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Your data matches 28 different statistics following compositions of up to 3 maps.
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Matching statistic: St001484
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> []
=> []
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> []
=> []
=> 0
[1,1,0,0]
=> [1,0,1,0]
=> [1]
=> [1]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> []
=> 0
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> [2]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1]
=> 0
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [2,1]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [3]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [3,2]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [3,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2,2]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [3,1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [2,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2,2,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [3,2,1]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [3]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [4,3]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [2]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [4,2]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [3,3]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [4,1,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [3,2]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [3,1,1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [2,2,2,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [4,3,2]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [4,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [3,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [2,1,1,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [4,3,1]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [4,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [2,2]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [2,2,2]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [4,2,2]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [3,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2,2,1,1]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,3,2]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [4,3,1,1]
=> 2
Description
The number of singletons of an integer partition.
A singleton in an integer partition is a part that appear precisely once.
Matching statistic: St000932
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 80%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1,0]
=> []
=> []
=> ? = 0
[1,0,1,0]
=> [1,1,0,0]
=> []
=> []
=> ? = 0
[1,1,0,0]
=> [1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,1,1,1,1]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [7,5]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,2,2,2,1]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [7,2,2,2,1,1]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,2,2,2]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [6,6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [7,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [7,5,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [7,6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,3,2]
=> ?
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0]
=> [7,2,2,2,1]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0,1,0]
=> [7,5,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> [6,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [6,6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> [7,5,4]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,2,1]
=> ?
=> ? = 3
[1,0,1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [7,6,2,2,2,1]
=> ?
=> ? = 3
[1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5,1,1,1]
=> ?
=> ? = 3
[1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,2,1,1,1]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,2,1,1]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> [6,6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0,1,0]
=> [7,5,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 0
[1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [7,4,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> [7,6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0,1,0]
=> [7,2,2,2]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
Description
The number of occurrences of the pattern UDU in a Dyck path.
The number of Dyck paths with statistic value 0 are counted by the Motzkin numbers [1].
Matching statistic: St001067
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 70%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001067: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> [1,0]
=> []
=> []
=> ? = 0
[1,0,1,0]
=> [1,1,0,0]
=> []
=> []
=> ? = 0
[1,1,0,0]
=> [1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,1,1,1,1]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [7,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,1,1,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [7,5]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,2,2,2,1]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,3,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [7,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [7,2,2,2,1,1]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0,1,0]
=> [7,6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [5,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [6,2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,6,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,2,2,2]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [6,6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [6,2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 2
Description
The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St000502
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 80%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1] => [1] => {{1}}
=> ? = 0
[1,0,1,0]
=> [1,2] => [1,2] => {{1},{2}}
=> 0
[1,1,0,0]
=> [2,1] => [2,1] => {{1,2}}
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 0
[1,1,1,0,0,0]
=> [3,1,2] => [3,1,2] => {{1,2,3}}
=> 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 0
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,4,2,3] => {{1},{2,3,4}}
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 0
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [4,2,1,3] => {{1,3,4},{2}}
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => {{1,2,3},{4}}
=> 2
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4,1,3,2] => {{1,2,4},{3}}
=> 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [4,3,2,1] => {{1,4},{2,3}}
=> 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => {{1,2,3,4}}
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,5,3,4] => {{1},{2},{3,4,5}}
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => {{1},{2,4,5},{3}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,4,2,3,5] => {{1},{2,3,4},{5}}
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,5,2,4,3] => {{1},{2,3,5},{4}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,5,2,3,4] => {{1},{2,3,4,5}}
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => {{1,2},{3,4},{5}}
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => {{1,2},{3,4,5}}
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => {{1,3},{2},{4},{5}}
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => {{1,4},{2},{3},{5}}
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => {{1,4,5},{2},{3}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => {{1,3,4},{2},{5}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => {{1,3,5},{2},{4}}
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => {{1,3,4,5},{2}}
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => {{1,2,3},{4},{5}}
=> 2
[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] => {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,8,7] => {{1},{2},{3},{4},{5},{6},{7,8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,5,7,6,8] => [1,2,3,4,5,7,6,8] => {{1},{2},{3},{4},{5},{6,7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => [1,2,3,4,5,8,7,6] => {{1},{2},{3},{4},{5},{6,8},{7}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,6,7] => [1,2,3,4,5,8,6,7] => {{1},{2},{3},{4},{5},{6,7,8}}
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,4,6,5,7,8] => [1,2,3,4,6,5,7,8] => {{1},{2},{3},{4},{5,6},{7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,4,6,5,8,7] => [1,2,3,4,6,5,8,7] => {{1},{2},{3},{4},{5,6},{7,8}}
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,4,6,7,5,8] => [1,2,3,4,7,6,5,8] => {{1},{2},{3},{4},{5,7},{6},{8}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => [1,2,3,4,8,6,7,5] => {{1},{2},{3},{4},{5,8},{6},{7}}
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,4,6,8,5,7] => [1,2,3,4,8,6,5,7] => {{1},{2},{3},{4},{5,7,8},{6}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,4,7,5,6,8] => [1,2,3,4,7,5,6,8] => {{1},{2},{3},{4},{5,6,7},{8}}
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,4,7,5,8,6] => [1,2,3,4,8,5,7,6] => {{1},{2},{3},{4},{5,6,8},{7}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,4,7,8,5,6] => [1,2,3,4,8,7,6,5] => {{1},{2},{3},{4},{5,8},{6,7}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,4,8,5,6,7] => [1,2,3,4,8,5,6,7] => {{1},{2},{3},{4},{5,6,7,8}}
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,3,5,4,6,7,8] => [1,2,3,5,4,6,7,8] => {{1},{2},{3},{4,5},{6},{7},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,3,5,4,6,8,7] => [1,2,3,5,4,6,8,7] => {{1},{2},{3},{4,5},{6},{7,8}}
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,3,5,4,7,6,8] => [1,2,3,5,4,7,6,8] => {{1},{2},{3},{4,5},{6,7},{8}}
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,3,5,4,7,8,6] => [1,2,3,5,4,8,7,6] => {{1},{2},{3},{4,5},{6,8},{7}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3,5,4,8,6,7] => [1,2,3,5,4,8,6,7] => {{1},{2},{3},{4,5},{6,7,8}}
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,3,5,6,4,7,8] => [1,2,3,6,5,4,7,8] => {{1},{2},{3},{4,6},{5},{7},{8}}
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,3,5,6,4,8,7] => [1,2,3,6,5,4,8,7] => {{1},{2},{3},{4,6},{5},{7,8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,3,5,6,7,4,8] => [1,2,3,7,5,6,4,8] => {{1},{2},{3},{4,7},{5},{6},{8}}
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => [1,2,3,8,5,6,7,4] => {{1},{2},{3},{4,8},{5},{6},{7}}
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,3,5,6,8,4,7] => [1,2,3,8,5,6,4,7] => {{1},{2},{3},{4,7,8},{5},{6}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,3,5,7,4,6,8] => [1,2,3,7,5,4,6,8] => {{1},{2},{3},{4,6,7},{5},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,3,5,7,4,8,6] => [1,2,3,8,5,4,7,6] => {{1},{2},{3},{4,6,8},{5},{7}}
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,3,5,7,8,4,6] => [1,2,3,8,5,7,6,4] => {{1},{2},{3},{4,8},{5},{6,7}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,3,6,4,5,7,8] => [1,2,3,6,4,5,7,8] => {{1},{2},{3},{4,5,6},{7},{8}}
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,3,6,4,5,8,7] => [1,2,3,6,4,5,8,7] => {{1},{2},{3},{4,5,6},{7,8}}
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,3,6,4,7,5,8] => [1,2,3,7,4,6,5,8] => {{1},{2},{3},{4,5,7},{6},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,3,6,4,7,8,5] => [1,2,3,8,4,6,7,5] => {{1},{2},{3},{4,5,8},{6},{7}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,3,6,4,8,5,7] => [1,2,3,8,4,6,5,7] => {{1},{2},{3},{4,5,7,8},{6}}
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,3,6,7,4,5,8] => [1,2,3,7,6,5,4,8] => {{1},{2},{3},{4,7},{5,6},{8}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,3,6,7,4,8,5] => [1,2,3,8,6,5,7,4] => {{1},{2},{3},{4,8},{5,6},{7}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,3,6,7,8,4,5] => [1,2,3,8,7,6,5,4] => {{1},{2},{3},{4,8},{5,7},{6}}
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,7,4,5,6,8] => [1,2,3,7,4,5,6,8] => {{1},{2},{3},{4,5,6,7},{8}}
=> ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,3,7,4,5,8,6] => [1,2,3,8,4,5,7,6] => {{1},{2},{3},{4,5,6,8},{7}}
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,3,7,4,8,5,6] => [1,2,3,8,4,7,6,5] => {{1},{2},{3},{4,5,8},{6,7}}
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,3,7,8,4,5,6] => [1,2,3,8,7,5,6,4] => {{1},{2},{3},{4,8},{5,6,7}}
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7,8] => [1,2,4,3,5,6,7,8] => {{1},{2},{3,4},{5},{6},{7},{8}}
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,2,4,3,5,6,8,7] => [1,2,4,3,5,6,8,7] => {{1},{2},{3,4},{5},{6},{7,8}}
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5,7,6,8] => [1,2,4,3,5,7,6,8] => {{1},{2},{3,4},{5},{6,7},{8}}
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,2,4,3,5,7,8,6] => [1,2,4,3,5,8,7,6] => {{1},{2},{3,4},{5},{6,8},{7}}
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,2,4,3,5,8,6,7] => [1,2,4,3,5,8,6,7] => {{1},{2},{3,4},{5},{6,7,8}}
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,6,5,7,8] => [1,2,4,3,6,5,7,8] => {{1},{2},{3,4},{5,6},{7},{8}}
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5,8,7] => [1,2,4,3,6,5,8,7] => {{1},{2},{3,4},{5,6},{7,8}}
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,2,4,3,6,7,5,8] => [1,2,4,3,7,6,5,8] => ?
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,2,4,3,6,7,8,5] => [1,2,4,3,8,6,7,5] => {{1},{2},{3,4},{5,8},{6},{7}}
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,2,4,3,6,8,5,7] => [1,2,4,3,8,6,5,7] => ?
=> ? = 2
Description
The number of successions of a set partitions.
This is the number of indices i such that i and i+1 belonging to the same block.
Matching statistic: St001189
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
St001189: Dyck paths ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 70%
Values
[1,0]
=> 0
[1,0,1,0]
=> 0
[1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> 0
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 0
[1,1,1,0,0,0]
=> 2
[1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> 0
[1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> ? = 3
Description
The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001223
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001223: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 60%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001223: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 60%
Values
[1,0]
=> [1,0]
=> []
=> []
=> ? = 0
[1,0,1,0]
=> [1,1,0,0]
=> []
=> []
=> ? = 0
[1,1,0,0]
=> [1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> []
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 0
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,1,1]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 3
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 0
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,2,2,1]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> ? = 3
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [3,3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,2,1,1]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> ? = 4
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> ? = 0
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 1
Description
Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless.
Matching statistic: St000441
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1] => 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 0
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 0
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,5,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,5,6,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,1,6] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 0
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,7,4,5,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [6,4,5,3,2,1,7] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,3,2,1] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,4,2,1] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,5,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,2,1] => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,1,5] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [7,4,5,3,2,1,6] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,3,2,1] => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,7,1] => ? = 0
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,6,4,7,3,2,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,4,2,1,6] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [6,4,3,2,1,5,7] => ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,6,3,4,2,1] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,6,7,3,4,2,1] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0,1,0]
=> [7,5,4,2,3,1,6] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,6,2,1] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,1,2] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,3,2,1] => ? = 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [4,5,3,2,1,6,7] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,2,1] => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [5,6,3,4,2,1,7] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [5,3,4,2,1,6,7] => ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,2,1] => ? = 4
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,3,1] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2,3,4,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,2,3,1] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,3,1,7] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [7,6,2,3,4,5,1] => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,6,4,2,3,1] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [5,6,7,4,2,3,1] => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,1,2,6] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2,4,5,1] => ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [6,5,2,3,4,1,7] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [6,7,5,2,3,4,1] => ? = 3
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [5,6,4,2,3,1,7] => ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [5,4,2,3,1,6,7] => ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,6,1] => ? = 4
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,1,4] => ? = 0
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [7,6,3,4,2,1,5] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [7,6,4,2,3,1,5] => ? = 1
Description
The number of successions of a permutation.
A succession of a permutation π is an index i such that π(i)+1=π(i+1). Successions are also known as ''small ascents'' or ''1-rises''.
Matching statistic: St000214
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000214: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [3,1,2] => 0
[1,1,1,0,0,0]
=> [3,1,2] => [2,3,1] => [3,2,1] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,3,4,2] => [1,4,3,2] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => [4,3,1,2] => 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => [4,2,1,3] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,4,1,2] => [4,1,3,2] => 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => [4,3,2,1] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,4,5,3] => [1,2,5,4,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,2,5,3] => [1,5,4,2,3] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,3,4,2,5] => [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,3,5,2,4] => [1,5,3,2,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,4,5,2,3] => [1,5,2,4,3] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,3,4,5,2] => [1,5,4,3,2] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,3,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => [2,1,5,4,3] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [5,1,2,3,4] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,1,2,5,3] => [5,4,1,2,3] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [3,1,4,2,5] => [4,3,1,2,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [3,1,5,2,4] => [5,3,1,2,4] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,1,5,2,3] => [5,1,2,4,3] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [3,1,4,5,2] => [5,4,3,1,2] => 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,4,6] => [1,2,3,6,4,7,5] => [1,2,3,7,6,4,5] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,4,7,5] => [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,4,5] => [1,2,3,6,7,4,5] => [1,2,3,7,4,6,5] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => [1,2,3,7,6,5,4] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => [1,2,4,3,7,6,5] => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => [1,2,6,3,4,5,7] => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [1,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => ? = 0
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,3,6] => [1,2,6,3,4,7,5] => [1,2,7,6,3,4,5] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,3,5,7] => [1,2,5,3,6,4,7] => [1,2,6,5,3,4,7] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,3,7,5] => [1,2,5,3,7,4,6] => [1,2,7,5,3,4,6] => ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,3,5] => [1,2,6,3,7,4,5] => [1,2,7,3,4,6,5] => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,3,5,6] => [1,2,5,3,6,7,4] => [1,2,7,6,5,3,4] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,3,4,6,7] => [1,2,4,5,3,6,7] => [1,2,5,4,3,6,7] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,3,4,7,6] => [1,2,4,5,3,7,6] => [1,2,5,4,3,7,6] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,3,6,4,7] => [1,2,4,6,3,5,7] => [1,2,6,4,3,5,7] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,3,6,7,4] => [1,2,4,7,3,5,6] => [1,2,7,4,3,5,6] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,3,7,4,6] => [1,2,4,6,3,7,5] => [1,2,7,6,4,3,5] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,3,4,7] => [1,2,5,6,3,4,7] => [1,2,6,3,5,4,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,3,7,4] => [1,2,5,7,3,4,6] => [1,2,7,3,5,4,6] => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,3,4] => [1,2,6,7,3,4,5] => [1,2,7,3,6,4,5] => ? = 0
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,3,4,6] => [1,2,5,6,3,7,4] => [1,2,7,6,3,5,4] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,3,4,5,7] => [1,2,4,5,6,3,7] => [1,2,6,5,4,3,7] => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,3,4,7,5] => [1,2,4,5,7,3,6] => [1,2,7,5,4,3,6] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,3,7,4,5] => [1,2,4,6,7,3,5] => [1,2,7,4,3,6,5] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,3,4,5] => [1,2,5,6,7,3,4] => [1,2,7,3,6,5,4] => ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,3,4,5,6] => [1,2,4,5,6,7,3] => [1,2,7,6,5,4,3] => ? = 4
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [1,3,2,4,7,5,6] => [1,3,2,4,7,5,6] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,5,6] => [1,3,2,4,6,7,5] => [1,3,2,4,7,6,5] => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [1,3,2,6,4,5,7] => [1,3,2,6,4,5,7] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,2,5,6,7,4] => [1,3,2,7,4,5,6] => [1,3,2,7,4,5,6] => ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,2,5,7,4,6] => [1,3,2,6,4,7,5] => [1,3,2,7,6,4,5] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,6,4,5,7] => [1,3,2,5,6,4,7] => [1,3,2,6,5,4,7] => ? = 3
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,2,6,4,7,5] => [1,3,2,5,7,4,6] => [1,3,2,7,5,4,6] => ? = 2
Description
The number of adjacencies of a permutation.
An adjacency of a permutation π is an index i such that π(i)−1=π(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].
Matching statistic: St000237
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 100%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => [3,2,1] => 0
[1,1,1,0,0,0]
=> [3,1,2] => [3,1,2] => [2,3,1] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,3,4,2] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [4,2,1,3] => [3,2,4,1] => 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => [2,3,1,4] => 2
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4,1,3,2] => [2,4,3,1] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [2,3,4,1] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,5,3,4] => [1,2,4,5,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [1,4,3,5,2] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,5,2,4,3] => [1,3,5,4,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,5,2,3,4] => [1,3,4,5,2] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [2,1,4,5,3] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [4,2,3,1,5] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [5,2,3,4,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => [4,2,3,5,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [3,2,4,1,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5,2,1,4,3] => [3,2,5,4,1] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [5,2,4,3,1] => [5,2,4,3,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [3,2,4,5,1] => 2
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => [1,2,3,4,7,6,5] => ? = 0
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,5,6] => [1,2,3,4,7,5,6] => [1,2,3,4,6,7,5] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => [1,2,3,5,4,6,7] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => [1,2,3,5,4,7,6] => ? = 2
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [1,2,3,6,5,4,7] => [1,2,3,6,5,4,7] => ? = 0
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [1,2,3,7,5,6,4] => [1,2,3,7,5,6,4] => ? = 0
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,4,6] => [1,2,3,7,5,4,6] => [1,2,3,6,5,7,4] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,4,5,7] => [1,2,3,6,4,5,7] => [1,2,3,5,6,4,7] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,4,7,5] => [1,2,3,7,4,6,5] => [1,2,3,5,7,6,4] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,4,5] => [1,2,3,7,6,5,4] => [1,2,3,7,6,5,4] => ? = 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,4,5,6] => [1,2,3,7,4,5,6] => [1,2,3,5,6,7,4] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => [1,2,4,3,5,7,6] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => [1,2,4,3,6,5,7] => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [1,2,4,3,7,6,5] => [1,2,4,3,7,6,5] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,5,6] => [1,2,4,3,7,5,6] => [1,2,4,3,6,7,5] => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [1,2,5,4,3,6,7] => [1,2,5,4,3,6,7] => ? = 0
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [1,2,5,4,3,7,6] => [1,2,5,4,3,7,6] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [1,2,6,4,5,3,7] => [1,2,6,4,5,3,7] => ? = 0
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [1,2,7,4,5,6,3] => [1,2,7,4,5,6,3] => ? = 0
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,3,6] => [1,2,7,4,5,3,6] => [1,2,6,4,5,7,3] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,3,5,7] => [1,2,6,4,3,5,7] => [1,2,5,4,6,3,7] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,3,7,5] => [1,2,7,4,3,6,5] => [1,2,5,4,7,6,3] => ? = 0
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,3,5] => [1,2,7,4,6,5,3] => [1,2,7,4,6,5,3] => ? = 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,3,5,6] => [1,2,7,4,3,5,6] => [1,2,5,4,6,7,3] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,3,4,6,7] => [1,2,5,3,4,6,7] => [1,2,4,5,3,6,7] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,3,4,7,6] => [1,2,5,3,4,7,6] => [1,2,4,5,3,7,6] => ? = 3
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,3,6,4,7] => [1,2,6,3,5,4,7] => [1,2,4,6,5,3,7] => ? = 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,3,6,7,4] => [1,2,7,3,5,6,4] => [1,2,4,7,5,6,3] => ? = 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,3,7,4,6] => [1,2,7,3,5,4,6] => [1,2,4,6,5,7,3] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,3,4,7] => [1,2,6,5,4,3,7] => [1,2,6,5,4,3,7] => ? = 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,3,7,4] => [1,2,7,5,4,6,3] => [1,2,7,5,4,6,3] => ? = 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,3,4] => [1,2,7,6,5,4,3] => [1,2,7,6,5,4,3] => ? = 0
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,3,4,6] => [1,2,7,5,4,3,6] => [1,2,6,5,4,7,3] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,3,4,5,7] => [1,2,6,3,4,5,7] => [1,2,4,5,6,3,7] => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,3,4,7,5] => [1,2,7,3,4,6,5] => [1,2,4,5,7,6,3] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,3,7,4,5] => [1,2,7,3,6,5,4] => [1,2,4,7,6,5,3] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,3,4,5] => [1,2,7,6,4,5,3] => [1,2,7,5,6,4,3] => ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,3,4,5,6] => [1,2,7,3,4,5,6] => [1,2,4,5,6,7,3] => ? = 4
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => [1,3,2,4,5,7,6] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [1,3,2,4,7,6,5] => [1,3,2,4,7,6,5] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,5,6] => [1,3,2,4,7,5,6] => [1,3,2,4,6,7,5] => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => [1,3,2,5,4,6,7] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => [1,3,2,5,4,7,6] => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [1,3,2,6,5,4,7] => [1,3,2,6,5,4,7] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,2,5,6,7,4] => [1,3,2,7,5,6,4] => [1,3,2,7,5,6,4] => ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,2,5,7,4,6] => [1,3,2,7,5,4,6] => [1,3,2,6,5,7,4] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,6,4,5,7] => [1,3,2,6,4,5,7] => [1,3,2,5,6,4,7] => ? = 3
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,2,6,4,7,5] => [1,3,2,7,4,6,5] => [1,3,2,5,7,6,4] => ? = 2
Description
The number of small exceedances.
This is the number of indices i such that πi=i+1.
Matching statistic: St001061
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001061: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 60%
St001061: Permutations ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 60%
Values
[1,0]
=> [1] => ? = 0
[1,0,1,0]
=> [1,2] => 0
[1,1,0,0]
=> [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => 0
[1,1,1,0,0,0]
=> [3,2,1] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 0
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 0
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 0
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 0
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 2
[1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,4,6,5,7,3,2] => ? = 2
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,4,6,7,5,3,2] => ? = 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,4,7,6,5,3,2] => ? = 3
[1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,4,3,2,6,7] => ? = 3
[1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,5,4,3,2,7,6] => ? = 4
[1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,5,4,3,6,2,7] => ? = 2
[1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,5,4,3,6,7,2] => ? = 2
[1,0,1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,5,4,3,7,6,2] => ? = 3
[1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,5,4,6,3,2,7] => ? = 2
[1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,5,4,6,3,7,2] => ? = 2
[1,0,1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,5,4,6,7,3,2] => ? = 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,5,4,7,6,3,2] => ? = 3
[1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,5,6,4,3,2,7] => ? = 2
[1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,5,6,4,3,7,2] => ? = 2
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,5,6,4,7,3,2] => ? = 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,5,6,7,4,3,2] => ? = 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? = 3
[1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,5,4,3,2,7] => ? = 4
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,6,5,4,3,7,2] => ? = 3
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,6,5,4,7,3,2] => ? = 3
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => ? = 3
[1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? = 3
[1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? = 5
[1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7] => ? = 1
[1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 2
[1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [2,1,3,4,6,5,7] => ? = 2
[1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [2,1,3,4,6,7,5] => ? = 1
[1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [2,1,3,4,7,6,5] => ? = 3
[1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => ? = 2
[1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => ? = 3
[1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,6,4,7] => ? = 1
[1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [2,1,3,5,6,7,4] => ? = 1
[1,1,0,0,1,0,1,1,0,1,1,0,0,0]
=> [2,1,3,5,7,6,4] => ? = 2
[1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,6,5,4,7] => ? = 3
[1,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [2,1,3,6,5,7,4] => ? = 2
[1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> [2,1,3,6,7,5,4] => ? = 2
[1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,7,6,5,4] => ? = 4
[1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 2
[1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,3,5,7,6] => ? = 3
[1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5,7] => ? = 3
[1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,3,6,7,5] => ? = 2
[1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,6,5] => ? = 4
[1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,5,3,6,7] => ? = 1
[1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [2,1,4,5,3,7,6] => ? = 2
[1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [2,1,4,5,6,3,7] => ? = 1
[1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [2,1,4,5,6,7,3] => ? = 1
[1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [2,1,4,5,7,6,3] => ? = 2
[1,1,0,0,1,1,0,1,1,0,0,0,1,0]
=> [2,1,4,6,5,3,7] => ? = 2
[1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [2,1,4,6,5,7,3] => ? = 1
Description
The number of indices that are both descents and recoils of a permutation.
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000931The number of occurrences of the pattern UUU in a Dyck path. St000247The number of singleton blocks of a set partition. St000925The number of topologically connected components of a set partition. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000366The number of double descents of a permutation. St000731The number of double exceedences of a permutation. St000365The number of double ascents of a permutation. St000732The number of double deficiencies of a permutation. St000039The number of crossings of a permutation. St000317The cycle descent number of a permutation. St001948The number of augmented double ascents of a permutation. St000392The length of the longest run of ones in a binary word. St000982The length of the longest constant subword. St001960The number of descents of a permutation minus one if its first entry is not one.
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