searching the database
Your data matches 7 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
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
St001115: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001115: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => 0
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 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]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 0
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 0
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 0
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => 0
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => 0
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 0
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => 0
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => 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
[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] => 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] => 0
[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] => 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => 0
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => 1
Description
The number of even descents of a permutation.
Matching statistic: St001092
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
St001092: Integer partitions ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> 0
[2]
=> [1,1]
=> 0
[1,1]
=> [2]
=> 1
[3]
=> [1,1,1]
=> 0
[2,1]
=> [2,1]
=> 1
[1,1,1]
=> [3]
=> 0
[4]
=> [1,1,1,1]
=> 0
[3,1]
=> [2,1,1]
=> 1
[2,2]
=> [2,2]
=> 1
[2,1,1]
=> [3,1]
=> 0
[1,1,1,1]
=> [4]
=> 1
[5]
=> [1,1,1,1,1]
=> 0
[4,1]
=> [2,1,1,1]
=> 1
[3,2]
=> [2,2,1]
=> 1
[3,1,1]
=> [3,1,1]
=> 0
[2,2,1]
=> [3,2]
=> 1
[2,1,1,1]
=> [4,1]
=> 1
[1,1,1,1,1]
=> [5]
=> 0
[6]
=> [1,1,1,1,1,1]
=> 0
[5,1]
=> [2,1,1,1,1]
=> 1
[4,2]
=> [2,2,1,1]
=> 1
[4,1,1]
=> [3,1,1,1]
=> 0
[3,3]
=> [2,2,2]
=> 1
[3,2,1]
=> [3,2,1]
=> 1
[3,1,1,1]
=> [4,1,1]
=> 1
[2,2,2]
=> [3,3]
=> 0
[2,2,1,1]
=> [4,2]
=> 2
[2,1,1,1,1]
=> [5,1]
=> 0
[1,1,1,1,1,1]
=> [6]
=> 1
[7]
=> [1,1,1,1,1,1,1]
=> 0
[6,1]
=> [2,1,1,1,1,1]
=> 1
[5,2]
=> [2,2,1,1,1]
=> 1
[5,1,1]
=> [3,1,1,1,1]
=> 0
[4,3]
=> [2,2,2,1]
=> 1
[4,2,1]
=> [3,2,1,1]
=> 1
[4,1,1,1]
=> [4,1,1,1]
=> 1
[3,3,1]
=> [3,2,2]
=> 1
[3,2,2]
=> [3,3,1]
=> 0
[3,2,1,1]
=> [4,2,1]
=> 2
[3,1,1,1,1]
=> [5,1,1]
=> 0
[2,2,2,1]
=> [4,3]
=> 1
[2,2,1,1,1]
=> [5,2]
=> 1
[2,1,1,1,1,1]
=> [6,1]
=> 1
[1,1,1,1,1,1,1]
=> [7]
=> 0
[8]
=> [1,1,1,1,1,1,1,1]
=> 0
[7,1]
=> [2,1,1,1,1,1,1]
=> 1
[6,2]
=> [2,2,1,1,1,1]
=> 1
[6,1,1]
=> [3,1,1,1,1,1]
=> 0
[5,3]
=> [2,2,2,1,1]
=> 1
[5,2,1]
=> [3,2,1,1,1]
=> 1
[5,5,2,2]
=> [4,4,2,2,2]
=> ? = 2
[8,7]
=> [2,2,2,2,2,2,2,1]
=> ? = 1
[7,5,3,1]
=> [4,3,3,2,2,1,1]
=> ? = 2
[9,8]
=> [2,2,2,2,2,2,2,2,1]
=> ? = 1
[7,5,5]
=> [3,3,3,3,3,1,1]
=> ? = 0
[5,4,4,2,2]
=> [5,5,3,3,1]
=> ? = 0
[6,5,5,4,3,2,1]
=> [7,6,5,4,3,1]
=> ? = 2
[6,6,4,4,3,2,1]
=> [7,6,5,4,2,2]
=> ? = 3
[6,6,5,3,3,2,1]
=> [7,6,5,3,3,2]
=> ? = 2
[6,5,4,3,3,2,1]
=> [7,6,5,3,2,1]
=> ? = 2
[6,6,5,4,2,2,1]
=> [7,6,4,4,3,2]
=> ? = 3
[6,6,5,4,3,1,1]
=> [7,5,5,4,3,2]
=> ? = 2
[6,5,4,3,2,1,1]
=> [7,5,4,3,2,1]
=> ? = 2
[6,6,2,1,1,1,1]
=> [7,3,2,2,2,2]
=> ? = 1
[6,6,5,4,3,2]
=> [6,6,5,4,3,2]
=> ? = 3
[7,5,5,4,3,2]
=> [6,6,5,4,3,1,1]
=> ? = 2
[7,6,4,4,3,2]
=> [6,6,5,4,2,2,1]
=> ? = 3
[6,5,5,4,3,1]
=> [6,5,5,4,3,1]
=> ? = 2
[7,5,4,3,2]
=> [5,5,4,3,2,1,1]
=> ? = 2
[7,5,4,3,1]
=> [5,4,4,3,2,1,1]
=> ? = 2
[7,6,5]
=> [3,3,3,3,3,2,1]
=> ? = 1
[8,7,6]
=> ?
=> ? = 1
[9,8,7]
=> ?
=> ? = 1
[9,8,7,6,5,4,3]
=> [7,7,7,6,5,4,3,2,1]
=> ? = 3
[8,7,6,5]
=> ?
=> ? = 2
[8,7,6,5,4]
=> ?
=> ? = 2
[9,8,7,6]
=> ?
=> ? = 2
[9,8,7,6,5]
=> ?
=> ? = 2
[9,8,7,6,5,4]
=> ?
=> ? = 3
Description
The number of distinct even parts of a partition.
See Section 3.3.1 of [1].
Matching statistic: St000257
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 78%●distinct values known / distinct values provided: 60%
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 78%●distinct values known / distinct values provided: 60%
Values
[1]
=> [1]
=> [1]
=> 0
[2]
=> [1,1]
=> [2]
=> 0
[1,1]
=> [2]
=> [1,1]
=> 1
[3]
=> [1,1,1]
=> [2,1]
=> 0
[2,1]
=> [2,1]
=> [1,1,1]
=> 1
[1,1,1]
=> [3]
=> [3]
=> 0
[4]
=> [1,1,1,1]
=> [4]
=> 0
[3,1]
=> [2,1,1]
=> [2,1,1]
=> 1
[2,2]
=> [2,2]
=> [1,1,1,1]
=> 1
[2,1,1]
=> [3,1]
=> [3,1]
=> 0
[1,1,1,1]
=> [4]
=> [2,2]
=> 1
[5]
=> [1,1,1,1,1]
=> [4,1]
=> 0
[4,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> 1
[3,2]
=> [2,2,1]
=> [1,1,1,1,1]
=> 1
[3,1,1]
=> [3,1,1]
=> [3,2]
=> 0
[2,2,1]
=> [3,2]
=> [3,1,1]
=> 1
[2,1,1,1]
=> [4,1]
=> [2,2,1]
=> 1
[1,1,1,1,1]
=> [5]
=> [5]
=> 0
[6]
=> [1,1,1,1,1,1]
=> [4,2]
=> 0
[5,1]
=> [2,1,1,1,1]
=> [4,1,1]
=> 1
[4,2]
=> [2,2,1,1]
=> [2,1,1,1,1]
=> 1
[4,1,1]
=> [3,1,1,1]
=> [3,2,1]
=> 0
[3,3]
=> [2,2,2]
=> [1,1,1,1,1,1]
=> 1
[3,2,1]
=> [3,2,1]
=> [3,1,1,1]
=> 1
[3,1,1,1]
=> [4,1,1]
=> [2,2,2]
=> 1
[2,2,2]
=> [3,3]
=> [6]
=> 0
[2,2,1,1]
=> [4,2]
=> [2,2,1,1]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [5,1]
=> 0
[1,1,1,1,1,1]
=> [6]
=> [3,3]
=> 1
[7]
=> [1,1,1,1,1,1,1]
=> [4,2,1]
=> 0
[6,1]
=> [2,1,1,1,1,1]
=> [4,1,1,1]
=> 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1,1,1]
=> 1
[5,1,1]
=> [3,1,1,1,1]
=> [4,3]
=> 0
[4,3]
=> [2,2,2,1]
=> [1,1,1,1,1,1,1]
=> 1
[4,2,1]
=> [3,2,1,1]
=> [3,2,1,1]
=> 1
[4,1,1,1]
=> [4,1,1,1]
=> [2,2,2,1]
=> 1
[3,3,1]
=> [3,2,2]
=> [3,1,1,1,1]
=> 1
[3,2,2]
=> [3,3,1]
=> [6,1]
=> 0
[3,2,1,1]
=> [4,2,1]
=> [2,2,1,1,1]
=> 2
[3,1,1,1,1]
=> [5,1,1]
=> [5,2]
=> 0
[2,2,2,1]
=> [4,3]
=> [3,2,2]
=> 1
[2,2,1,1,1]
=> [5,2]
=> [5,1,1]
=> 1
[2,1,1,1,1,1]
=> [6,1]
=> [3,3,1]
=> 1
[1,1,1,1,1,1,1]
=> [7]
=> [7]
=> 0
[8]
=> [1,1,1,1,1,1,1,1]
=> [8]
=> 0
[7,1]
=> [2,1,1,1,1,1,1]
=> [4,2,1,1]
=> 1
[6,2]
=> [2,2,1,1,1,1]
=> [4,1,1,1,1]
=> 1
[6,1,1]
=> [3,1,1,1,1,1]
=> [4,3,1]
=> 0
[5,3]
=> [2,2,2,1,1]
=> [2,1,1,1,1,1,1]
=> 1
[5,2,1]
=> [3,2,1,1,1]
=> [3,2,1,1,1]
=> 1
[5,5,2,2]
=> [4,4,2,2,2]
=> [2,2,2,2,1,1,1,1,1,1]
=> ? = 2
[6,5,4]
=> [3,3,3,3,2,1]
=> [12,1,1,1]
=> ? = 1
[6,3,3,3]
=> [4,4,4,1,1,1]
=> [2,2,2,2,2,2,2,1]
=> ? = 1
[7,5,3,1]
=> [4,3,3,2,2,1,1]
=> [6,2,2,2,1,1,1,1]
=> ? = 2
[9,8]
=> [2,2,2,2,2,2,2,2,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[7,5,5]
=> [3,3,3,3,3,1,1]
=> [12,3,2]
=> ? = 0
[6,5,3,3]
=> [4,4,4,2,2,1]
=> [2,2,2,2,2,2,1,1,1,1,1]
=> ? = 2
[5,4,4,2,2]
=> [5,5,3,3,1]
=> [10,6,1]
=> ? = 0
[]
=> []
=> ?
=> ? = 0
[6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> ?
=> ? = 3
[6,5,4,3,2]
=> [5,5,4,3,2,1]
=> ?
=> ? = 2
[6,5,4,2,2]
=> [5,5,3,3,2,1]
=> ?
=> ? = 1
[6,4,4,2,2]
=> [5,5,3,3,1,1]
=> ?
=> ? = 0
[6,5,4,3]
=> [4,4,4,3,2,1]
=> ?
=> ? = 2
[7,6,5,4,3,2,1]
=> [7,6,5,4,3,2,1]
=> ?
=> ? = 3
[6,5,5,4,3,2,1]
=> [7,6,5,4,3,1]
=> ?
=> ? = 2
[6,6,4,4,3,2,1]
=> [7,6,5,4,2,2]
=> ?
=> ? = 3
[6,6,5,3,3,2,1]
=> [7,6,5,3,3,2]
=> ?
=> ? = 2
[6,5,4,3,3,2,1]
=> [7,6,5,3,2,1]
=> ?
=> ? = 2
[6,6,5,4,2,2,1]
=> [7,6,4,4,3,2]
=> ?
=> ? = 3
[6,6,5,4,3,1,1]
=> [7,5,5,4,3,2]
=> ?
=> ? = 2
[6,5,4,3,2,1,1]
=> [7,5,4,3,2,1]
=> ?
=> ? = 2
[6,6,2,1,1,1,1]
=> [7,3,2,2,2,2]
=> ?
=> ? = 1
[7,6,5,4,3,2]
=> [6,6,5,4,3,2,1]
=> ?
=> ? = 3
[6,6,5,4,3,2]
=> [6,6,5,4,3,2]
=> ?
=> ? = 3
[7,5,5,4,3,2]
=> [6,6,5,4,3,1,1]
=> ?
=> ? = 2
[7,6,4,4,3,2]
=> [6,6,5,4,2,2,1]
=> ?
=> ? = 3
[6,6,2,2,2,2]
=> [6,6,2,2,2,2]
=> ?
=> ? = 2
[7,6,5,4,3,1]
=> [6,5,5,4,3,2,1]
=> ?
=> ? = 3
[6,5,5,4,3,1]
=> [6,5,5,4,3,1]
=> ?
=> ? = 2
[7,6,5,3,2,1]
=> [6,5,4,3,3,2,1]
=> ?
=> ? = 3
[7,5,4,3,2,1]
=> [6,5,4,3,2,1,1]
=> ?
=> ? = 3
[7,6,5,4,3]
=> [5,5,5,4,3,2,1]
=> ?
=> ? = 2
[7,5,4,3,2]
=> [5,5,4,3,2,1,1]
=> ?
=> ? = 2
[7,5,4,3,1]
=> [5,4,4,3,2,1,1]
=> ?
=> ? = 2
[7,6,5,4]
=> [4,4,4,4,3,2,1]
=> ?
=> ? = 2
[7,6,5]
=> [3,3,3,3,3,2,1]
=> ?
=> ? = 1
[5,5,5,5]
=> [4,4,4,4,4]
=> ?
=> ? = 1
[6,6,6,6]
=> [4,4,4,4,4,4]
=> ?
=> ? = 1
[6,6,6]
=> [3,3,3,3,3,3]
=> ?
=> ? = 0
[8,7,6]
=> ?
=> ?
=> ? = 1
[9,8,7]
=> ?
=> ?
=> ? = 1
[7,7,7]
=> [3,3,3,3,3,3,3]
=> ?
=> ? = 0
[8,7,6,5,4,3,2,1]
=> [8,7,6,5,4,3,2,1]
=> ?
=> ? = 4
[9,8,7,6,5,4,3,2,1]
=> [9,8,7,6,5,4,3,2,1]
=> ?
=> ? = 4
[8,7,6,5,4,3,2]
=> [7,7,6,5,4,3,2,1]
=> ?
=> ? = 3
[9,8,7,6,5,4,3,2]
=> [8,8,7,6,5,4,3,2,1]
=> ?
=> ? = 4
[8,7,6,5,4,3]
=> [6,6,6,5,4,3,2,1]
=> ?
=> ? = 3
[5,5,5,5,5]
=> [5,5,5,5,5]
=> ?
=> ? = 0
[9,8,7,6,5,4,3]
=> [7,7,7,6,5,4,3,2,1]
=> ?
=> ? = 3
Description
The number of distinct parts of a partition that occur at least twice.
See Section 3.3.1 of [2].
Matching statistic: St001114
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001114: Permutations ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 80%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001114: Permutations ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 80%
Values
[1]
=> []
=> []
=> [] => ? = 0
[2]
=> []
=> []
=> [] => ? = 0
[1,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[3]
=> []
=> []
=> [] => ? = 0
[2,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[4]
=> []
=> []
=> [] => ? = 0
[3,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[5]
=> []
=> []
=> [] => ? = 0
[4,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[6]
=> []
=> []
=> [] => ? = 0
[5,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 1
[7]
=> []
=> []
=> [] => ? = 0
[6,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 0
[8]
=> []
=> []
=> [] => ? = 0
[7,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 1
[9]
=> []
=> []
=> [] => ? = 0
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 0
[2,2,1,1,1,1,1]
=> [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
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? = 0
[10]
=> []
=> []
=> [] => ? = 0
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 0
[3,2,1,1,1,1,1]
=> [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
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? = 1
[2,2,2,1,1,1,1]
=> [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] => ? = 0
[2,2,1,1,1,1,1,1]
=> [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] => ? = 2
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? = 0
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => ? = 1
[6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[2,2,2,2,2,2]
=> [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
[7,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? = 1
[7,7]
=> [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] => ? = 1
[8,7]
=> [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] => ? = 1
[5,5,5]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 0
[8,8]
=> [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
[4,4,4,4]
=> [4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [5,6,7,1,2,3,4] => ? = 1
[4,3,3,2,2,1,1]
=> [3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [4,5,3,6,2,7,1] => ? = 0
[9,8]
=> [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
[7,5,5]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => ? = 0
[]
=> ?
=> ?
=> ? => ? = 0
[7,6,5,4,3,2,1]
=> [6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1] => ? = 3
[6,5,5,4,3,2,1]
=> [5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => ? = 2
[6,6,4,4,3,2,1]
=> [6,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => ? = 3
[6,6,5,3,3,2,1]
=> [6,5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => ? = 2
[6,5,4,3,3,2,1]
=> [5,4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,7,3,2,1] => ? = 2
[6,6,5,4,2,2,1]
=> [6,5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,2,1] => ? = 3
[6,6,5,4,3,1,1]
=> [6,5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,3,1] => ? = 2
[6,5,4,3,2,1,1]
=> [5,4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,7,1] => ? = 2
[6,6,2,1,1,1,1]
=> [6,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,4,5,6,1] => ? = 1
[7,6,5,4,3,2]
=> [6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,1,2] => ? = 3
[6,6,5,4,3,2]
=> [6,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,1,2] => ? = 3
[7,5,5,4,3,2]
=> [5,5,4,3,2]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,1,2] => ? = 2
[7,6,4,4,3,2]
=> [6,4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,1,2] => ? = 3
[6,6,2,2,2,2]
=> [6,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,1,2] => ? = 2
Description
The number of odd descents of a permutation.
Matching statistic: St001151
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00215: Set partitions —Wachs-White⟶ Set partitions
St001151: Set partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 80%
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00215: Set partitions —Wachs-White⟶ Set partitions
St001151: Set partitions ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 80%
Values
[1]
=> [1,0,1,0]
=> {{1},{2}}
=> {{1},{2}}
=> 1 = 0 + 1
[2]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> {{1},{2,3}}
=> 1 = 0 + 1
[1,1]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> {{1,2},{3}}
=> 2 = 1 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 1 = 0 + 1
[2,1]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 2 = 1 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> 1 = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> 1 = 0 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 2 = 1 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> 1 = 0 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> {{1,2,3,4},{5}}
=> 2 = 1 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> {{1},{2,3,4,5,6}}
=> 1 = 0 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> {{1},{2,4},{3,5}}
=> 2 = 1 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 2 = 1 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> 1 = 0 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> {{1,3},{2,4},{5}}
=> 2 = 1 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3,4,5,6}}
=> {{1,2,3,4,5},{6}}
=> 1 = 0 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6},{7}}
=> {{1},{2,3,4,5,6,7}}
=> 1 = 0 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> {{1,2,3,5},{4},{6}}
=> {{1},{2,4},{3,5,6}}
=> 2 = 1 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> {{1},{2,5},{3,4}}
=> 2 = 1 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> 1 = 0 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> 2 = 1 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 2 = 1 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> {{1,4},{2,3},{5}}
=> 2 = 1 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> 1 = 0 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> {{1,2,4},{3},{5}}
=> 3 = 2 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> {{1},{2,3,4,6},{5}}
=> {{1,3},{2,4,5},{6}}
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,6,7}}
=> {{1,2,3,4,5,6},{7}}
=> 2 = 1 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7},{8}}
=> {{1},{2,3,4,5,6,7,8}}
=> ? = 0 + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,6},{5},{7}}
=> {{1},{2,4},{3,5,6,7}}
=> 2 = 1 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> {{1,2,5},{3,4},{6}}
=> {{1},{2,5},{3,4,6}}
=> 2 = 1 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> {{1,2,4,5},{3},{6}}
=> {{1},{2,3,5},{4,6}}
=> 1 = 0 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> 2 = 1 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> {{1},{2,5},{3},{4}}
=> 2 = 1 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> 2 = 1 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> {{1,2},{3,5},{4}}
=> 2 = 1 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> 1 = 0 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> {{1,4},{2},{3},{5}}
=> 3 = 2 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> {{1},{2,3,6},{4,5}}
=> {{1,4},{2,3,5},{6}}
=> 1 = 0 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> {{1,2,3},{4},{5}}
=> 2 = 1 + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> {{1},{2,3,5,6},{4}}
=> {{1,2,4},{3,5},{6}}
=> 2 = 1 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> {{1},{2,3,4,5,7},{6}}
=> {{1,3},{2,4,5,6},{7}}
=> 2 = 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]
=> {{1},{2,3,4,5,6,7,8}}
=> {{1,2,3,4,5,6,7},{8}}
=> ? = 0 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7,8},{9}}
=> {{1},{2,3,4,5,6,7,8,9}}
=> ? = 0 + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,7},{6},{8}}
=> {{1},{2,4},{3,5,6,7,8}}
=> ? = 1 + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> {{1,2,3,6},{4,5},{7}}
=> {{1},{2,5},{3,4,6,7}}
=> 2 = 1 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,5,6},{4},{7}}
=> {{1},{2,3,5},{4,6,7}}
=> 1 = 0 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> {{1,5},{2,3,4},{6}}
=> {{1},{2,6},{3,4,5}}
=> 2 = 1 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> {{1,2,5},{3},{4},{6}}
=> {{1},{2,5},{3},{4,6}}
=> 2 = 1 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> {{1,3,4,5},{2},{6}}
=> {{1},{2,3,4,6},{5}}
=> 2 = 1 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> {{1,2,3,4},{5,6}}
=> {{1,2},{3,4,5,6}}
=> 2 = 1 + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> {{1},{2},{3,5},{4}}
=> 2 = 1 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> 1 = 0 + 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},{2,3,4,5,6,8},{7}}
=> {{1,3},{2,4,5,6,7},{8}}
=> ? = 0 + 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]
=> {{1},{2,3,4,5,6,7,8,9}}
=> {{1,2,3,4,5,6,7,8},{9}}
=> ? = 1 + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7,8,9},{10}}
=> {{1},{2,3,4,5,6,7,8,9,10}}
=> ? = 0 + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,8},{7},{9}}
=> {{1},{2,4},{3,5,6,7,8,9}}
=> ? = 1 + 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,7},{5,6},{8}}
=> {{1},{2,5},{3,4,6,7,8}}
=> ? = 1 + 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,6,7},{5},{8}}
=> {{1},{2,3,5},{4,6,7,8}}
=> ? = 0 + 1
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> {{1},{2,3,4,5,8},{6,7}}
=> {{1,4},{2,3,5,6,7},{8}}
=> ? = 0 + 1
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> {{1},{2,3,4,5,7,8},{6}}
=> {{1,2,4},{3,5,6,7},{8}}
=> ? = 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},{2,3,4,5,6,7,9},{8}}
=> {{1,3},{2,4,5,6,7,8},{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]
=> {{1},{2,3,4,5,6,7,8,9,10}}
=> {{1,2,3,4,5,6,7,8,9},{10}}
=> ? = 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]
=> {{1,2,3,4,5,6,7,8,9,10},{11}}
=> {{1},{2,3,4,5,6,7,8,9,10,11}}
=> ? = 0 + 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,6,7,9},{8},{10}}
=> {{1},{2,4},{3,5,6,7,8,9,10}}
=> ? = 1 + 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,8},{6,7},{9}}
=> {{1},{2,5},{3,4,6,7,8,9}}
=> ? = 1 + 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> {{1,2,3,4,5,7,8},{6},{9}}
=> {{1},{2,3,5},{4,6,7,8,9}}
=> ? = 0 + 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> {{1,2,3,7},{4,5,6},{8}}
=> {{1},{2,6},{3,4,5,7,8}}
=> ? = 1 + 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,7},{5},{6},{8}}
=> {{1},{2,5},{3},{4,6,7,8}}
=> ? = 1 + 1
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> {{1,2,3,5,6,7},{4},{8}}
=> {{1},{2,3,4,6},{5,7,8}}
=> ? = 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},{2,3,4,8},{5,6,7}}
=> {{1,5},{2,3,4,6,7},{8}}
=> ? = 0 + 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},{2,3,4,5,8},{6},{7}}
=> {{1,4},{2},{3,5,6,7},{8}}
=> ? = 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]
=> {{1},{2,3,4,5,6,9},{7,8}}
=> {{1,4},{2,3,5,6,7,8},{9}}
=> ? = 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]
=> {{1},{2,3,4,6,7,8},{5}}
=> {{1,2,3,5},{4,6,7},{8}}
=> ? = 0 + 1
[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]
=> {{1},{2,3,4,5,6,8,9},{7}}
=> {{1,2,4},{3,5,6,7,8},{9}}
=> ? = 2 + 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},{2,3,4,5,6,7,8,10},{9}}
=> {{1,3},{2,4,5,6,7,8,9},{10}}
=> ? = 0 + 1
[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},{2,3,4,5,6,7,8,9,10,11}}
=> {{1,2,3,4,5,6,7,8,9,10},{11}}
=> ? = 1 + 1
[6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> {{1,2,3,4,5,6},{7,8}}
=> {{1,2},{3,4,5,6,7,8}}
=> ? = 1 + 1
[2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,2},{3,4,5,6,7,8}}
=> {{1,2,3,4,5,6},{7,8}}
=> ? = 1 + 1
[7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> {{1},{2},{3,4,5,6,7,8}}
=> ? = 1 + 1
[7,7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> {{1,2,3,4,5,6,7},{8,9}}
=> {{1,2},{3,4,5,6,7,8,9}}
=> ? = 1 + 1
[8,7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6,7},{8},{9}}
=> {{1},{2},{3,4,5,6,7,8,9}}
=> ? = 1 + 1
[5,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> {{1,2,3,4,5},{6,7,8}}
=> {{1,2,3},{4,5,6,7,8}}
=> ? = 0 + 1
[8,8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> {{1,2,3,4,5,6,7,8},{9,10}}
=> {{1,2},{3,4,5,6,7,8,9,10}}
=> ? = 1 + 1
[7,5,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> {{1,7},{2,6},{3,5},{4},{8}}
=> {{1},{2,8},{3,7},{4,6},{5}}
=> ? = 2 + 1
[4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> {{1,2,3,4},{5,6,7,8}}
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 1 + 1
[4,3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> {{1},{2,4,6,8},{3},{5},{7}}
=> {{1,3},{2,5},{4,7},{6},{8}}
=> ? = 0 + 1
[9,8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6,7,8},{9},{10}}
=> {{1},{2},{3,4,5,6,7,8,9,10}}
=> ? = 1 + 1
[7,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> {{1,2,3,4,5},{6,7},{8}}
=> {{1},{2,3},{4,5,6,7,8}}
=> ? = 0 + 1
[]
=> []
=> {}
=> {}
=> ? = 0 + 1
[7,6,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 3 + 1
[6,5,5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3},{4},{5},{6,8},{7}}
=> {{1,3},{2},{4},{5},{6},{7},{8}}
=> ? = 2 + 1
[6,6,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> {{1},{2},{3},{4},{5,6},{7,8}}
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> ? = 3 + 1
[6,6,5,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5},{6},{7,8}}
=> {{1,2},{3},{4,5},{6},{7},{8}}
=> ? = 2 + 1
[6,5,4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> {{1},{2},{3},{4,8},{5},{6},{7}}
=> {{1,5},{2},{3},{4},{6},{7},{8}}
=> ? = 2 + 1
[6,6,5,4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4},{5},{6},{7,8}}
=> {{1,2},{3},{4},{5,6},{7},{8}}
=> ? = 3 + 1
[6,6,5,4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> {{1},{2,3},{4},{5},{6},{7,8}}
=> {{1,2},{3},{4},{5},{6,7},{8}}
=> ? = 2 + 1
[6,5,4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> {{1},{2,8},{3},{4},{5},{6},{7}}
=> {{1,7},{2},{3},{4},{5},{6},{8}}
=> ? = 2 + 1
[6,6,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> {{1},{2,3,4,6},{5},{7,8}}
=> {{1,2},{3,5},{4,6,7},{8}}
=> ? = 1 + 1
Description
The number of blocks with odd minimum.
See [[St000746]] for the analogous statistic on perfect matchings.
Matching statistic: St000256
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 60%
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000256: Integer partitions ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 60%
Values
[1]
=> [1]
=> [1]
=> [1]
=> 0
[2]
=> [1,1]
=> [2]
=> [1,1]
=> 0
[1,1]
=> [2]
=> [1,1]
=> [2]
=> 1
[3]
=> [1,1,1]
=> [2,1]
=> [2,1]
=> 0
[2,1]
=> [2,1]
=> [1,1,1]
=> [3]
=> 1
[1,1,1]
=> [3]
=> [3]
=> [1,1,1]
=> 0
[4]
=> [1,1,1,1]
=> [4]
=> [1,1,1,1]
=> 0
[3,1]
=> [2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[2,2]
=> [2,2]
=> [1,1,1,1]
=> [4]
=> 1
[2,1,1]
=> [3,1]
=> [3,1]
=> [2,1,1]
=> 0
[1,1,1,1]
=> [4]
=> [2,2]
=> [2,2]
=> 1
[5]
=> [1,1,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 0
[4,1]
=> [2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1
[3,2]
=> [2,2,1]
=> [1,1,1,1,1]
=> [5]
=> 1
[3,1,1]
=> [3,1,1]
=> [3,2]
=> [2,2,1]
=> 0
[2,2,1]
=> [3,2]
=> [3,1,1]
=> [3,1,1]
=> 1
[2,1,1,1]
=> [4,1]
=> [2,2,1]
=> [3,2]
=> 1
[1,1,1,1,1]
=> [5]
=> [5]
=> [1,1,1,1,1]
=> 0
[6]
=> [1,1,1,1,1,1]
=> [4,2]
=> [2,2,1,1]
=> 0
[5,1]
=> [2,1,1,1,1]
=> [4,1,1]
=> [3,1,1,1]
=> 1
[4,2]
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [5,1]
=> 1
[4,1,1]
=> [3,1,1,1]
=> [3,2,1]
=> [3,2,1]
=> 0
[3,3]
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [6]
=> 1
[3,2,1]
=> [3,2,1]
=> [3,1,1,1]
=> [4,1,1]
=> 1
[3,1,1,1]
=> [4,1,1]
=> [2,2,2]
=> [3,3]
=> 1
[2,2,2]
=> [3,3]
=> [6]
=> [1,1,1,1,1,1]
=> 0
[2,2,1,1]
=> [4,2]
=> [2,2,1,1]
=> [4,2]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [5,1]
=> [2,1,1,1,1]
=> 0
[1,1,1,1,1,1]
=> [6]
=> [3,3]
=> [2,2,2]
=> 1
[7]
=> [1,1,1,1,1,1,1]
=> [4,2,1]
=> [3,2,1,1]
=> 0
[6,1]
=> [2,1,1,1,1,1]
=> [4,1,1,1]
=> [4,1,1,1]
=> 1
[5,2]
=> [2,2,1,1,1]
=> [2,1,1,1,1,1]
=> [6,1]
=> 1
[5,1,1]
=> [3,1,1,1,1]
=> [4,3]
=> [2,2,2,1]
=> 0
[4,3]
=> [2,2,2,1]
=> [1,1,1,1,1,1,1]
=> [7]
=> 1
[4,2,1]
=> [3,2,1,1]
=> [3,2,1,1]
=> [4,2,1]
=> 1
[4,1,1,1]
=> [4,1,1,1]
=> [2,2,2,1]
=> [4,3]
=> 1
[3,3,1]
=> [3,2,2]
=> [3,1,1,1,1]
=> [5,1,1]
=> 1
[3,2,2]
=> [3,3,1]
=> [6,1]
=> [2,1,1,1,1,1]
=> 0
[3,2,1,1]
=> [4,2,1]
=> [2,2,1,1,1]
=> [5,2]
=> 2
[3,1,1,1,1]
=> [5,1,1]
=> [5,2]
=> [2,2,1,1,1]
=> 0
[2,2,2,1]
=> [4,3]
=> [3,2,2]
=> [3,3,1]
=> 1
[2,2,1,1,1]
=> [5,2]
=> [5,1,1]
=> [3,1,1,1,1]
=> 1
[2,1,1,1,1,1]
=> [6,1]
=> [3,3,1]
=> [3,2,2]
=> 1
[1,1,1,1,1,1,1]
=> [7]
=> [7]
=> [1,1,1,1,1,1,1]
=> 0
[8]
=> [1,1,1,1,1,1,1,1]
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0
[7,1]
=> [2,1,1,1,1,1,1]
=> [4,2,1,1]
=> [4,2,1,1]
=> 1
[6,2]
=> [2,2,1,1,1,1]
=> [4,1,1,1,1]
=> [5,1,1,1]
=> 1
[6,1,1]
=> [3,1,1,1,1,1]
=> [4,3,1]
=> [3,2,2,1]
=> 0
[5,3]
=> [2,2,2,1,1]
=> [2,1,1,1,1,1,1]
=> [7,1]
=> 1
[5,2,1]
=> [3,2,1,1,1]
=> [3,2,1,1,1]
=> [5,2,1]
=> 1
[6,5]
=> [2,2,2,2,2,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> ? = 1
[5,4,2]
=> [3,3,2,2,1]
=> [6,1,1,1,1,1]
=> [6,1,1,1,1,1]
=> ? = 1
[5,4,1,1]
=> [4,2,2,2,1]
=> [2,2,1,1,1,1,1,1,1]
=> [9,2]
=> ? = 2
[5,3,3]
=> [3,3,3,1,1]
=> [6,3,2]
=> [3,3,2,1,1,1]
=> ? = 0
[5,3,2,1]
=> [4,3,2,1,1]
=> [3,2,2,2,1,1]
=> [6,4,1]
=> ? = 2
[5,3,1,1,1]
=> [5,2,2,1,1]
=> [5,2,1,1,1,1]
=> [6,2,1,1,1]
=> ? = 1
[5,2,2,2]
=> [4,4,1,1,1]
=> [2,2,2,2,2,1]
=> [6,5]
=> ? = 1
[5,2,2,1,1]
=> [5,3,1,1,1]
=> [5,3,2,1]
=> [4,3,2,1,1]
=> ? = 0
[4,4,3]
=> [3,3,3,2]
=> [6,3,1,1]
=> [4,2,2,1,1,1]
=> ? = 1
[4,4,2,1]
=> [4,3,2,2]
=> [3,2,2,1,1,1,1]
=> [7,3,1]
=> ? = 2
[4,4,1,1,1]
=> [5,2,2,2]
=> [5,1,1,1,1,1,1]
=> [7,1,1,1,1]
=> ? = 1
[4,3,3,1]
=> [4,3,3,1]
=> [6,2,2,1]
=> [4,3,1,1,1,1]
=> ? = 1
[4,3,2,2]
=> [4,4,2,1]
=> [2,2,2,2,1,1,1]
=> [7,4]
=> ? = 2
[4,3,2,1,1]
=> [5,3,2,1]
=> [5,3,1,1,1]
=> [5,2,2,1,1]
=> ? = 1
[4,2,2,2,1]
=> [5,4,1,1]
=> [5,2,2,2]
=> [4,4,1,1,1]
=> ? = 1
[3,3,3,2]
=> [4,4,3]
=> [3,2,2,2,2]
=> [5,5,1]
=> ? = 1
[3,3,3,1,1]
=> [5,3,3]
=> [6,5]
=> [2,2,2,2,2,1]
=> ? = 0
[3,3,2,2,1]
=> [5,4,2]
=> [5,2,2,1,1]
=> [5,3,1,1,1]
=> ? = 2
[6,6]
=> [2,2,2,2,2,2]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [12]
=> ? = 1
[5,4,3]
=> [3,3,3,2,1]
=> [6,3,1,1,1]
=> [5,2,2,1,1,1]
=> ? = 1
[5,4,2,1]
=> [4,3,2,2,1]
=> [3,2,2,1,1,1,1,1]
=> [8,3,1]
=> ? = 2
[5,4,1,1,1]
=> [5,2,2,2,1]
=> [5,1,1,1,1,1,1,1]
=> [8,1,1,1,1]
=> ? = 1
[5,3,3,1]
=> [4,3,3,1,1]
=> [6,2,2,2]
=> [4,4,1,1,1,1]
=> ? = 1
[5,3,2,2]
=> [4,4,2,1,1]
=> [2,2,2,2,2,1,1]
=> [7,5]
=> ? = 2
[5,3,2,1,1]
=> [5,3,2,1,1]
=> [5,3,2,1,1]
=> [5,3,2,1,1]
=> ? = 1
[5,2,2,2,1]
=> [5,4,1,1,1]
=> [5,2,2,2,1]
=> [5,4,1,1,1]
=> ? = 1
[4,4,4]
=> [3,3,3,3]
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[4,4,3,1]
=> [4,3,3,2]
=> [6,2,2,1,1]
=> [5,3,1,1,1,1]
=> ? = 2
[4,4,2,2]
=> [4,4,2,2]
=> [2,2,2,2,1,1,1,1]
=> [8,4]
=> ? = 2
[4,4,2,1,1]
=> [5,3,2,2]
=> [5,3,1,1,1,1]
=> [6,2,2,1,1]
=> ? = 1
[4,3,3,2]
=> [4,4,3,1]
=> [3,2,2,2,2,1]
=> [6,5,1]
=> ? = 1
[4,3,3,1,1]
=> [5,3,3,1]
=> [6,5,1]
=> [3,2,2,2,2,1]
=> ? = 0
[4,3,2,2,1]
=> [5,4,2,1]
=> [5,2,2,1,1,1]
=> [6,3,1,1,1]
=> ? = 2
[3,3,3,2,1]
=> [5,4,3]
=> [5,3,2,2]
=> [4,4,2,1,1]
=> ? = 1
[2,2,2,2,2,2]
=> [6,6]
=> [3,3,3,3]
=> [4,4,4]
=> ? = 1
[7,6]
=> [2,2,2,2,2,2,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [13]
=> ? = 1
[5,4,3,1]
=> [4,3,3,2,1]
=> [6,2,2,1,1,1]
=> [6,3,1,1,1,1]
=> ? = 2
[5,4,2,2]
=> [4,4,2,2,1]
=> [2,2,2,2,1,1,1,1,1]
=> [9,4]
=> ? = 2
[5,4,2,1,1]
=> [5,3,2,2,1]
=> [5,3,1,1,1,1,1]
=> [7,2,2,1,1]
=> ? = 1
[5,3,3,2]
=> [4,4,3,1,1]
=> [3,2,2,2,2,2]
=> [6,6,1]
=> ? = 1
[5,3,3,1,1]
=> [5,3,3,1,1]
=> [6,5,2]
=> [3,3,2,2,2,1]
=> ? = 0
[5,3,2,2,1]
=> [5,4,2,1,1]
=> [5,2,2,2,1,1]
=> [6,4,1,1,1]
=> ? = 2
[4,4,3,2]
=> [4,4,3,2]
=> [3,2,2,2,2,1,1]
=> [7,5,1]
=> ? = 2
[4,4,3,1,1]
=> [5,3,3,2]
=> [6,5,1,1]
=> [4,2,2,2,2,1]
=> ? = 1
[4,4,2,2,1]
=> [5,4,2,2]
=> [5,2,2,1,1,1,1]
=> [7,3,1,1,1]
=> ? = 2
[4,3,3,3]
=> [4,4,4,1]
=> [2,2,2,2,2,2,1]
=> [7,6]
=> ? = 1
[4,3,3,2,1]
=> [5,4,3,1]
=> [5,3,2,2,1]
=> [5,4,2,1,1]
=> ? = 1
[7,7]
=> [2,2,2,2,2,2,2]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [14]
=> ? = 1
[6,4,4]
=> [3,3,3,3,1,1]
=> [12,2]
=> [2,2,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[5,5,2,2]
=> [4,4,2,2,2]
=> [2,2,2,2,1,1,1,1,1,1]
=> [10,4]
=> ? = 2
Description
The number of parts from which one can substract 2 and still get an integer partition.
Matching statistic: St000214
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 60%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 60%
Values
[1]
=> [[1]]
=> [1] => [1] => 0
[2]
=> [[1,2]]
=> [1,2] => [1,2] => 0
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 1
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,2,3] => 0
[2,1]
=> [[1,3],[2]]
=> [2,1,3] => [2,1,3] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [2,3,1] => 0
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 0
[3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [4,1,3,2] => 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => 0
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [3,2,4,1] => 1
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1
[3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [4,1,3,2,5] => 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => 0
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,5,1,4,3] => 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [3,2,4,1,5] => 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [3,4,2,5,1] => 0
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [4,1,3,2,5,6] => 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [2,3,1,4,5,6] => 0
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [6,1,5,2,4,3] => 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [2,5,1,4,3,6] => 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [3,2,4,1,5,6] => 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [3,4,6,1,5,2] => 0
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => [3,2,6,1,5,4] => 2
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => [3,4,2,5,1,6] => 0
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [4,3,5,2,6,1] => 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [4,1,3,2,5,6,7] => ? = 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [2,3,1,4,5,6,7] => 0
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [6,1,5,2,4,3,7] => ? = 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => [2,5,1,4,3,6,7] => ? = 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => [3,2,4,1,5,6,7] => ? = 1
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => [2,7,1,6,3,5,4] => ? = 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => [3,4,6,1,5,2,7] => ? = 0
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => [3,2,6,1,5,4,7] => ? = 2
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => [3,4,2,5,1,6,7] => ? = 0
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => [5,7,2,4,1,6,3] => ? = 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => [3,4,2,7,1,6,5] => ? = 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => [4,3,5,2,6,1,7] => ? = 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [4,5,3,6,2,7,1] => ? = 0
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => 0
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => [4,1,3,2,5,6,7,8] => ? = 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => [2,3,1,4,5,6,7,8] => ? = 0
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => [6,1,5,2,4,3,7,8] => ? = 1
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => [2,5,1,4,3,6,7,8] => ? = 1
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => [3,2,4,1,5,6,7,8] => ? = 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [8,1,7,2,6,3,5,4] => ? = 1
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => [2,7,1,6,3,5,4,8] => ? = 1
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => [3,4,6,1,5,2,7,8] => ? = 0
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => [3,2,6,1,5,4,7,8] => ? = 2
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => [3,4,2,5,1,6,7,8] => ? = 0
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => [3,4,8,1,7,2,6,5] => ? = 1
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => [3,2,8,1,7,4,6,5] => ? = 2
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => [5,7,2,4,1,6,3,8] => ? = 1
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => [3,4,2,7,1,6,5,8] => ? = 1
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => [4,3,5,2,6,1,7,8] => ? = 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [6,3,5,4,8,1,7,2] => ? = 1
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4] => [3,6,8,2,5,1,7,4] => ? = 0
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6] => [4,3,5,2,8,1,7,6] => ? = 2
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => [4,5,3,6,2,7,1,8] => ? = 0
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => [5,4,6,3,7,2,8,1] => 1
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => 0
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => 1
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [3,4,1,2,5,6,7,8,9] => [4,1,3,2,5,6,7,8,9] => ? = 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [3,2,1,4,5,6,7,8,9] => [2,3,1,4,5,6,7,8,9] => ? = 0
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [4,5,6,1,2,3,7,8,9] => [6,1,5,2,4,3,7,8,9] => ? = 1
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [4,2,5,1,3,6,7,8,9] => [2,5,1,4,3,6,7,8,9] => ? = 1
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => [3,2,4,1,5,6,7,8,9] => ? = 1
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4,9] => [8,1,7,2,6,3,5,4,9] => ? = 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8,9] => [2,7,1,6,3,5,4,8,9] => ? = 1
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8,9] => [3,4,6,1,5,2,7,8,9] => ? = 0
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8,9] => [3,2,6,1,5,4,7,8,9] => ? = 2
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => [3,4,2,5,1,6,7,8,9] => ? = 0
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [6,2,7,8,9,1,3,4,5] => [2,9,1,8,3,7,4,6,5] => ? = 1
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5,9] => [3,4,8,1,7,2,6,5,9] => ? = 1
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5,9] => [3,2,8,1,7,4,6,5,9] => ? = 2
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8,9] => [5,7,2,4,1,6,3,8,9] => ? = 1
[4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8,9] => [3,4,2,7,1,6,5,8,9] => ? = 1
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9] => [4,3,5,2,6,1,7,8,9] => ? = 1
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3] => [4,5,6,9,1,8,2,7,3] => ? = 0
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6] => [5,9,2,4,1,8,3,7,6] => ? = 2
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6] => [3,4,2,9,1,8,5,7,6] => ? = 1
[10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => 0
[9,1]
=> [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9,10] => 1
[]
=> []
=> [] => [] => 0
Description
The number of adjacencies of a permutation.
An adjacency of a permutation $\pi$ is an index $i$ such that $\pi(i)-1 = \pi(i+1)$. Adjacencies are also known as ''small descents''.
This can be also described as an occurrence of the bivincular pattern ([2,1], {((0,1),(1,0),(1,1),(1,2),(2,1)}), i.e., the middle row and the middle column are shaded, see [3].
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!