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Your data matches 32 different statistics following compositions of up to 3 maps.
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(click to perform a complete search on your data)
Matching statistic: St000445
(load all 34 compositions to match this statistic)
(load all 34 compositions to match this statistic)
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 2
[1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> 1
[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]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> 3
[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]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> 3
[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]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> 0
Description
The number of rises of length 1 of a Dyck path.
Matching statistic: St001126
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001126: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001126: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [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]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
Description
Number of simple module that are 1-regular in the corresponding Nakayama algebra.
Matching statistic: St000475
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> 1
[1,0,1,0]
=> [1,1] => [1,1]
=> 2
[1,1,0,0]
=> [2] => [2]
=> 0
[1,0,1,0,1,0]
=> [1,1,1] => [1,1,1]
=> 3
[1,0,1,1,0,0]
=> [1,2] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [2,1] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [3] => [3]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [3,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [2,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [2,2]
=> 0
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [2,1,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [2,1,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,2] => [2,2]
=> 0
[1,1,1,0,0,0,1,0]
=> [3,1] => [3,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [3,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [3,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [4]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,1,1,1,1]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [3,1,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [2,2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [2,1,1,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,1,1,1]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [2,2,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [2,1,1,1]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [2,1,1,1]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [2,2,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,2,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [2,2,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [2,2,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [3,2]
=> 0
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000932
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000932: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 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: St000247
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> {{1}}
=> ? = 1
[1,0,1,0]
=> [1,0,1,0]
=> {{1},{2}}
=> 2
[1,1,0,0]
=> [1,1,0,0]
=> {{1,2}}
=> 0
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 3
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 0
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 2
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 0
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 0
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> {{1,5},{2,4},{3}}
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 2
Description
The number of singleton blocks of a set partition.
Matching statistic: St000248
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000248: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000248: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> {{1}}
=> ? = 1
[1,0,1,0]
=> [1,1,0,0]
=> {{1,2}}
=> 2
[1,1,0,0]
=> [1,0,1,0]
=> {{1},{2}}
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 0
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 0
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> {{1,2,3,5},{4}}
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> {{1,2,5},{3,4}}
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> {{1,5},{2,3,4}}
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> {{1,5},{2,4},{3}}
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 2
Description
The number of anti-singletons of a set partition.
An anti-singleton of a set partition $S$ is an index $i$ such that $i$ and $i+1$ (considered cyclically) are both in the same block of $S$.
For noncrossing set partitions, this is also the number of singletons of the image of $S$ under the Kreweras complement.
Matching statistic: St000674
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> ? = 1
[1,0,1,0]
=> [1,1] => [1,0,1,0]
=> 2
[1,1,0,0]
=> [2] => [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 0
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 2
Description
The number of hills of a Dyck path.
A hill is a peak with up step starting and down step ending at height zero.
Matching statistic: St000986
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000986: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000986: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1
[1,0,1,0]
=> [1,1] => [2] => ([],2)
=> 2
[1,1,0,0]
=> [2] => [1,1] => ([(0,1)],2)
=> 0
[1,0,1,0,1,0]
=> [1,1,1] => [3] => ([],3)
=> 3
[1,0,1,1,0,0]
=> [1,2] => [1,2] => ([(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,1,1,0,0,0]
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => ([],4)
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,3] => ([(2,3)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,1,1,0,0,0,1,0]
=> [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,0,1,0,0,0]
=> [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => ([],5)
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [1,4] => ([(3,4)],5)
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,1,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
Description
The multiplicity of the eigenvalue zero of the adjacency matrix of the graph.
Matching statistic: St001484
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001484: Integer partitions ⟶ ℤResult quality: 57% ●values known / values provided: 57%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 4
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 0
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 0
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> 0
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2,1]
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> ? = 4
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,5,3,2,2,1]
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,2,1]
=> ? = 4
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,1,1]
=> ? = 2
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,1,1]
=> ? = 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,1]
=> ? = 4
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,1,1]
=> ? = 2
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,4,2,1,1]
=> ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,1]
=> ? = 4
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,3,2,1]
=> ? = 5
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4,3,2,1]
=> ? = 5
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,3,2,1]
=> ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,4,3,2,1]
=> ? = 5
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,5,4,4,3,2,1]
=> ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4,3,2,1]
=> ? = 4
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,3,2,1]
=> ? = 5
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [6,6,5,3,3,2,1]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,2,1]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,5,3,3,2,1]
=> ? = 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [5,5,5,3,3,2,1]
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,3,2,1]
=> ? = 5
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [6,6,4,3,3,2,1]
=> ? = 3
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,3,2,1]
=> ? = 5
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,5,4,3,3,2,1]
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [7,4,4,3,3,2,1]
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [6,4,4,3,3,2,1]
=> ? = 3
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,3,2,1]
=> ? = 3
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,4,4,3,3,2,1]
=> ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [7,6,3,3,3,2,1]
=> ? = 4
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [6,6,3,3,3,2,1]
=> ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [7,5,3,3,3,2,1]
=> ? = 4
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [5,5,3,3,3,2,1]
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,4,3,3,3,2,1]
=> ? = 4
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [5,4,3,3,3,2,1]
=> ? = 4
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,3,2,1]
=> ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,3,2,1]
=> ? = 3
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [5,3,3,3,3,2,1]
=> ? = 3
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,3,3,3,2,1]
=> ? = 3
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,2,1]
=> ? = 5
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,2,2,1]
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [7,5,5,4,2,2,1]
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,2,2,1]
=> ? = 3
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,2,2,1]
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4,2,2,1]
=> ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,2,2,1]
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [7,5,4,4,2,2,1]
=> ? = 3
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,2,2,1]
=> ? = 3
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [5,5,4,4,2,2,1]
=> ? = 1
Description
The number of singletons of an integer partition.
A singleton in an integer partition is a part that appear precisely once.
Matching statistic: St000022
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 100%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1] => 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,2] => 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2,1] => 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[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]
=> [1,2,3,4,5] => 5
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 0
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 0
[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]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [4,2,3,1,5,6,7] => ? = 5
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,4,3,2,5,6,7] => ? = 5
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => ? = 5
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7] => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [2,4,3,1,5,6,7] => ? = 4
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [3,2,4,1,5,6,7] => ? = 4
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,3,4,2,5,6,7] => ? = 4
[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]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [5,2,3,4,1,6,7] => ? = 5
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [5,3,2,4,1,6,7] => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,4,3,2,6,7] => ? = 3
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0,1,0]
=> [2,5,4,3,1,6,7] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,5,3,4,2,6,7] => ? = 5
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [5,2,4,3,1,6,7] => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => ? = 5
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => ? = 5
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,5,4,3,6,7] => ? = 3
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,2,1,5,4,6,7] => ? = 3
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => ? = 3
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,4,6,7] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0,1,0]
=> [2,5,3,4,1,6,7] => ? = 4
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [5,3,4,2,1,6,7] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [3,2,5,4,1,6,7] => ? = 4
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> [4,2,3,5,1,6,7] => ? = 4
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [4,3,2,5,1,6,7] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,3,5,4,2,6,7] => ? = 4
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,4,3,5,2,6,7] => ? = 4
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => ? = 4
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [2,1,4,5,3,6,7] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [2,3,5,4,1,6,7] => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> [2,4,3,5,1,6,7] => ? = 3
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [3,2,4,5,1,6,7] => ? = 3
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,3,4,5,2,6,7] => ? = 3
[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]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [6,2,3,4,5,1,7] => ? = 5
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [6,3,2,4,5,1,7] => ? = 3
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,6,4,3,5,2,7] => ? = 3
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [2,6,4,3,5,1,7] => ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,5,4,3,2,7] => ? = 3
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => ? = 3
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,6,5,4,3,7] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0,1,0]
=> [3,2,6,5,4,1,7] => ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,3,6,5,4,2,7] => ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [2,3,6,5,4,1,7] => ? = 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,6,3,4,5,2,7] => ? = 5
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [6,2,4,3,5,1,7] => ? = 3
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,6,3,5,4,2,7] => ? = 3
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0,1,0]
=> [2,6,3,5,4,1,7] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,6,4,5,3,7] => ? = 5
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,5,4,1,7] => ? = 3
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => ? = 5
Description
The number of fixed points of a permutation.
The following 22 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000215The number of adjacencies of a permutation, zero appended. St000895The number of ones on the main diagonal of an alternating sign matrix. St000502The number of successions of a set partitions. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St000221The number of strong fixed points of a permutation. St000241The number of cyclical small excedances. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St000894The trace of an alternating sign matrix. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000441The number of successions of a permutation. St000237The number of small exceedances. St000214The number of adjacencies of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St000239The number of small weak excedances. St001061The number of indices that are both descents and recoils of a permutation. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001223Number 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. St000731The number of double exceedences of a permutation. St001948The number of augmented double ascents of a permutation.
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