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Your data matches 117 different statistics following compositions of up to 3 maps.
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(click to perform a complete search on your data)
Matching statistic: St001137
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001137: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001137: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 0
[1,1] => [1,0,1,0]
=> 0
[2] => [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> 0
[3] => [1,1,1,0,0,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[1,3] => [1,0,1,1,1,0,0,0]
=> 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> 0
[4] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 0
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 1
Description
Number of simple modules that are 3-regular in the corresponding Nakayama algebra.
Matching statistic: St000661
(load all 29 compositions to match this statistic)
(load all 29 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000661: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
St000661: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> ? = 0
[1,1] => [1,0,1,0]
=> 0
[2] => [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> 0
[1,2] => [1,0,1,1,0,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> 0
[3] => [1,1,1,0,0,0]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 0
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 1
[3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
Description
The number of rises of length 3 of a Dyck path.
Matching statistic: St001141
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001141: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
St001141: Dyck paths ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> ? = 0
[1,1] => [1,0,1,0]
=> 0
[2] => [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> 0
[1,2] => [1,0,1,1,0,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> 0
[3] => [1,1,1,0,0,0]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 0
[1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[2,2] => [1,1,0,0,1,1,0,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 0
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 1
[3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
Description
The number of occurrences of hills of size 3 in a Dyck path.
A hill of size three is a subpath beginning at height zero, consisting of three up steps followed by three down steps.
Matching statistic: St001960
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1] => ? = 0
[1,1] => [2] => [1,1,0,0]
=> [1,2] => 0
[2] => [1] => [1,0]
=> [1] => ? = 0
[1,1,1] => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,2] => [1,1] => [1,0,1,0]
=> [2,1] => 0
[2,1] => [1,1] => [1,0,1,0]
=> [2,1] => 0
[3] => [1] => [1,0]
=> [1] => ? = 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 0
[1,3] => [1,1] => [1,0,1,0]
=> [2,1] => 0
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 0
[2,2] => [2] => [1,1,0,0]
=> [1,2] => 0
[3,1] => [1,1] => [1,0,1,0]
=> [2,1] => 0
[4] => [1] => [1,0]
=> [1] => ? = 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 0
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 0
[1,4] => [1,1] => [1,0,1,0]
=> [2,1] => 0
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 0
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[2,3] => [1,1] => [1,0,1,0]
=> [2,1] => 0
[3,1,1] => [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 0
[3,2] => [1,1] => [1,0,1,0]
=> [2,1] => 0
[4,1] => [1,1] => [1,0,1,0]
=> [2,1] => 0
[5] => [1] => [1,0]
=> [1] => ? = 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ? ∊ {0,2}
[1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1
[1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 1
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1
[1,1,4] => [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 0
[1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 0
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 1
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 0
[1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0
[1,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 0
[1,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 0
[1,5] => [1,1] => [1,0,1,0]
=> [2,1] => 0
[2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 0
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 1
[2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 0
[2,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 0
[2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[2,2,2] => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
[2,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 0
[2,4] => [1,1] => [1,0,1,0]
=> [2,1] => 0
[3,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0
[6] => [1] => [1,0]
=> [1] => ? ∊ {0,2}
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => ? ∊ {0,0,0,0,0,1,2,2}
[1,1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => ? ∊ {0,0,0,0,0,1,2,2}
[1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => ? ∊ {0,0,0,0,0,1,2,2}
[1,1,1,2,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => ? ∊ {0,0,0,0,0,1,2,2}
[1,1,2,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => ? ∊ {0,0,0,0,0,1,2,2}
[1,2,1,1,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => ? ∊ {0,0,0,0,0,1,2,2}
[2,1,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => ? ∊ {0,0,0,0,0,1,2,2}
[7] => [1] => [1,0]
=> [1] => ? ∊ {0,0,0,0,0,1,2,2}
[1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,1,1,2] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,1,2,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,2,3,4,6,7,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,1,3] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,2,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,2,3,5,6,4,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,2,2] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,3,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,2,3,5,6,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,2,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,2,4,5,3,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,2,1,2] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,2,4,5,6,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,2,2,1] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,3,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,2,4,5,3,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,2,1,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,3,4,2,5,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,2,1,1,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,2,1,2,1] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,2,2,1,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,3,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,1,1,1,1] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,1,4,5,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,1,1,2] => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,1,2,1] => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,2,1,1] => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,2,1,1,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,1,1,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,1,1,1,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,1,2,1] => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,5,6,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,2,1,1] => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,2,1,1,1] => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,2,1,1,1,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[8] => [1] => [1,0]
=> [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Matching statistic: St000649
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000649: Permutations ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000649: Permutations ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0,1,0]
=> [2,1] => 0
[1,1] => [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[2] => [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0
[1,1,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[1,2] => [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[1,1,1,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0
[1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1
[2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0
[2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
[3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 1
[4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 0
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 0
[1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 0
[1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 0
[1,1,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1
[1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 0
[1,2,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 0
[1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1
[1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 0
[2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 0
[2,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 0
[2,2,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 0
[2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 1
[3,1,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1
[3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 1
[4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 0
[5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 0
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {0,0}
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 0
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 0
[1,1,1,3] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 0
[1,1,2,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 0
[1,1,3,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[1,1,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 0
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 0
[1,2,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 0
[1,2,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 0
[1,2,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 1
[1,3,1,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1
[1,3,2] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 1
[1,4,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 0
[1,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => 0
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => 0
[2,1,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 0
[2,1,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 0
[2,1,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 1
[6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => ? ∊ {0,0}
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1}
[1,1,1,1,1,2] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1}
[1,1,1,1,2,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1}
[1,1,1,2,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1}
[1,1,2,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1}
[1,2,1,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1}
[1,6] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,1,1}
[2,1,1,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,1,1}
[6,1] => [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,1,1}
[7] => [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => ? ∊ {0,0,0,0,0,0,0,0,1,1}
[1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,1,2,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,1,3] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,2,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,2,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,1,3,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,2,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,2,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,2,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,1,3,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,2,1,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,2,1,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,2,1,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,2,2,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,3,1,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,1,6] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,1,1,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,1,1,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,1,1,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,1,2,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,2,1,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,1,1,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,6,1] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,7] => [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,1,1,1,1,1] => [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,1,1,1,2] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,1,1,2,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,1,2,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,2,1,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,2,1,1,1,1] => [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,4,2,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,6] => [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,1,1,1,1] => [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,7,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[6,1,1] => [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[6,2] => [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[7,1] => [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[8] => [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
Description
The number of 3-excedences of a permutation.
This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+3$.
Matching statistic: St001465
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [1] => 0
[1,1] => [1,0,1,0]
=> [1,2] => [1,2] => 0
[2] => [1,1,0,0]
=> [2,1] => [1,2] => 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0
[1,2] => [1,0,1,1,0,0]
=> [1,3,2] => [1,2,3] => 0
[2,1] => [1,1,0,0,1,0]
=> [2,1,3] => [1,2,3] => 0
[3] => [1,1,1,0,0,0]
=> [3,2,1] => [1,3,2] => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,3,4] => 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,2,3,4] => 0
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,4,3] => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,2,3,4] => 0
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,2,3,4] => 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,3,2,4] => 1
[4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,4,2,3] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,4,5] => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,3,4,5] => 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,3,5,4] => 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,2,3,4,5] => 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,3,4,5] => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,4,3,5] => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,2,5,3,4] => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,2,3,4,5] => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,2,3,4,5] => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,2,3,4,5] => 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,2,3,5,4] => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,3,2,4,5] => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [1,3,2,4,5] => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [1,4,2,3,5] => 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [1,5,2,4,3] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [1,2,3,4,5,6] => 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [1,2,3,4,5,6] => 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [1,2,3,4,6,5] => 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [1,2,3,4,5,6] => 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => [1,2,3,4,5,6] => 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => [1,2,3,5,4,6] => 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => [1,2,3,6,4,5] => 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => [1,2,3,4,5,6] => 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => [1,2,3,4,5,6] => 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => [1,2,3,4,5,6] => 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => [1,2,3,4,6,5] => 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => [1,2,4,3,5,6] => 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => [1,2,4,3,5,6] => 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => [1,2,5,3,4,6] => 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [1,2,6,3,5,4] => 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => [1,2,3,4,5,6] => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => [1,2,3,4,5,6] => 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => [1,2,3,4,5,6] => 0
[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,4,3,2,1,6,7] => [1,5,2,4,3,6,7] => ? ∊ {0,0,0,0}
[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,4,3,2,1,7,6] => [1,5,2,4,3,6,7] => ? ∊ {0,0,0,0}
[6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => [1,6,2,5,3,4,7] => ? ∊ {0,0,0,0}
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,6,5,4,3,2,1] => [1,7,2,6,3,5,4] => ? ∊ {0,0,0,0}
[1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,3,6,5,4,7,8] => [1,2,3,4,6,5,7,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,3,6,5,4,8,7] => [1,2,3,4,6,5,7,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,7,6,5,4,8] => [1,2,3,4,7,5,6,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,8,7,6,5,4] => [1,2,3,4,8,5,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,2,4,3,7,6,5,8] => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,2,5,4,3,7,6,8] => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,5,4,3,8,7,6] => [1,2,3,5,4,6,8,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,6,5,4,3,8,7] => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,5,4,6,8,7] => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,2,6,5,4,7,8] => [1,2,3,4,6,5,7,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2,6,5,4,8,7] => [1,2,3,4,6,5,7,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,3,2,7,6,5,4,8] => [1,2,3,4,7,5,6,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,3,2,8,7,6,5,4] => [1,2,3,4,8,5,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,4,3,2,5,6,7,8] => [1,2,4,3,5,6,7,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,4,3,2,5,6,8,7] => [1,2,4,3,5,6,7,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,4,3,2,5,7,6,8] => [1,2,4,3,5,6,7,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,4,3,2,5,8,7,6] => [1,2,4,3,5,6,8,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,4,3,2,6,5,7,8] => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,4,3,2,6,5,8,7] => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,7,6,5,8] => [1,2,4,3,5,7,6,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,4,3,2,8,7,6,5] => [1,2,4,3,5,8,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,5,4,3,2,6,7,8] => [1,2,5,3,4,6,7,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,5,4,3,2,6,8,7] => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,5,4,3,2,7,6,8] => [1,2,5,3,4,6,7,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,5,4,3,2,8,7,6] => [1,2,5,3,4,6,8,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,6,5,4,3,2,7,8] => [1,2,6,3,5,4,7,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,6,5,4,3,2,8,7] => [1,2,6,3,5,4,7,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,7,6,5,4,3,2,8] => [1,2,7,3,6,4,5,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => [1,2,8,3,7,4,6,5] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,7,6,5,8] => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6,8] => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,6,5,4,7,8] => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,3,6,5,4,8,7] => [1,2,3,4,6,5,7,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,7,6,5,4,8] => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,8,7,6,5,4] => [1,2,3,4,8,5,7,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,3,5,7,6,8] => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,7,6,5,8] => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3,8,7,6] => [1,2,3,5,4,6,8,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,6,5,4,3,7,8] => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [3,2,1,4,5,8,7,6] => [1,3,2,4,5,6,8,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [3,2,1,4,6,5,7,8] => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> [3,2,1,4,6,5,8,7] => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [3,2,1,5,4,6,8,7] => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [3,2,1,5,4,8,7,6] => ? => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,6,5,4,7,8] => [1,3,2,4,6,5,7,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,6,5,4,8,7] => [1,3,2,4,6,5,7,8] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
Description
The number of adjacent transpositions in the cycle decomposition of a permutation.
Matching statistic: St001940
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001940: Integer partitions ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001940: Integer partitions ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
=> []
=> ? = 0
[1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0}
[2] => [[2],[]]
=> []
=> ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0}
[1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0}
[2,1] => [[2,2],[1]]
=> [1]
=> 1
[3] => [[3],[]]
=> []
=> ? ∊ {0,0,0}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0}
[1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
[1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
[2,2] => [[3,2],[1]]
=> [1]
=> 1
[3,1] => [[3,3],[2]]
=> [2]
=> 0
[4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1
[1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[2,3] => [[4,2],[1]]
=> [1]
=> 1
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
[3,2] => [[4,3],[2]]
=> [2]
=> 0
[4,1] => [[4,4],[3]]
=> [3]
=> 0
[5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> 1
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 0
[1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 0
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 1
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> 0
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> 0
[1,5] => [[5,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 0
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 0
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 2
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 1
[2,4] => [[5,2],[1]]
=> [1]
=> 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 0
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 1
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 0
[3,3] => [[5,3],[2]]
=> [2]
=> 0
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 0
[4,2] => [[5,4],[3]]
=> [3]
=> 0
[5,1] => [[5,5],[4]]
=> [4]
=> 0
[6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,1}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,2}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,2}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> 1
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,2}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> 0
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> 0
[1,1,1,4] => [[4,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,2}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> 0
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> 0
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> 1
[1,1,5] => [[5,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,2}
[1,6] => [[6,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,2}
[7] => [[7],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,2}
[1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,1,1,3] => [[3,1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,1,4] => [[4,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,2,3] => [[4,2,1,1,1],[1]]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,3,1,1] => [[3,3,3,1,1,1],[2,2]]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,5] => [[5,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,6] => [[6,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,1,1,3] => [[4,2,2,2,1],[1,1,1]]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,1,4] => [[5,2,2,1],[1,1]]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,4,1] => [[5,5,2,1],[4,1]]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,5] => [[6,2,1],[1]]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,3,1,1,1,1] => [[3,3,3,3,3,1],[2,2,2,2]]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,3,4] => [[6,3,1],[2]]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
Description
The number of distinct parts that are equal to their multiplicity in the integer partition.
Matching statistic: St000257
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
=> []
=> ?
=> ? = 0
[1,1] => [[1,1],[]]
=> []
=> ?
=> ? ∊ {0,0}
[2] => [[2],[]]
=> []
=> ?
=> ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1}
[1,2] => [[2,1],[]]
=> []
=> ?
=> ? ∊ {0,0,1}
[2,1] => [[2,2],[1]]
=> [1]
=> [1]
=> 0
[3] => [[3],[]]
=> []
=> ?
=> ? ∊ {0,0,1}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1}
[1,1,2] => [[2,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> [1]
=> 0
[1,3] => [[3,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1,1]
=> 1
[2,2] => [[3,2],[1]]
=> [1]
=> [1]
=> 0
[3,1] => [[3,3],[2]]
=> [2]
=> [2]
=> 0
[4] => [[4],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> [1]
=> 0
[1,1,3] => [[3,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1]
=> 1
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> [1]
=> 0
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> [2]
=> 0
[1,4] => [[4,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,1]
=> 1
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1,1]
=> 1
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [3]
=> 0
[2,3] => [[4,2],[1]]
=> [1]
=> [1]
=> 0
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [4]
=> 0
[3,2] => [[4,3],[2]]
=> [2]
=> [2]
=> 0
[4,1] => [[4,4],[3]]
=> [3]
=> [2,1]
=> 0
[5] => [[5],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> [1]
=> 0
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1]
=> 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> [1]
=> 0
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> [2]
=> 0
[1,1,4] => [[4,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,1]
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1]
=> 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [3]
=> 0
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> [1]
=> 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [4]
=> 0
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> [2]
=> 0
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> [2,1]
=> 0
[1,5] => [[5,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1,1]
=> 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,1]
=> 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [3,1]
=> 0
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1,1]
=> 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [5]
=> 0
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [3]
=> 0
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [2,1,1]
=> 1
[2,4] => [[5,2],[1]]
=> [1]
=> [1]
=> 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [6]
=> 0
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [4]
=> 0
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [4,1]
=> 0
[3,3] => [[5,3],[2]]
=> [2]
=> [2]
=> 0
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [3,2,1]
=> 0
[4,2] => [[5,4],[3]]
=> [3]
=> [2,1]
=> 0
[5,1] => [[5,5],[4]]
=> [4]
=> [2,2]
=> 1
[6] => [[6],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,1,2}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,2,2,2}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,2,2,2}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> [1]
=> 0
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,2,2,2}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [1,1]
=> 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> [1]
=> 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> [2]
=> 0
[1,1,1,4] => [[4,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,2,2,2}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1,1]
=> 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1]
=> 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [3]
=> 0
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> [1]
=> 0
[1,1,5] => [[5,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,2,2,2}
[1,6] => [[6,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,2,2,2}
[7] => [[7],[]]
=> []
=> ?
=> ? ∊ {0,0,0,1,2,2,2}
[1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,1,3] => [[3,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,1,4] => [[4,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,2,3] => [[4,2,1,1,1],[1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,3,1,1] => [[3,3,3,1,1,1],[2,2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,1,5] => [[5,1,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,1,6] => [[6,1,1],[]]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,1,3] => [[4,2,2,2,1],[1,1,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,1,4] => [[5,2,2,1],[1,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,4,1] => [[5,5,2,1],[4,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,5] => [[6,2,1],[1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,1,1,1,1] => [[3,3,3,3,3,1],[2,2,2,2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,4] => [[6,3,1],[2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
Description
The number of distinct parts of a partition that occur at least twice.
See Section 3.3.1 of [2].
Matching statistic: St001022
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001022: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001022: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
=> []
=> []
=> ? = 0
[1,1] => [[1,1],[]]
=> []
=> []
=> ? ∊ {0,0}
[2] => [[2],[]]
=> []
=> []
=> ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,1}
[1,2] => [[2,1],[]]
=> []
=> []
=> ? ∊ {0,0,1}
[2,1] => [[2,2],[1]]
=> [1]
=> [1,0]
=> 0
[3] => [[3],[]]
=> []
=> []
=> ? ∊ {0,0,1}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,1}
[1,1,2] => [[2,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,1}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,3] => [[3,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,1}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,2] => [[3,2],[1]]
=> [1]
=> [1,0]
=> 0
[3,1] => [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[4] => [[4],[]]
=> []
=> []
=> ? ∊ {0,0,1,1}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,3] => [[3,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[1,4] => [[4,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,3] => [[4,2],[1]]
=> [1]
=> [1,0]
=> 0
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[3,2] => [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[4,1] => [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 1
[5] => [[5],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[1,1,4] => [[4,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 1
[1,5] => [[5,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,4] => [[5,2],[1]]
=> [1]
=> [1,0]
=> 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 0
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 0
[3,3] => [[5,3],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 0
[4,2] => [[5,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 1
[5,1] => [[5,5],[4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
[6] => [[6],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,1,1,2,2}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,1,1,2,2}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,1,1,2,2}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[1,1,1,4] => [[4,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,1,1,2,2}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,5] => [[5,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,1,1,2,2}
[1,6] => [[6,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,1,1,2,2}
[7] => [[7],[]]
=> []
=> []
=> ? ∊ {0,0,0,1,1,2,2}
[1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,1,1,1,3] => [[3,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,1,1,4] => [[4,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,1,2,3] => [[4,2,1,1,1],[1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,1,3,1,1] => [[3,3,3,1,1,1],[2,2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,1,5] => [[5,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,1,6] => [[6,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,2,1,1,3] => [[4,2,2,2,1],[1,1,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,2,1,4] => [[5,2,2,1],[1,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,2,4,1] => [[5,5,2,1],[4,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,2,5] => [[6,2,1],[1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,3,1,1,1,1] => [[3,3,3,3,3,1],[2,2,2,2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
[1,3,4] => [[6,3,1],[2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2}
Description
Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001113
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001113: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001113: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
[1] => [[1],[]]
=> []
=> []
=> ? = 0
[1,1] => [[1,1],[]]
=> []
=> []
=> ? ∊ {0,0}
[2] => [[2],[]]
=> []
=> []
=> ? ∊ {0,0}
[1,1,1] => [[1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,1}
[1,2] => [[2,1],[]]
=> []
=> []
=> ? ∊ {0,0,1}
[2,1] => [[2,2],[1]]
=> [1]
=> [1,0]
=> 0
[3] => [[3],[]]
=> []
=> []
=> ? ∊ {0,0,1}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,1}
[1,1,2] => [[2,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,1}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,3] => [[3,1],[]]
=> []
=> []
=> ? ∊ {0,0,1,1}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,2] => [[3,2],[1]]
=> [1]
=> [1,0]
=> 0
[3,1] => [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[4] => [[4],[]]
=> []
=> []
=> ? ∊ {0,0,1,1}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,3] => [[3,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[1,4] => [[4,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,3] => [[4,2],[1]]
=> [1]
=> [1,0]
=> 0
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[3,2] => [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[4,1] => [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 1
[5] => [[5],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[1,1,4] => [[4,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 1
[1,5] => [[5,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,4] => [[5,2],[1]]
=> [1]
=> [1,0]
=> 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 0
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 0
[3,3] => [[5,3],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 0
[4,2] => [[5,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 1
[5,1] => [[5,5],[4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 1
[6] => [[6],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[1,1,1,4] => [[4,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,5] => [[5,1,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[1,6] => [[6,1],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[7] => [[7],[]]
=> []
=> []
=> ? ∊ {0,1,1,1,1,2,2}
[1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,1,1,3] => [[3,1,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,1,4] => [[4,1,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,2,3] => [[4,2,1,1,1],[1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,3,1,1] => [[3,3,3,1,1,1],[2,2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,1,5] => [[5,1,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,1,6] => [[6,1,1],[]]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,1,1,3] => [[4,2,2,2,1],[1,1,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,1,4] => [[5,2,2,1],[1,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,4,1] => [[5,5,2,1],[4,1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,2,5] => [[6,2,1],[1]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,3,1,1,1,1] => [[3,3,3,3,3,1],[2,2,2,2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
[1,3,4] => [[6,3,1],[2]]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2}
Description
Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra.
The following 107 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St000884The number of isolated descents of a permutation. St000658The number of rises of length 2 of a Dyck path. St000929The constant term of the character polynomial of an integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000313The number of degree 2 vertices of a graph. St000648The number of 2-excedences of a permutation. St001172The number of 1-rises at odd height of a Dyck path. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001587Half of the largest even part of an integer partition. St000731The number of double exceedences of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St000850The number of 1/2-balanced pairs in a poset. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000366The number of double descents of a permutation. St000534The number of 2-rises of a permutation. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000137The Grundy value of an integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001939The number of parts that are equal to their multiplicity in the integer partition. St000732The number of double deficiencies of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000260The radius of a connected graph. St000699The toughness times the least common multiple of 1,. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St000650The number of 3-rises of a permutation. St001826The maximal number of leaves on a vertex of a graph. St001071The beta invariant of the graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001877Number of indecomposable injective modules with projective dimension 2. St001323The independence gap of a graph. St001271The competition number of a graph. St000379The number of Hamiltonian cycles in a graph. St001479The number of bridges of a graph. St001061The number of indices that are both descents and recoils of a permutation. St001871The number of triconnected components of a graph. St000274The number of perfect matchings of a graph. St000310The minimal degree of a vertex of a graph. St001827The number of two-component spanning forests of a graph. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001128The exponens consonantiae of a partition. St001399The distinguishing number of a poset. St000633The size of the automorphism group of a poset. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000664The number of right ropes of a permutation. St001728The number of invisible descents of a permutation. St001964The interval resolution global dimension of a poset. St001570The minimal number of edges to add to make a graph Hamiltonian. St000456The monochromatic index of a connected graph. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001845The number of join irreducibles minus the rank of a lattice. St001651The Frankl number of a lattice. St000031The number of cycles in the cycle decomposition of a permutation. St001981The size of the largest square of zeros in the top left corner of an alternating sign matrix.
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