,1,1,1,1]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[.,[[[.,.],.],.]],.]
=> 1 = 0 + 1
[8]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> 2 = 1 + 1
[7,1]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[[.,.],.]]]]]],.]
=> 1 = 0 + 1
[6,2]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> 3 = 2 + 1
[6,1,1]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> 2 = 1 + 1
[5,3]
=> [5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[.,[[.,[.,[.,.]]],.]],.]
=> 1 = 0 + 1
[5,2,1]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[.,[[.,.],[.,[.,.]]]],.]
=> 2 = 1 + 1
[11]
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]],.]
=> ? = 0 + 1
[9,2]
=> [9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]],.]
=> ? = 1 + 1
[9,1,1]
=> [9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]],.]
=> ? = 0 + 1
[7,4]
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[[.,[.,.]],[.,.]]]]],.]
=> ? = 1 + 1
[7,3,1]
=> [7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [[.,[.,[.,[[.,[[.,.],.]],.]]]],.]
=> ? = 0 + 1
[7,2,2]
=> [7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[.,[.,[[.,.],[.,[.,[.,.]]]]]],.]
=> ? = 1 + 1
[7,2,1,1]
=> [7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[[.,.],[[.,.],.]]]]],.]
=> ? = 1 + 1
[7,1,1,1,1]
=> [7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [[.,[.,[[.,[.,[.,[.,.]]]],.]]],.]
=> ? = 0 + 1
[6,2,2,1]
=> [3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
=> ? = 2 + 1
[5,5,1]
=> [10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]],.]
=> ? = 0 + 1
[5,2,2,2]
=> [5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [[.,.],[.,[[.,[.,[.,[.,.]]]],.]]]
=> ? = 1 + 1
[4,4,2,1]
=> [2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
=> ? = 2 + 1
[4,3,2,2]
=> [3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
=> ? = 2 + 1
[4,2,2,2,1]
=> [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]
=> [[.,.],[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]]
=> ? = 2 + 1
[4,2,2,1,1,1]
=> [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]
=> [[.,.],[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> ? = 2 + 1
[3,2,2,2,2]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]
=> ? = 1 + 1
[3,2,2,2,1,1]
=> [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]
=> [[.,.],[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> ? = 1 + 1
[3,1,1,1,1,1,1,1,1]
=> [8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]],.]
=> ? = 0 + 1
[2,2,2,2,2,1]
=> [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]
=> ?
=> ? = 1 + 1
[2,2,2,2,1,1,1]
=> [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]
=> [[.,.],[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]
=> ? = 1 + 1
[2,2,2,1,1,1,1,1]
=> [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]
=> [[.,.],[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> ? = 1 + 1
[2,2,1,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [[.,.],[.,[.,[[.,[[.,.],.]],.]]]]
=> ? = 1 + 1
[2,1,1,1,1,1,1,1,1,1]
=> [8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]],.]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]],.]
=> ? = 0 + 1
[12]
=> [6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],[.,.]]
=> ? = 1 + 1
[11,1]
=> [11,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]],.]
=> ? = 0 + 1
[9,3]
=> [9,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]],.]
=> ? = 0 + 1
[9,2,1]
=> [9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]],.]
=> ? = 1 + 1
[9,1,1,1]
=> [9,2,1]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]],.]
=> ? = 0 + 1
[7,5]
=> [7,5]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [[.,[[.,[.,[.,[.,[.,.]]]]],.]],.]
=> ? = 0 + 1
[7,4,1]
=> [7,2,2,1]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[[[.,.],.],[.,.]]]]],.]
=> ? = 1 + 1
[7,3,2]
=> [7,3,1,1]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
=> [[.,[.,[.,[[[.,.],[.,.]],.]]]],.]
=> ? = 1 + 1
[7,3,1,1]
=> [7,3,2]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [[.,[.,[.,[[[.,[.,.]],.],.]]]],.]
=> ? = 0 + 1
[7,2,2,1]
=> [7,1,1,1,1,1]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[.,[[.,.],[.,[.,[.,[.,.]]]]]],.]
=> ? = 1 + 1
[7,2,1,1,1]
=> [7,2,1,1,1]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[.,[.,[[.,.],[.,[[.,.],.]]]]],.]
=> ? = 1 + 1
[7,1,1,1,1,1]
=> [7,4,1]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [[.,[.,[[.,[.,[[.,.],.]]],.]]],.]
=> ? = 0 + 1
[6,2,2,2]
=> [3,3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[[.,[.,.]],[.,.]]]]]]
=> ? = 2 + 1
[6,2,2,1,1]
=> [3,3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [[.,.],[.,[.,[[[.,.],.],[.,.]]]]]
=> ? = 2 + 1
[5,5,2]
=> [10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]]],.]
=> ? = 1 + 1
[5,5,1,1]
=> [10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]],.]
=> ? = 0 + 1
[5,2,2,2,1]
=> [5,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [[.,.],[.,[.,[[.,[.,[.,[.,.]]]],.]]]]
=> ? = 1 + 1
[5,2,2,1,1,1]
=> [5,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [[.,.],[.,[[.,[.,[[.,.],.]]],.]]]
=> ? = 1 + 1
[4,4,4]
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> ? = 1 + 1
[4,4,2,2]
=> [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]
=> [[.,.],[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> ? = 2 + 1
[4,4,2,1,1]
=> [2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> ? = 2 + 1
[4,3,2,2,1]
=> [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]
=> [[.,.],[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> ? = 2 + 1
[4,3,2,1,1,1]
=> [3,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [[.,.],[.,[[.,.],[.,[[.,.],.]]]]]
=> ? = 2 + 1
[4,2,2,2,2]
=> [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]
=> [[.,.],[.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]]]
=> ? = 2 + 1
[4,2,2,2,1,1]
=> [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]
=> [[.,.],[.,[.,[.,[.,[[.,.],[.,[.,.]]]]]]]]
=> ? = 2 + 1
[4,2,2,1,1,1,1]
=> [4,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [[.,.],[.,[.,[[[.,.],[.,.]],.]]]]
=> ? = 2 + 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: St000196
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 44%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 44%
Values
[1]
=> [1]
=> [1,0,1,0]
=> [[.,.],.]
=> 0
[2]
=> [1,1]
=> [1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 1
[1,1]
=> [2]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 0
[3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [[.,[.,[.,.]]],.]
=> 0
[2,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[.,.],[.,[.,.]]]
=> 1
[1,1,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [[[.,.],.],.]
=> 0
[4]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> 1
[3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[.,[[.,.],.]],.]
=> 0
[2,2]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[.,.],[.,[.,[.,.]]]]
=> 1
[2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[.,.],[[.,.],.]]
=> 1
[1,1,1,1]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> 0
[5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> 0
[4,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[[.,.],.],[.,.]]
=> 1
[3,2]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[[.,.],[.,.]],.]
=> 1
[3,1,1]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> 0
[2,2,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> 1
[2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> 1
[1,1,1,1,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[.,[.,[[.,.],.]]],.]
=> 0
[6]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[.,[.,[.,.]]],[.,.]]
=> 1
[5,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[.,[.,[.,[[.,.],.]]]],.]
=> 0
[4,2]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[.,.],[[.,.],[.,.]]]
=> 2
[4,1,1]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[.,[.,.]],[.,[.,.]]]
=> 1
[3,3]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> 0
[3,2,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[.,.],[[.,[.,.]],.]]
=> 1
[3,1,1,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[[[.,.],.],.],.]
=> 0
[2,2,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,[.,.]]]]]]
=> 1
[2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[.,.],[.,[.,[[.,.],.]]]]
=> 1
[2,1,1,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[.,[[.,.],[.,.]]],.]
=> 1
[1,1,1,1,1,1]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> 0
[7]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
=> ? = 0
[6,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> 1
[5,2]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[.,[.,[[.,.],[.,.]]]],.]
=> 1
[5,1,1]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[.,[.,[[.,[.,.]],.]]],.]
=> 0
[4,3]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 1
[4,2,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> 2
[4,1,1,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[[.,.],.],[.,[.,.]]]
=> 1
[3,3,1]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[[.,.],.]]]]],.]
=> 0
[3,2,2]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[.,.],[.,[[.,[.,.]],.]]]
=> 1
[3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 1
[3,1,1,1,1]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[[.,[.,[.,.]]],.],.]
=> 0
[2,2,2,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ? = 1
[2,2,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[[.,.],.]]]]]
=> 1
[2,1,1,1,1,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],.]
=> 1
[1,1,1,1,1,1,1]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[.,[[[.,.],.],.]],.]
=> 0
[8]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> 1
[7,1]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[[.,.],.]]]]]],.]
=> ? = 0
[6,2]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> 2
[6,1,1]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> 1
[5,3]
=> [5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[.,[[.,[.,[.,.]]],.]],.]
=> 0
[5,2,1]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[.,[[.,.],[.,[.,.]]]],.]
=> 1
[5,1,1,1]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[.,[.,[[[.,.],.],.]]],.]
=> 0
[4,4]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[.,[.,.]],[.,[.,[.,.]]]]
=> 1
[4,3,1]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 1
[2,2,2,2]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
=> ? = 1
[2,2,2,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> ? = 1
[1,1,1,1,1,1,1,1]
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
=> ? = 0
[9]
=> [9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]],.]
=> ? = 0
[7,2]
=> [7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[[.,.],[.,.]]]]]],.]
=> ? = 1
[7,1,1]
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[[.,[.,.]],.]]]]],.]
=> ? = 0
[4,2,2,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> ? = 2
[3,2,2,2]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[[.,[.,.]],.]]]]]
=> ? = 1
[2,2,2,2,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]
=> ? = 1
[2,2,2,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
=> ? = 1
[1,1,1,1,1,1,1,1,1]
=> [8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]],.]
=> ? = 0
[9,1]
=> [9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]],.]
=> ? = 0
[7,3]
=> [7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [[.,[.,[.,[[.,[.,[.,.]]],.]]]],.]
=> ? = 0
[7,2,1]
=> [7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[[.,.],[.,[.,.]]]]]],.]
=> ? = 1
[7,1,1,1]
=> [7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[[[.,.],.],.]]]]],.]
=> ? = 0
[5,5]
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]],.]
=> ? = 0
[4,2,2,2]
=> [2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,[[.,.],[.,.]]]]]]]
=> ? = 2
[4,2,2,1,1]
=> [2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> ? = 2
[3,2,2,2,1]
=> [3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,[[.,[.,.]],.]]]]]]
=> ? = 1
[3,2,2,1,1,1]
=> [3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[[[.,.],.],.]]]]]
=> ? = 1
[2,2,2,2,2]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]
=> ? = 1
[2,2,2,2,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
=> ? = 1
[2,2,2,1,1,1,1]
=> [4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [[.,.],[.,[.,[[.,[.,[.,.]]],.]]]]
=> ? = 1
[2,1,1,1,1,1,1,1,1]
=> [8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]],.]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1]
=> [8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]],.]
=> ? = 0
[11]
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]]]],.]
=> ? = 0
[9,2]
=> [9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]],.]
=> ? = 1
[9,1,1]
=> [9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]],.]
=> ? = 0
[7,4]
=> [7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[[.,[.,.]],[.,.]]]]],.]
=> ? = 1
[7,3,1]
=> [7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [[.,[.,[.,[[.,[[.,.],.]],.]]]],.]
=> ? = 0
[7,2,2]
=> [7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[.,[.,[[.,.],[.,[.,[.,.]]]]]],.]
=> ? = 1
[7,2,1,1]
=> [7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[[.,.],[[.,.],.]]]]],.]
=> ? = 1
[7,1,1,1,1]
=> [7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [[.,[.,[[.,[.,[.,[.,.]]]],.]]],.]
=> ? = 0
[6,2,2,1]
=> [3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [[.,.],[.,[.,[[.,[.,.]],[.,.]]]]]
=> ? = 2
[5,5,1]
=> [10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]],.]
=> ? = 0
[5,2,2,2]
=> [5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [[.,.],[.,[[.,[.,[.,[.,.]]]],.]]]
=> ? = 1
[4,4,2,1]
=> [2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
=> ? = 2
[4,3,2,2]
=> [3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [[.,.],[.,[.,[[.,.],[[.,.],.]]]]]
=> ? = 2
[4,2,2,2,1]
=> [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]
=> [[.,.],[.,[.,[.,[.,[.,[[.,.],[.,.]]]]]]]]
=> ? = 2
[4,2,2,1,1,1]
=> [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]
=> [[.,.],[.,[.,[.,[[.,.],[.,[.,.]]]]]]]
=> ? = 2
[3,2,2,2,2]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0]
=> [[.,.],[.,[.,[.,[.,[.,[[.,[.,.]],.]]]]]]]
=> ? = 1
[3,2,2,2,1,1]
=> [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]
=> [[.,.],[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> ? = 1
[3,1,1,1,1,1,1,1,1]
=> [8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]],.]
=> ? = 0
[2,2,2,2,2,1]
=> [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]
=> ?
=> ? = 1
[2,2,2,2,1,1,1]
=> [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]
=> [[.,.],[.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]]
=> ? = 1
[2,2,2,1,1,1,1,1]
=> [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]
=> [[.,.],[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> ? = 1
[2,2,1,1,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [[.,.],[.,[.,[[.,[[.,.],.]],.]]]]
=> ? = 1
Description
The number of occurrences of the contiguous pattern {{{[[.,.],[.,.]]}}} in a binary tree.
Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by $2^{n-2}$.
Matching statistic: St001115
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St001115: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 56%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St001115: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 56%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [2,1] => 0
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [3,1,2] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,1,2,3] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1,4,2] => 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,4,2,3] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [4,2,1,3] => 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [3,1,2,5,4] => 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,3,2] => 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,1,3,2] => 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [4,3,1,2] => 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [5,2,1,3,4] => 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [6,1,2,3,4,5] => 0
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [3,1,2,4,6,5] => 0
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,4,2,5,3] => 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [4,1,2,5,3] => 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,2,5,3,4] => 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [5,1,3,2,4] => 0
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,5,2,3,4] => 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [5,3,1,2,4] => 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [6,2,1,3,4,5] => 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] => [7,1,2,3,4,5,6] => 0
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [1,2,3,4,5,6,8,7] => 0
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => [3,1,2,4,5,7,6] => 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => [1,4,2,3,6,5] => 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => [4,1,2,3,6,5] => 0
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,3,1,5,2] => 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [5,1,2,4,3] => 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,1,5,2,3] => 0
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,5,3,2,4] => 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [5,3,2,1,4] => 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [6,1,3,2,4,5] => 0
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [5,4,1,2,3] => 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [6,3,1,2,4,5] => 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => [7,2,1,3,4,5,6] => 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [8,1,2,3,4,5,6,7] => 0
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,9,8] => 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => [3,1,2,4,5,6,8,7] => ? = 0
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => [1,4,2,3,5,7,6] => 2
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => [4,1,2,3,5,7,6] => 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,2,5,3,6,4] => 0
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => [4,3,1,2,6,5] => 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => [5,1,2,3,6,4] => 0
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => [7,1,3,2,4,5,6] => ? = 0
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [9,2,1,3,4,5,6,7,8] => [3,1,2,4,5,6,7,9,8] => ? = 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [8,3,1,2,4,5,6,7] => [1,4,2,3,5,6,8,7] => ? = 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [8,2,3,1,4,5,6,7] => [4,1,2,3,5,6,8,7] => ? = 0
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [7,3,2,1,4,5,6] => [4,3,1,2,5,7,6] => ? = 2
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,6,7,1] => [7,1,2,4,3,5,6] => ? = 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [4,2,3,5,6,7,8,1] => [8,1,3,2,4,5,6,7] => ? = 0
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,1,3,4,5,6,7,8,9] => [3,1,2,4,5,6,7,8,10,9] => ? = 0
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [9,3,1,2,4,5,6,7,8] => [1,4,2,3,5,6,7,9,8] => ? = 2
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [9,2,3,1,4,5,6,7,8] => [4,1,2,3,5,6,7,9,8] => ? = 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [8,4,1,2,3,5,6,7] => [1,2,5,3,4,6,8,7] => ? = 0
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [8,3,2,1,4,5,6,7] => [4,3,1,2,5,6,8,7] => ? = 1
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [7,4,2,1,3,5,6] => [4,1,5,2,3,7,6] => ? = 1
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,4,1,5,6] => [5,3,1,2,4,7,6] => ? = 2
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [6,2,3,4,5,7,1] => [7,1,2,3,5,4,6] => ? = 0
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [5,3,2,4,6,7,1] => [7,3,1,4,2,5,6] => ? = 2
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [5,2,3,4,6,7,8,1] => [8,1,2,4,3,5,6,7] => ? = 1
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,5,2,3,6,7,1] => [7,1,4,2,3,5,6] => ? = 0
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [4,2,3,5,6,7,8,9,1] => [9,1,3,2,4,5,6,7,8] => ? = 0
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [12,1,2,3,4,5,6,7,8,9,10,11] => [1,2,3,4,5,6,7,8,9,10,12,11] => ? = 0
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [11,2,1,3,4,5,6,7,8,9,10] => [3,1,2,4,5,6,7,8,9,11,10] => ? = 1
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [10,3,1,2,4,5,6,7,8,9] => [1,4,2,3,5,6,7,8,10,9] => ? = 1
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,3,1,4,5,6,7,8,9] => [4,1,2,3,5,6,7,8,10,9] => ? = 0
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [9,4,1,2,3,5,6,7,8] => [1,2,5,3,4,6,7,9,8] => ? = 1
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,2,1,4,5,6,7,8] => ? => ? = 2
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [9,2,3,4,1,5,6,7,8] => [5,1,2,3,4,6,7,9,8] => ? = 1
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [8,5,1,2,3,4,6,7] => [1,2,3,6,4,5,8,7] => ? = 1
[7,3,1]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [8,4,2,1,3,5,6,7] => [4,1,5,2,3,6,8,7] => ? = 0
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [8,3,4,1,2,5,6,7] => [1,5,2,3,4,6,8,7] => ? = 1
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,2,4,1,5,6,7] => ? => ? = 1
[6,4,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [7,5,2,1,3,4,6] => [4,1,2,6,3,7,5] => ? = 2
[6,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [7,4,3,1,2,5,6] => [1,5,4,2,3,7,6] => ? = 2
[6,3,1,1]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [7,4,2,3,1,5,6] => [5,1,4,2,3,7,6] => ? = 1
[6,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [7,3,4,2,1,5,6] => [5,4,1,2,3,7,6] => ? = 2
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,4,5,1,6] => [6,3,1,2,4,7,5] => ? = 2
[6,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,6,1] => [7,1,2,3,4,6,5] => ? = 1
[5,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [6,7,2,1,3,4,5] => [4,1,2,3,7,5,6] => ? = 0
[5,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [6,3,2,4,5,7,1] => [7,3,1,2,5,4,6] => ? = 1
[5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [6,2,3,4,5,7,8,1] => [8,1,2,3,5,4,6,7] => ? = 0
[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] => [7,1,4,3,2,5,6] => ? = 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] => [7,4,1,3,2,5,6] => ? = 2
[4,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [5,3,2,4,6,7,8,1] => [8,3,1,4,2,5,6,7] => ? = 2
[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
[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] => [7,4,3,1,2,5,6] => ? = 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] => [8,1,4,2,3,5,6,7] => ? = 0
[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] => [1,7,3,2,4,5,6] => ? = 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] => [7,5,2,1,3,4,6] => ? = 1
[3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [4,3,5,2,6,7,8,1] => [8,4,2,1,3,5,6,7] => ? = 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] => [9,3,2,1,4,5,6,7,8] => ? = 1
Description
The number of even descents of a permutation.
Matching statistic: St000256
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 33%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 33%
Values
[1]
=> [1]
=> [1]
=> 0
[2]
=> [1,1]
=> [2]
=> 1
[1,1]
=> [2]
=> [1,1]
=> 0
[3]
=> [3]
=> [1,1,1]
=> 0
[2,1]
=> [1,1,1]
=> [3]
=> 1
[1,1,1]
=> [2,1]
=> [2,1]
=> 0
[4]
=> [2,2]
=> [2,2]
=> 1
[3,1]
=> [3,1]
=> [2,1,1]
=> 0
[2,2]
=> [1,1,1,1]
=> [4]
=> 1
[2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[1,1,1,1]
=> [4]
=> [1,1,1,1]
=> 0
[5]
=> [5]
=> [1,1,1,1,1]
=> 0
[4,1]
=> [2,2,1]
=> [3,2]
=> 1
[3,2]
=> [3,1,1]
=> [3,1,1]
=> 1
[3,1,1]
=> [3,2]
=> [2,2,1]
=> 0
[2,2,1]
=> [1,1,1,1,1]
=> [5]
=> 1
[2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1
[1,1,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 0
[6]
=> [3,3]
=> [2,2,2]
=> 1
[5,1]
=> [5,1]
=> [2,1,1,1,1]
=> 0
[4,2]
=> [2,2,1,1]
=> [4,2]
=> 2
[4,1,1]
=> [2,2,2]
=> [3,3]
=> 1
[3,3]
=> [6]
=> [1,1,1,1,1,1]
=> 0
[3,2,1]
=> [3,1,1,1]
=> [4,1,1]
=> 1
[3,1,1,1]
=> [3,2,1]
=> [3,2,1]
=> 0
[2,2,2]
=> [1,1,1,1,1,1]
=> [6]
=> 1
[2,2,1,1]
=> [2,1,1,1,1]
=> [5,1]
=> 1
[2,1,1,1,1]
=> [4,1,1]
=> [3,1,1,1]
=> 1
[1,1,1,1,1,1]
=> [4,2]
=> [2,2,1,1]
=> 0
[7]
=> [7]
=> [1,1,1,1,1,1,1]
=> 0
[6,1]
=> [3,3,1]
=> [3,2,2]
=> 1
[5,2]
=> [5,1,1]
=> [3,1,1,1,1]
=> 1
[5,1,1]
=> [5,2]
=> [2,2,1,1,1]
=> 0
[4,3]
=> [3,2,2]
=> [3,3,1]
=> 1
[4,2,1]
=> [2,2,1,1,1]
=> [5,2]
=> 2
[4,1,1,1]
=> [2,2,2,1]
=> [4,3]
=> 1
[3,3,1]
=> [6,1]
=> [2,1,1,1,1,1]
=> 0
[3,2,2]
=> [3,1,1,1,1]
=> [5,1,1]
=> 1
[3,2,1,1]
=> [3,2,1,1]
=> [4,2,1]
=> 1
[3,1,1,1,1]
=> [4,3]
=> [2,2,2,1]
=> 0
[2,2,2,1]
=> [1,1,1,1,1,1,1]
=> [7]
=> 1
[2,2,1,1,1]
=> [2,1,1,1,1,1]
=> [6,1]
=> 1
[2,1,1,1,1,1]
=> [4,1,1,1]
=> [4,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> [4,2,1]
=> [3,2,1,1]
=> 0
[8]
=> [4,4]
=> [2,2,2,2]
=> 1
[7,1]
=> [7,1]
=> [2,1,1,1,1,1,1]
=> 0
[6,2]
=> [3,3,1,1]
=> [4,2,2]
=> 2
[6,1,1]
=> [3,3,2]
=> [3,3,2]
=> 1
[5,3]
=> [5,3]
=> [2,2,2,1,1]
=> 0
[5,2,1]
=> [5,1,1,1]
=> [4,1,1,1,1]
=> 1
[11]
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[10,1]
=> [5,5,1]
=> [3,2,2,2,2]
=> ? = 1
[9,2]
=> [9,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> ? = 1
[9,1,1]
=> [9,2]
=> [2,2,1,1,1,1,1,1,1]
=> ? = 0
[8,3]
=> [4,4,3]
=> [3,3,3,2]
=> ? = 1
[8,2,1]
=> [4,4,1,1,1]
=> [5,2,2,2]
=> ? = 2
[8,1,1,1]
=> [4,4,2,1]
=> [4,3,2,2]
=> ? = 1
[7,4]
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> ? = 1
[7,3,1]
=> [7,3,1]
=> [3,2,2,1,1,1,1]
=> ? = 0
[7,2,2]
=> [7,1,1,1,1]
=> [5,1,1,1,1,1,1]
=> ? = 1
[7,2,1,1]
=> [7,2,1,1]
=> [4,2,1,1,1,1,1]
=> ? = 1
[7,1,1,1,1]
=> [7,4]
=> [2,2,2,2,1,1,1]
=> ? = 0
[6,5]
=> [5,3,3]
=> [3,3,3,1,1]
=> ? = 1
[6,4,1]
=> [3,3,2,2,1]
=> [5,4,2]
=> ? = 2
[6,3,2]
=> [3,3,3,1,1]
=> [5,3,3]
=> ? = 2
[6,3,1,1]
=> [3,3,3,2]
=> [4,4,3]
=> ? = 1
[6,2,2,1]
=> [3,3,1,1,1,1,1]
=> [7,2,2]
=> ? = 2
[6,2,1,1,1]
=> [3,3,2,1,1,1]
=> [6,3,2]
=> ? = 2
[6,1,1,1,1,1]
=> [4,3,3,1]
=> [4,3,3,1]
=> ? = 1
[5,5,1]
=> [10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> ? = 0
[5,4,2]
=> [5,2,2,1,1]
=> [5,3,1,1,1]
=> ? = 2
[5,4,1,1]
=> [5,2,2,2]
=> [4,4,1,1,1]
=> ? = 1
[5,3,3]
=> [6,5]
=> [2,2,2,2,2,1]
=> ? = 0
[5,3,2,1]
=> [5,3,1,1,1]
=> [5,2,2,1,1]
=> ? = 1
[5,3,1,1,1]
=> [5,3,2,1]
=> [4,3,2,1,1]
=> ? = 0
[5,2,2,2]
=> [5,1,1,1,1,1,1]
=> [7,1,1,1,1]
=> ? = 1
[5,2,2,1,1]
=> [5,2,1,1,1,1]
=> [6,2,1,1,1]
=> ? = 1
[5,2,1,1,1,1]
=> [5,4,1,1]
=> [4,2,2,2,1]
=> ? = 1
[5,1,1,1,1,1,1]
=> [5,4,2]
=> [3,3,2,2,1]
=> ? = 0
[4,4,3]
=> [3,2,2,2,2]
=> [5,5,1]
=> ? = 1
[4,4,2,1]
=> [2,2,2,2,1,1,1]
=> [7,4]
=> ? = 2
[4,4,1,1,1]
=> [2,2,2,2,2,1]
=> [6,5]
=> ? = 1
[4,3,3,1]
=> [6,2,2,1]
=> [4,3,1,1,1,1]
=> ? = 1
[4,3,2,2]
=> [3,2,2,1,1,1,1]
=> [7,3,1]
=> ? = 2
[4,3,2,1,1]
=> [3,2,2,2,1,1]
=> [6,4,1]
=> ? = 2
[4,3,1,1,1,1]
=> [4,3,2,2]
=> [4,4,2,1]
=> ? = 1
[4,2,2,2,1]
=> [2,2,1,1,1,1,1,1,1]
=> [9,2]
=> ? = 2
[4,2,2,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [8,3]
=> ? = 2
[4,2,1,1,1,1,1]
=> [4,2,2,1,1,1]
=> [6,3,1,1]
=> ? = 2
[4,1,1,1,1,1,1,1]
=> [4,2,2,2,1]
=> [5,4,1,1]
=> ? = 1
[3,3,3,2]
=> [6,3,1,1]
=> [4,2,2,1,1,1]
=> ? = 1
[3,3,3,1,1]
=> [6,3,2]
=> [3,3,2,1,1,1]
=> ? = 0
[3,3,2,2,1]
=> [6,1,1,1,1,1]
=> [6,1,1,1,1,1]
=> ? = 1
[3,3,2,1,1,1]
=> [6,2,1,1,1]
=> [5,2,1,1,1,1]
=> ? = 1
[3,3,1,1,1,1,1]
=> [6,4,1]
=> [3,2,2,2,1,1]
=> ? = 0
[3,2,2,2,2]
=> [3,1,1,1,1,1,1,1,1]
=> [9,1,1]
=> ? = 1
[3,2,2,2,1,1]
=> [3,2,1,1,1,1,1,1]
=> [8,2,1]
=> ? = 1
[3,2,2,1,1,1,1]
=> [4,3,1,1,1,1]
=> [6,2,2,1]
=> ? = 1
[3,2,1,1,1,1,1,1]
=> [4,3,2,1,1]
=> [5,3,2,1]
=> ? = 1
[3,1,1,1,1,1,1,1,1]
=> [8,3]
=> [2,2,2,1,1,1,1,1]
=> ? = 0
Description
The number of parts from which one can substract 2 and still get an integer partition.
Matching statistic: St001114
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St001114: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 33%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St001114: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 33%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [1,2] => 0
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [3,1,2] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [1,2,3] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [3,4,1,2] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => [2,1,3] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [3,4,5,1,2] => 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,1,3] => 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,1,2,3] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,1,3,4] => 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [3,4,5,6,1,2] => 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [2,4,5,1,3] => 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [4,2,1,3] => 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,3,1,4] => 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1,2,4] => 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,1,3,4,5] => 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [3,4,5,6,7,1,2] => ? = 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [2,4,5,6,1,3] => 0
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [4,2,5,1,3] => 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,5,1,4] => 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [4,5,1,2,3] => 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [3,2,1,4] => 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,3,1,4,5] => 0
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,1,2,3,4] => 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,2,4,5] => 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [2,1,3,4,5,6] => 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] => [1,2,3,4,5,6,7] => 0
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [3,4,5,6,7,8,1,2] => ? = 0
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => [2,4,5,6,7,1,3] => ? = 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => [4,2,5,6,1,3] => 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => [2,3,5,6,1,4] => 0
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [4,5,2,1,3] => 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,2,5,1,4] => 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,4,1,5] => 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,5,1,2,4] => 0
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [5,2,1,3,4] => 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [3,2,1,4,5] => 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [2,3,1,4,5,6] => 0
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [4,1,2,3,5] => 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [3,1,2,4,5,6] => 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => [2,1,3,4,5,6,7] => ? = 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [1,2,3,4,5,6,7,8] => ? = 0
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [3,4,5,6,7,8,9,1,2] => ? = 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => [2,4,5,6,7,8,1,3] => ? = 0
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => [4,2,5,6,7,1,3] => ? = 2
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => [2,3,5,6,7,1,4] => ? = 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [4,5,2,6,1,3] => 0
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => [3,2,5,6,1,4] => 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => [2,3,4,6,1,5] => 0
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [4,5,6,1,2,3] => 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [3,5,2,1,4] => 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [5,2,3,1,4] => 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,4,1,5] => 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,6,1] => [2,3,4,1,5,6] => 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [5,3,1,2,4] => 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,4,1,2,5] => 0
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,2,1,3,5] => 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => [2,3,1,4,5,6,7] => ? = 0
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => [3,1,2,4,5,6,7] => ? = 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => [2,1,3,4,5,6,7,8] => ? = 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => [1,2,3,4,5,6,7,8,9] => ? = 0
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [3,4,5,6,7,8,9,10,1,2] => ? = 0
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [9,2,1,3,4,5,6,7,8] => [2,4,5,6,7,8,9,1,3] => ? = 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [8,3,1,2,4,5,6,7] => [4,2,5,6,7,8,1,3] => ? = 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [8,2,3,1,4,5,6,7] => [2,3,5,6,7,8,1,4] => ? = 0
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [7,4,1,2,3,5,6] => [4,5,2,6,7,1,3] => ? = 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [7,3,2,1,4,5,6] => [3,2,5,6,7,1,4] => ? = 2
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,1,5,6] => [2,3,4,6,7,1,5] => ? = 1
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,6,7,1] => [2,3,4,1,5,6,7] => ? = 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,6,7,1] => [3,2,1,4,5,6,7] => ? = 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [4,2,3,5,6,7,8,1] => [2,3,1,4,5,6,7,8] => ? = 0
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [4,1,2,3,5,6,7] => ? = 1
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,1] => [3,1,2,4,5,6,7,8] => ? = 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,1] => [2,1,3,4,5,6,7,8,9] => ? = 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [1,2,3,4,5,6,7,8,9,10] => ? = 0
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [3,4,5,6,7,8,9,10,11,1,2] => ? = 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,1,3,4,5,6,7,8,9] => [2,4,5,6,7,8,9,10,1,3] => ? = 0
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [9,3,1,2,4,5,6,7,8] => [4,2,5,6,7,8,9,1,3] => ? = 2
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [9,2,3,1,4,5,6,7,8] => [2,3,5,6,7,8,9,1,4] => ? = 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [8,4,1,2,3,5,6,7] => [4,5,2,6,7,8,1,3] => ? = 0
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [8,3,2,1,4,5,6,7] => [3,2,5,6,7,8,1,4] => ? = 1
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [8,2,3,4,1,5,6,7] => [2,3,4,6,7,8,1,5] => ? = 0
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [7,5,1,2,3,4,6] => [4,5,6,2,7,1,3] => ? = 2
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [7,4,2,1,3,5,6] => [3,5,2,6,7,1,4] => ? = 1
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [7,3,4,1,2,5,6] => [5,2,3,6,7,1,4] => ? = 2
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,4,1,5,6] => [3,2,4,6,7,1,5] => ? = 2
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,1,6] => [2,3,4,5,7,1,6] => ? = 1
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => [4,5,6,7,1,2,3] => ? = 0
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [6,2,3,4,5,7,1] => [2,3,4,5,1,6,7] => ? = 0
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [5,3,2,4,6,7,1] => [3,2,4,1,5,6,7] => ? = 2
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [5,2,3,4,6,7,8,1] => [2,3,4,1,5,6,7,8] => ? = 1
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,5,2,3,6,7,1] => [3,4,1,2,5,6,7] => ? = 0
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,6,7,1] => [4,2,1,3,5,6,7] => ? = 1
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,5,6,7,8,1] => [3,2,1,4,5,6,7,8] => ? = 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [4,2,3,5,6,7,8,9,1] => [2,3,1,4,5,6,7,8,9] => ? = 0
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => [7,1,2,3,4,5,6] => ? = 1
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => [5,1,2,3,4,6,7] => ? = 1
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [3,4,5,2,6,7,8,1] => [4,1,2,3,5,6,7,8] => ? = 1
Description
The number of odd descents of a permutation.
Matching statistic: St001151
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St001151: Set partitions ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 33%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St001151: Set partitions ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 33%
Values
[1]
=> [1,0,1,0]
=> [3,1,2] => {{1,3},{2}}
=> 1 = 0 + 1
[2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => {{1,2,4},{3}}
=> 2 = 1 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => {{1,3,4},{2}}
=> 1 = 0 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => {{1,2,3,5},{4}}
=> 1 = 0 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => {{1,4},{2},{3}}
=> 2 = 1 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => {{1,3,4,5},{2}}
=> 1 = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => {{1,2,3,4,6},{5}}
=> 2 = 1 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => {{1,5},{2,3},{4}}
=> 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => {{1,2,4,5},{3}}
=> 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => {{1,5},{2},{3,4}}
=> 2 = 1 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => {{1,3,4,5,6},{2}}
=> 1 = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,3,4,5,7,1,6] => {{1,2,3,4,5,7},{6}}
=> 1 = 0 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => {{1,6},{2,3,4},{5}}
=> 2 = 1 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => {{1,2,5},{3},{4}}
=> 2 = 1 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => {{1,3,5},{2},{4}}
=> 1 = 0 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => {{1,4,5},{2},{3}}
=> 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => {{1,6},{2},{3,4,5}}
=> 2 = 1 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => {{1,3,4,5,6,7},{2}}
=> 1 = 0 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => {{1,2,3,4,5,6,8},{7}}
=> ? = 1 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,4,5,1,2,6] => {{1,7},{2,3,4,5},{6}}
=> 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => {{1,2,6},{3,4},{5}}
=> 3 = 2 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => {{1,4,6},{2,3},{5}}
=> 2 = 1 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => {{1,2,3,5,6},{4}}
=> 1 = 0 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => {{1,5},{2},{3},{4}}
=> 2 = 1 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => {{1,3,6},{2},{4,5}}
=> 1 = 0 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => {{1,2,4,5,6},{3}}
=> 2 = 1 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => {{1,5,6},{2},{3,4}}
=> 2 = 1 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => {{1,7},{2},{3,4,5,6}}
=> 2 = 1 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => {{1,3,4,5,6,7,8},{2}}
=> ? = 0 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,9,1,8] => {{1,2,3,4,5,6,7,9},{8}}
=> ? = 0 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [8,3,4,5,6,1,2,7] => {{1,8},{2,3,4,5,6},{7}}
=> ? = 1 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [2,7,4,5,1,3,6] => {{1,2,7},{3,4,5},{6}}
=> 2 = 1 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => {{1,5,7},{2,3,4},{6}}
=> 1 = 0 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => {{1,2,3,6},{4},{5}}
=> 2 = 1 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => {{1,6},{2,4},{3},{5}}
=> 3 = 2 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => {{1,3,4,6},{2},{5}}
=> 2 = 1 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => {{1,5,6},{2,3},{4}}
=> 1 = 0 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => {{1,2,6},{3},{4,5}}
=> 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => {{1,6},{2},{3,5},{4}}
=> 2 = 1 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,7,5,6,2,4] => {{1,3,7},{2},{4,5,6}}
=> 1 = 0 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => {{1,4,5,6},{2},{3}}
=> 2 = 1 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [6,1,4,5,2,7,3] => {{1,6,7},{2},{3,4,5}}
=> 2 = 1 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [8,1,4,5,6,7,2,3] => {{1,8},{2},{3,4,5,6,7}}
=> ? = 1 + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,2] => {{1,3,4,5,6,7,8,9},{2}}
=> ? = 0 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,10,1,9] => {{1,2,3,4,5,6,7,8,10},{9}}
=> ? = 1 + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [9,3,4,5,6,7,1,2,8] => {{1,9},{2,3,4,5,6,7},{8}}
=> ? = 0 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [2,8,4,5,6,1,3,7] => {{1,2,8},{3,4,5,6},{7}}
=> ? = 2 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,3,4,5,1,8,2,7] => {{1,6,8},{2,3,4,5},{7}}
=> ? = 1 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [2,3,7,5,1,4,6] => {{1,2,3,7},{4,5},{6}}
=> 1 = 0 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [7,5,4,1,2,3,6] => {{1,7},{2,5},{3,4},{6}}
=> 2 = 1 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => {{1,4,5,7},{2,3},{6}}
=> 1 = 0 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,3,4,6,1,7,5] => {{1,2,3,4,6,7},{5}}
=> 2 = 1 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => {{1,6},{2,3},{4},{5}}
=> 2 = 1 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => {{1,2,4,6},{3},{5}}
=> 3 = 2 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => {{1,6},{2},{3,4},{5}}
=> 3 = 2 + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [3,1,4,7,6,2,5] => {{1,3,4,7},{2},{5,6}}
=> 2 = 1 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => {{1,2,5,6},{3},{4}}
=> 2 = 1 + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => {{1,3,5,6},{2},{4}}
=> 1 = 0 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => {{1,6},{2},{3},{4,5}}
=> 2 = 1 + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => {{1,7},{2},{3,6},{4,5}}
=> 2 = 1 + 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [3,1,8,5,6,7,2,4] => {{1,3,8},{2},{4,5,6,7}}
=> ? = 0 + 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [7,1,4,5,6,2,8,3] => {{1,7,8},{2},{3,4,5,6}}
=> ? = 1 + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [9,1,4,5,6,7,8,2,3] => {{1,9},{2},{3,4,5,6,7,8}}
=> ? = 1 + 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,2] => {{1,3,4,5,6,7,8,9,10},{2}}
=> ? = 0 + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,9,11,1,10] => {{1,2,3,4,5,6,7,8,9,11},{10}}
=> ? = 0 + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [10,3,4,5,6,7,8,1,2,9] => {{1,10},{2,3,4,5,6,7,8},{9}}
=> ? = 1 + 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [2,9,4,5,6,7,1,3,8] => {{1,2,9},{3,4,5,6,7},{8}}
=> ? = 1 + 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [7,3,4,5,6,1,9,2,8] => {{1,7,9},{2,3,4,5,6},{8}}
=> ? = 0 + 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [2,3,8,5,6,1,4,7] => {{1,2,3,8},{4,5,6},{7}}
=> ? = 1 + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [8,6,4,5,1,2,3,7] => {{1,8},{2,6},{3,4,5},{7}}
=> ? = 2 + 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [5,3,4,1,6,8,2,7] => {{1,5,6,8},{2,3,4},{7}}
=> ? = 1 + 1
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [3,1,4,8,6,7,2,5] => {{1,3,4,8},{2},{5,6,7}}
=> ? = 1 + 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [8,1,7,5,6,2,3,4] => {{1,8},{2},{3,7},{4,5,6}}
=> ? = 1 + 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [3,1,9,5,6,7,8,2,4] => {{1,3,9},{2},{4,5,6,7,8}}
=> ? = 0 + 1
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,1,4,5,2,7,8,3] => {{1,6,7,8},{2},{3,4,5}}
=> ? = 1 + 1
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [8,1,4,5,6,7,2,9,3] => {{1,8,9},{2},{3,4,5,6,7}}
=> ? = 1 + 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [10,1,4,5,6,7,8,9,2,3] => {{1,10},{2},{3,4,5,6,7,8,9}}
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,9,10,11,2] => {{1,3,4,5,6,7,8,9,10,11},{2}}
=> ? = 0 + 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,9,10,12,1,11] => {{1,2,3,4,5,6,7,8,9,10,12},{11}}
=> ? = 1 + 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [11,3,4,5,6,7,8,9,1,2,10] => {{1,11},{2,3,4,5,6,7,8,9},{10}}
=> ? = 0 + 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [2,10,4,5,6,7,8,1,3,9] => {{1,2,10},{3,4,5,6,7,8},{9}}
=> ? = 2 + 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [8,3,4,5,6,7,1,10,2,9] => {{1,8,10},{2,3,4,5,6,7},{9}}
=> ? = 1 + 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [2,3,9,5,6,7,1,4,8] => {{1,2,3,9},{4,5,6,7},{8}}
=> ? = 0 + 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [9,7,4,5,6,1,2,3,8] => {{1,9},{2,7},{3,4,5,6},{8}}
=> ? = 1 + 1
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [6,3,4,5,1,7,9,2,8] => {{1,6,7,9},{2,3,4,5},{8}}
=> ? = 0 + 1
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [2,3,4,8,6,1,5,7] => {{1,2,3,4,8},{5,6},{7}}
=> ? = 2 + 1
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [8,3,6,5,1,2,4,7] => {{1,8},{2,3,6},{4,5},{7}}
=> ? = 1 + 1
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [2,6,4,5,1,8,3,7] => {{1,2,6,8},{3,4,5},{7}}
=> ? = 2 + 1
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [8,5,4,1,6,2,3,7] => {{1,8},{2,5,6},{3,4},{7}}
=> ? = 2 + 1
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [4,3,1,5,6,8,2,7] => {{1,4,5,6,8},{2,3},{7}}
=> ? = 1 + 1
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [2,3,4,5,7,1,8,6] => {{1,2,3,4,5,7,8},{6}}
=> ? = 0 + 1
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [3,1,4,5,8,7,2,6] => {{1,3,4,5,8},{2},{6,7}}
=> ? = 0 + 1
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [8,1,4,7,6,2,3,5] => {{1,8},{2},{3,4,7},{5,6}}
=> ? = 2 + 1
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3,1,4,9,6,7,8,2,5] => {{1,3,4,9},{2},{5,6,7,8}}
=> ? = 1 + 1
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [3,1,7,5,6,2,8,4] => {{1,3,7,8},{2},{4,5,6}}
=> ? = 0 + 1
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [8,1,6,5,2,7,3,4] => {{1,8},{2},{3,6,7},{4,5}}
=> ? = 1 + 1
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [9,1,8,5,6,7,2,3,4] => {{1,9},{2},{3,8},{4,5,6,7}}
=> ? = 1 + 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [3,1,10,5,6,7,8,9,2,4] => {{1,3,10},{2},{4,5,6,7,8,9}}
=> ? = 0 + 1
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,4,1,5,6,7,8,3] => {{1,2,4,5,6,7,8},{3}}
=> ? = 1 + 1
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [5,1,4,2,6,7,8,3] => {{1,5,6,7,8},{2},{3,4}}
=> ? = 1 + 1
Description
The number of blocks with odd minimum.
See [[St000746]] for the analogous statistic on perfect matchings.
Matching statistic: St000023
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 33%
Values
[1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[2]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[1,1]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0
[3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 0
[2,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,1,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[4]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1
[3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 0
[2,2]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
[1,1,1,1]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 0
[5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 0
[4,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[3,2]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1
[3,1,1]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 0
[2,2,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 1
[2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
[1,1,1,1,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 0
[6]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
[5,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => 0
[4,2]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2
[4,1,1]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
[3,3]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 0
[3,2,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[3,1,1,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
[2,2,2]
=> [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
[2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 1
[2,1,1,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 1
[1,1,1,1,1,1]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 0
[7]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => ? = 0
[6,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 1
[5,2]
=> [5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => 1
[5,1,1]
=> [5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => 0
[4,3]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 1
[4,2,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => 2
[4,1,1,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 1
[3,3,1]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? = 0
[3,2,2]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => 1
[3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 1
[3,1,1,1,1]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 0
[2,2,2,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 1
[2,2,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? = 1
[2,1,1,1,1,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[1,1,1,1,1,1,1]
=> [4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 0
[8]
=> [4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => 1
[7,1]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => ? = 0
[6,2]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 2
[6,1,1]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 1
[5,3]
=> [5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 0
[5,2,1]
=> [5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => 1
[5,1,1,1]
=> [5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => 0
[4,4]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => 1
[4,3,1]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 1
[4,2,2]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? = 2
[4,2,1,1]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => 2
[4,1,1,1,1]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 1
[3,3,2]
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => ? = 1
[3,3,1,1]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => ? = 0
[3,2,2,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => ? = 1
[3,2,1,1,1]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,6,1] => 1
[3,1,1,1,1,1]
=> [4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 0
[2,2,2,2]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? = 1
[2,2,2,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? = 1
[1,1,1,1,1,1,1,1]
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => ? = 0
[9]
=> [9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => ? = 0
[7,2]
=> [7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [8,2,3,1,4,5,6,7] => ? = 1
[7,1,1]
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [8,3,1,2,4,5,6,7] => ? = 0
[4,2,2,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,1] => ? = 2
[4,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => ? = 2
[3,3,3]
=> [6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [7,4,1,2,3,5,6] => ? = 0
[3,3,2,1]
=> [6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,1,5,6] => ? = 1
[3,3,1,1,1]
=> [6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [7,3,2,1,4,5,6] => ? = 0
[3,2,2,2]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [4,2,3,5,6,7,8,1] => ? = 1
[3,2,2,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,6,7,1] => ? = 1
[2,2,2,2,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => ? = 1
[2,2,2,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,1] => ? = 1
[2,2,1,1,1,1,1]
=> [4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,6,7,1] => ? = 1
[1,1,1,1,1,1,1,1,1]
=> [8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [9,2,1,3,4,5,6,7,8] => ? = 0
[10]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 1
[9,1]
=> [9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,1,3,4,5,6,7,8,9] => ? = 0
[7,3]
=> [7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [8,4,1,2,3,5,6,7] => ? = 0
[7,2,1]
=> [7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [8,2,3,4,1,5,6,7] => ? = 1
[7,1,1,1]
=> [7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [8,3,2,1,4,5,6,7] => ? = 0
[6,2,2]
=> [3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,5,2,3,6,7,1] => ? = 2
[5,5]
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => ? = 0
[5,2,2,1]
=> [5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [6,2,3,4,5,7,1] => ? = 1
[4,4,2]
=> [2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,2,7,1] => ? = 2
[4,4,1,1]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => ? = 1
[4,3,3]
=> [6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [7,3,4,1,2,5,6] => ? = 1
[4,3,2,1]
=> [3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,5,2,6,7,1] => ? = 2
[4,2,2,2]
=> [2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,1] => ? = 2
[4,2,2,1,1]
=> [2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [3,4,5,2,6,7,8,1] => ? = 2
[3,3,3,1]
=> [6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [7,4,2,1,3,5,6] => ? = 0
[3,3,2,2]
=> [6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,1,6] => ? = 1
[3,3,2,1,1]
=> [6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,4,1,5,6] => ? = 1
[3,3,1,1,1,1]
=> [6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [7,5,1,2,3,4,6] => ? = 0
[3,2,2,2,1]
=> [3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [4,2,3,5,6,7,8,9,1] => ? = 1
[3,2,2,1,1,1]
=> [3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [4,3,2,5,6,7,8,1] => ? = 1
[2,2,2,2,2]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => ? = 1
[2,2,2,2,1,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => ? = 1
Description
The number of inner peaks of a permutation.
The number of peaks including the boundary is [[St000092]].
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000779The tier of a permutation. St000099The number of valleys of a permutation, including the boundary. St000659The number of rises of length at least 2 of a Dyck path. St001960The number of descents of a permutation minus one if its first entry is not one. St001729The number of visible descents of a permutation.
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