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Your data matches 21 different statistics following compositions of up to 3 maps.
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Matching statistic: St000667
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(load all 3 compositions to match this statistic)
St000667: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 2
[1,1]
=> 1
[3]
=> 3
[2,1]
=> 1
[1,1,1]
=> 1
[4]
=> 4
[3,1]
=> 1
[2,2]
=> 2
[2,1,1]
=> 1
[1,1,1,1]
=> 1
[5]
=> 5
[4,1]
=> 1
[3,2]
=> 1
[3,1,1]
=> 1
[2,2,1]
=> 1
[2,1,1,1]
=> 1
[1,1,1,1,1]
=> 1
[6]
=> 6
[5,1]
=> 1
[4,2]
=> 2
[4,1,1]
=> 1
[3,3]
=> 3
[3,2,1]
=> 1
[3,1,1,1]
=> 1
[2,2,2]
=> 2
[2,2,1,1]
=> 1
[2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 1
[7]
=> 7
[6,1]
=> 1
[5,2]
=> 1
[5,1,1]
=> 1
[4,3]
=> 1
[4,2,1]
=> 1
[4,1,1,1]
=> 1
[3,3,1]
=> 1
[3,2,2]
=> 1
[3,2,1,1]
=> 1
[3,1,1,1,1]
=> 1
[2,2,2,1]
=> 1
[2,2,1,1,1]
=> 1
[2,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> 1
[8]
=> 8
[7,1]
=> 1
[6,2]
=> 2
[6,1,1]
=> 1
[5,3]
=> 1
[5,2,1]
=> 1
Description
The greatest common divisor of the parts of the partition.
Matching statistic: St001063
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001063: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 77%●distinct values known / distinct values provided: 17%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001063: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 77%●distinct values known / distinct values provided: 17%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1
[2]
=> []
=> ?
=> ?
=> ? ∊ {1,2}
[1,1]
=> [1]
=> []
=> []
=> ? ∊ {1,2}
[3]
=> []
=> ?
=> ?
=> ? ∊ {1,3}
[2,1]
=> [1]
=> []
=> []
=> ? ∊ {1,3}
[1,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[4]
=> []
=> ?
=> ?
=> ? ∊ {1,2,4}
[3,1]
=> [1]
=> []
=> []
=> ? ∊ {1,2,4}
[2,2]
=> [2]
=> []
=> []
=> ? ∊ {1,2,4}
[2,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[5]
=> []
=> ?
=> ?
=> ? ∊ {1,1,5}
[4,1]
=> [1]
=> []
=> []
=> ? ∊ {1,1,5}
[3,2]
=> [2]
=> []
=> []
=> ? ∊ {1,1,5}
[3,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[2,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[6]
=> []
=> ?
=> ?
=> ? ∊ {2,2,3,6}
[5,1]
=> [1]
=> []
=> []
=> ? ∊ {2,2,3,6}
[4,2]
=> [2]
=> []
=> []
=> ? ∊ {2,2,3,6}
[4,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[3,3]
=> [3]
=> []
=> []
=> ? ∊ {2,2,3,6}
[3,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[2,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[7]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,7}
[6,1]
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,7}
[5,2]
=> [2]
=> []
=> []
=> ? ∊ {1,1,1,7}
[5,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[4,3]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,7}
[4,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[3,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[3,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[8]
=> []
=> ?
=> ?
=> ? ∊ {2,2,2,4,8}
[7,1]
=> [1]
=> []
=> []
=> ? ∊ {2,2,2,4,8}
[6,2]
=> [2]
=> []
=> []
=> ? ∊ {2,2,2,4,8}
[6,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[5,3]
=> [3]
=> []
=> []
=> ? ∊ {2,2,2,4,8}
[5,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[4,4]
=> [4]
=> []
=> []
=> ? ∊ {2,2,2,4,8}
[4,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[4,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,3,2]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[9]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,3,3,9}
[8,1]
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,3,3,9}
[7,2]
=> [2]
=> []
=> []
=> ? ∊ {1,1,1,3,3,9}
[7,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[6,3]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,3,3,9}
[6,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[5,4]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,3,3,9}
[5,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[5,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,4,1]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,3,3,9}
[10]
=> []
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,2,5,10}
[9,1]
=> [1]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,5,10}
[8,2]
=> [2]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,5,10}
[7,3]
=> [3]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,5,10}
[6,4]
=> [4]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,5,10}
[5,5]
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,5,10}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,5,10}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,5,10}
[11]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,11}
[10,1]
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,11}
[9,2]
=> [2]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,11}
[8,3]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,11}
[7,4]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,11}
[6,5]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,11}
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,11}
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,11}
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,11}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,11}
[12]
=> []
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,4,4,6,12}
[11,1]
=> [1]
=> []
=> []
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,4,4,6,12}
Description
Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra.
Matching statistic: St001064
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001064: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 77%●distinct values known / distinct values provided: 17%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001064: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 77%●distinct values known / distinct values provided: 17%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1
[2]
=> []
=> ?
=> ?
=> ? ∊ {1,2}
[1,1]
=> [1]
=> []
=> []
=> ? ∊ {1,2}
[3]
=> []
=> ?
=> ?
=> ? ∊ {1,3}
[2,1]
=> [1]
=> []
=> []
=> ? ∊ {1,3}
[1,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[4]
=> []
=> ?
=> ?
=> ? ∊ {1,2,4}
[3,1]
=> [1]
=> []
=> []
=> ? ∊ {1,2,4}
[2,2]
=> [2]
=> []
=> []
=> ? ∊ {1,2,4}
[2,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[5]
=> []
=> ?
=> ?
=> ? ∊ {1,1,5}
[4,1]
=> [1]
=> []
=> []
=> ? ∊ {1,1,5}
[3,2]
=> [2]
=> []
=> []
=> ? ∊ {1,1,5}
[3,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[2,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[6]
=> []
=> ?
=> ?
=> ? ∊ {2,2,3,6}
[5,1]
=> [1]
=> []
=> []
=> ? ∊ {2,2,3,6}
[4,2]
=> [2]
=> []
=> []
=> ? ∊ {2,2,3,6}
[4,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[3,3]
=> [3]
=> []
=> []
=> ? ∊ {2,2,3,6}
[3,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[2,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[7]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,7}
[6,1]
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,7}
[5,2]
=> [2]
=> []
=> []
=> ? ∊ {1,1,1,7}
[5,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[4,3]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,7}
[4,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[3,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[3,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[8]
=> []
=> ?
=> ?
=> ? ∊ {2,2,2,4,8}
[7,1]
=> [1]
=> []
=> []
=> ? ∊ {2,2,2,4,8}
[6,2]
=> [2]
=> []
=> []
=> ? ∊ {2,2,2,4,8}
[6,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[5,3]
=> [3]
=> []
=> []
=> ? ∊ {2,2,2,4,8}
[5,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[4,4]
=> [4]
=> []
=> []
=> ? ∊ {2,2,2,4,8}
[4,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[4,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,3,2]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[9]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,3,3,9}
[8,1]
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,3,3,9}
[7,2]
=> [2]
=> []
=> []
=> ? ∊ {1,1,1,3,3,9}
[7,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[6,3]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,3,3,9}
[6,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[5,4]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,3,3,9}
[5,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[5,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,4,1]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,3,3,9}
[10]
=> []
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,2,5,10}
[9,1]
=> [1]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,5,10}
[8,2]
=> [2]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,5,10}
[7,3]
=> [3]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,5,10}
[6,4]
=> [4]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,5,10}
[5,5]
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,5,10}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,5,10}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {2,2,2,2,2,2,5,10}
[11]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,11}
[10,1]
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,11}
[9,2]
=> [2]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,11}
[8,3]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,11}
[7,4]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,11}
[6,5]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,11}
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,11}
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,11}
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,11}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,11}
[12]
=> []
=> ?
=> ?
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,4,4,6,12}
[11,1]
=> [1]
=> []
=> []
=> ? ∊ {1,2,2,2,2,2,2,3,3,3,4,4,6,12}
Description
Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules.
Matching statistic: St001722
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 8% ●values known / values provided: 77%●distinct values known / distinct values provided: 8%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 8% ●values known / values provided: 77%●distinct values known / distinct values provided: 8%
Values
[1]
=> []
=> ?
=> ? => ? = 1
[2]
=> []
=> ?
=> ? => ? ∊ {1,2}
[1,1]
=> [1]
=> []
=> ? => ? ∊ {1,2}
[3]
=> []
=> ?
=> ? => ? ∊ {1,3}
[2,1]
=> [1]
=> []
=> ? => ? ∊ {1,3}
[1,1,1]
=> [1,1]
=> [1]
=> 1 => 1
[4]
=> []
=> ?
=> ? => ? ∊ {1,2,4}
[3,1]
=> [1]
=> []
=> ? => ? ∊ {1,2,4}
[2,2]
=> [2]
=> []
=> ? => ? ∊ {1,2,4}
[2,1,1]
=> [1,1]
=> [1]
=> 1 => 1
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 11 => 1
[5]
=> []
=> ?
=> ? => ? ∊ {1,1,5}
[4,1]
=> [1]
=> []
=> ? => ? ∊ {1,1,5}
[3,2]
=> [2]
=> []
=> ? => ? ∊ {1,1,5}
[3,1,1]
=> [1,1]
=> [1]
=> 1 => 1
[2,2,1]
=> [2,1]
=> [1]
=> 1 => 1
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 11 => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 1
[6]
=> []
=> ?
=> ? => ? ∊ {2,2,3,6}
[5,1]
=> [1]
=> []
=> ? => ? ∊ {2,2,3,6}
[4,2]
=> [2]
=> []
=> ? => ? ∊ {2,2,3,6}
[4,1,1]
=> [1,1]
=> [1]
=> 1 => 1
[3,3]
=> [3]
=> []
=> ? => ? ∊ {2,2,3,6}
[3,2,1]
=> [2,1]
=> [1]
=> 1 => 1
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 11 => 1
[2,2,2]
=> [2,2]
=> [2]
=> 0 => 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 11 => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 1
[7]
=> []
=> ?
=> ? => ? ∊ {1,1,1,7}
[6,1]
=> [1]
=> []
=> ? => ? ∊ {1,1,1,7}
[5,2]
=> [2]
=> []
=> ? => ? ∊ {1,1,1,7}
[5,1,1]
=> [1,1]
=> [1]
=> 1 => 1
[4,3]
=> [3]
=> []
=> ? => ? ∊ {1,1,1,7}
[4,2,1]
=> [2,1]
=> [1]
=> 1 => 1
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 11 => 1
[3,3,1]
=> [3,1]
=> [1]
=> 1 => 1
[3,2,2]
=> [2,2]
=> [2]
=> 0 => 1
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 11 => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 01 => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 11111 => 1
[8]
=> []
=> ?
=> ? => ? ∊ {2,2,2,4,8}
[7,1]
=> [1]
=> []
=> ? => ? ∊ {2,2,2,4,8}
[6,2]
=> [2]
=> []
=> ? => ? ∊ {2,2,2,4,8}
[6,1,1]
=> [1,1]
=> [1]
=> 1 => 1
[5,3]
=> [3]
=> []
=> ? => ? ∊ {2,2,2,4,8}
[5,2,1]
=> [2,1]
=> [1]
=> 1 => 1
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 11 => 1
[4,4]
=> [4]
=> []
=> ? => ? ∊ {2,2,2,4,8}
[4,3,1]
=> [3,1]
=> [1]
=> 1 => 1
[4,2,2]
=> [2,2]
=> [2]
=> 0 => 1
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> 11 => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 1
[3,3,2]
=> [3,2]
=> [2]
=> 0 => 1
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> 11 => 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 01 => 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 1
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 00 => 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 011 => 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 11111 => 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 111111 => 1
[9]
=> []
=> ?
=> ? => ? ∊ {1,1,1,3,3,9}
[8,1]
=> [1]
=> []
=> ? => ? ∊ {1,1,1,3,3,9}
[7,2]
=> [2]
=> []
=> ? => ? ∊ {1,1,1,3,3,9}
[7,1,1]
=> [1,1]
=> [1]
=> 1 => 1
[6,3]
=> [3]
=> []
=> ? => ? ∊ {1,1,1,3,3,9}
[6,2,1]
=> [2,1]
=> [1]
=> 1 => 1
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 11 => 1
[5,4]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,3,3,9}
[5,3,1]
=> [3,1]
=> [1]
=> 1 => 1
[5,2,2]
=> [2,2]
=> [2]
=> 0 => 1
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> 11 => 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 1
[4,4,1]
=> [4,1]
=> [1]
=> 1 => 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 1111111 => ? ∊ {1,1,1,3,3,9}
[10]
=> []
=> ?
=> ? => ? ∊ {2,2,2,2,2,2,5,10}
[9,1]
=> [1]
=> []
=> ? => ? ∊ {2,2,2,2,2,2,5,10}
[8,2]
=> [2]
=> []
=> ? => ? ∊ {2,2,2,2,2,2,5,10}
[7,3]
=> [3]
=> []
=> ? => ? ∊ {2,2,2,2,2,2,5,10}
[6,4]
=> [4]
=> []
=> ? => ? ∊ {2,2,2,2,2,2,5,10}
[5,5]
=> [5]
=> []
=> ? => ? ∊ {2,2,2,2,2,2,5,10}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 1111111 => ? ∊ {2,2,2,2,2,2,5,10}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> 11111111 => ? ∊ {2,2,2,2,2,2,5,10}
[11]
=> []
=> ?
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,11}
[10,1]
=> [1]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,11}
[9,2]
=> [2]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,11}
[8,3]
=> [3]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,11}
[7,4]
=> [4]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,11}
[6,5]
=> [5]
=> []
=> ? => ? ∊ {1,1,1,1,1,1,1,1,1,11}
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 1111111 => ? ∊ {1,1,1,1,1,1,1,1,1,11}
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> 1111111 => ? ∊ {1,1,1,1,1,1,1,1,1,11}
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> 11111111 => ? ∊ {1,1,1,1,1,1,1,1,1,11}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> 111111111 => ? ∊ {1,1,1,1,1,1,1,1,1,11}
[12]
=> []
=> ?
=> ? => ? ∊ {2,2,2,2,2,2,2,3,3,3,4,4,6,12}
[11,1]
=> [1]
=> []
=> ? => ? ∊ {2,2,2,2,2,2,2,3,3,3,4,4,6,12}
Description
The number of minimal chains with small intervals between a binary word and the top element.
A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. Let $P$ be the lattice on binary words of length $n$, where the covering elements of a word are obtained by replacing a valley with a peak. An interval $[w_1, w_2]$ in $P$ is small if $w_2$ is obtained from $w_1$ by replacing some valleys with peaks.
This statistic counts the number of chains $w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length.
For example, there are two such chains for the word $0110$:
$$ 0110 < 1011 < 1101 < 1110 < 1111 $$
and
$$ 0110 < 1010 < 1101 < 1110 < 1111. $$
Matching statistic: St001256
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001256: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 76%●distinct values known / distinct values provided: 17%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001256: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 76%●distinct values known / distinct values provided: 17%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1
[2]
=> []
=> ?
=> ?
=> ? ∊ {1,2}
[1,1]
=> [1]
=> []
=> []
=> ? ∊ {1,2}
[3]
=> []
=> ?
=> ?
=> ? ∊ {1,3}
[2,1]
=> [1]
=> []
=> []
=> ? ∊ {1,3}
[1,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 1
[4]
=> []
=> ?
=> ?
=> ? ∊ {1,2,4}
[3,1]
=> [1]
=> []
=> []
=> ? ∊ {1,2,4}
[2,2]
=> [2]
=> []
=> []
=> ? ∊ {1,2,4}
[2,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[5]
=> []
=> ?
=> ?
=> ? ∊ {1,1,5}
[4,1]
=> [1]
=> []
=> []
=> ? ∊ {1,1,5}
[3,2]
=> [2]
=> []
=> []
=> ? ∊ {1,1,5}
[3,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 1
[2,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[6]
=> []
=> ?
=> ?
=> ? ∊ {2,2,3,6}
[5,1]
=> [1]
=> []
=> []
=> ? ∊ {2,2,3,6}
[4,2]
=> [2]
=> []
=> []
=> ? ∊ {2,2,3,6}
[4,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 1
[3,3]
=> [3]
=> []
=> []
=> ? ∊ {2,2,3,6}
[3,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[7]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,7}
[6,1]
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,7}
[5,2]
=> [2]
=> []
=> []
=> ? ∊ {1,1,1,7}
[5,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 1
[4,3]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,7}
[4,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,3,1]
=> [3,1]
=> [1]
=> [1,0]
=> 1
[3,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[8]
=> []
=> ?
=> ?
=> ? ∊ {2,2,2,4,8}
[7,1]
=> [1]
=> []
=> []
=> ? ∊ {2,2,2,4,8}
[6,2]
=> [2]
=> []
=> []
=> ? ∊ {2,2,2,4,8}
[6,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 1
[5,3]
=> [3]
=> []
=> []
=> ? ∊ {2,2,2,4,8}
[5,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[4,4]
=> [4]
=> []
=> []
=> ? ∊ {2,2,2,4,8}
[4,3,1]
=> [3,1]
=> [1]
=> [1,0]
=> 1
[4,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,3,2]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[9]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,3,3,9}
[8,1]
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,3,3,9}
[7,2]
=> [2]
=> []
=> []
=> ? ∊ {1,1,1,3,3,9}
[7,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 1
[6,3]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,3,3,9}
[6,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[5,4]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,3,3,9}
[5,3,1]
=> [3,1]
=> [1]
=> [1,0]
=> 1
[5,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[4,4,1]
=> [4,1]
=> [1]
=> [1,0]
=> 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,3,3,9}
[10]
=> []
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,2,5,10}
[9,1]
=> [1]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,5,10}
[8,2]
=> [2]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,5,10}
[7,3]
=> [3]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,5,10}
[6,4]
=> [4]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,5,10}
[5,5]
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,5,10}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,5,10}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,5,10}
[11]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,11}
[10,1]
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,11}
[9,2]
=> [2]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,11}
[8,3]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,11}
[7,4]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,11}
[6,5]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,11}
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,11}
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,11}
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,11}
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,11}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,11}
[12]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,3,3,3,4,4,6,12}
Description
Number of simple reflexive modules that are 2-stable reflexive.
See Definition 3.1. in the reference for the definition of 2-stable reflexive.
Matching statistic: St001493
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001493: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 76%●distinct values known / distinct values provided: 17%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001493: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 76%●distinct values known / distinct values provided: 17%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1
[2]
=> []
=> ?
=> ?
=> ? ∊ {1,2}
[1,1]
=> [1]
=> []
=> []
=> ? ∊ {1,2}
[3]
=> []
=> ?
=> ?
=> ? ∊ {1,3}
[2,1]
=> [1]
=> []
=> []
=> ? ∊ {1,3}
[1,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 1
[4]
=> []
=> ?
=> ?
=> ? ∊ {1,2,4}
[3,1]
=> [1]
=> []
=> []
=> ? ∊ {1,2,4}
[2,2]
=> [2]
=> []
=> []
=> ? ∊ {1,2,4}
[2,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[5]
=> []
=> ?
=> ?
=> ? ∊ {1,1,5}
[4,1]
=> [1]
=> []
=> []
=> ? ∊ {1,1,5}
[3,2]
=> [2]
=> []
=> []
=> ? ∊ {1,1,5}
[3,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 1
[2,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[6]
=> []
=> ?
=> ?
=> ? ∊ {2,2,3,6}
[5,1]
=> [1]
=> []
=> []
=> ? ∊ {2,2,3,6}
[4,2]
=> [2]
=> []
=> []
=> ? ∊ {2,2,3,6}
[4,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 1
[3,3]
=> [3]
=> []
=> []
=> ? ∊ {2,2,3,6}
[3,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[7]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,7}
[6,1]
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,7}
[5,2]
=> [2]
=> []
=> []
=> ? ∊ {1,1,1,7}
[5,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 1
[4,3]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,7}
[4,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,3,1]
=> [3,1]
=> [1]
=> [1,0]
=> 1
[3,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[8]
=> []
=> ?
=> ?
=> ? ∊ {2,2,2,4,8}
[7,1]
=> [1]
=> []
=> []
=> ? ∊ {2,2,2,4,8}
[6,2]
=> [2]
=> []
=> []
=> ? ∊ {2,2,2,4,8}
[6,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 1
[5,3]
=> [3]
=> []
=> []
=> ? ∊ {2,2,2,4,8}
[5,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[4,4]
=> [4]
=> []
=> []
=> ? ∊ {2,2,2,4,8}
[4,3,1]
=> [3,1]
=> [1]
=> [1,0]
=> 1
[4,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,3,2]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[9]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,3,3,9}
[8,1]
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,3,3,9}
[7,2]
=> [2]
=> []
=> []
=> ? ∊ {1,1,1,3,3,9}
[7,1,1]
=> [1,1]
=> [1]
=> [1,0]
=> 1
[6,3]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,3,3,9}
[6,2,1]
=> [2,1]
=> [1]
=> [1,0]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[5,4]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,3,3,9}
[5,3,1]
=> [3,1]
=> [1]
=> [1,0]
=> 1
[5,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 1
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[4,4,1]
=> [4,1]
=> [1]
=> [1,0]
=> 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,3,3,9}
[10]
=> []
=> ?
=> ?
=> ? ∊ {2,2,2,2,2,2,5,10}
[9,1]
=> [1]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,5,10}
[8,2]
=> [2]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,5,10}
[7,3]
=> [3]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,5,10}
[6,4]
=> [4]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,5,10}
[5,5]
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,2,2,5,10}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,5,10}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {2,2,2,2,2,2,5,10}
[11]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,11}
[10,1]
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,11}
[9,2]
=> [2]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,11}
[8,3]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,11}
[7,4]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,11}
[6,5]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,11}
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,11}
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,11}
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,11}
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,11}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,11}
[12]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2,2,3,3,3,4,4,6,12}
Description
The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra.
Matching statistic: St001487
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 8% ●values known / values provided: 65%●distinct values known / distinct values provided: 8%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 8% ●values known / values provided: 65%●distinct values known / distinct values provided: 8%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1
[2]
=> []
=> ?
=> ?
=> ? ∊ {1,2}
[1,1]
=> [1]
=> []
=> [[],[]]
=> ? ∊ {1,2}
[3]
=> []
=> ?
=> ?
=> ? ∊ {1,3}
[2,1]
=> [1]
=> []
=> [[],[]]
=> ? ∊ {1,3}
[1,1,1]
=> [1,1]
=> [1]
=> [[1],[]]
=> 1
[4]
=> []
=> ?
=> ?
=> ? ∊ {1,2,4}
[3,1]
=> [1]
=> []
=> [[],[]]
=> ? ∊ {1,2,4}
[2,2]
=> [2]
=> []
=> [[],[]]
=> ? ∊ {1,2,4}
[2,1,1]
=> [1,1]
=> [1]
=> [[1],[]]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[5]
=> []
=> ?
=> ?
=> ? ∊ {1,1,5}
[4,1]
=> [1]
=> []
=> [[],[]]
=> ? ∊ {1,1,5}
[3,2]
=> [2]
=> []
=> [[],[]]
=> ? ∊ {1,1,5}
[3,1,1]
=> [1,1]
=> [1]
=> [[1],[]]
=> 1
[2,2,1]
=> [2,1]
=> [1]
=> [[1],[]]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[6]
=> []
=> ?
=> ?
=> ? ∊ {2,2,3,6}
[5,1]
=> [1]
=> []
=> [[],[]]
=> ? ∊ {2,2,3,6}
[4,2]
=> [2]
=> []
=> [[],[]]
=> ? ∊ {2,2,3,6}
[4,1,1]
=> [1,1]
=> [1]
=> [[1],[]]
=> 1
[3,3]
=> [3]
=> []
=> [[],[]]
=> ? ∊ {2,2,3,6}
[3,2,1]
=> [2,1]
=> [1]
=> [[1],[]]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[2,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[7]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,7}
[6,1]
=> [1]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,7}
[5,2]
=> [2]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,7}
[5,1,1]
=> [1,1]
=> [1]
=> [[1],[]]
=> 1
[4,3]
=> [3]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,7}
[4,2,1]
=> [2,1]
=> [1]
=> [[1],[]]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[3,3,1]
=> [3,1]
=> [1]
=> [[1],[]]
=> 1
[3,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[2,1],[]]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1
[8]
=> []
=> ?
=> ?
=> ? ∊ {1,2,2,2,4,8}
[7,1]
=> [1]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,4,8}
[6,2]
=> [2]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,4,8}
[6,1,1]
=> [1,1]
=> [1]
=> [[1],[]]
=> 1
[5,3]
=> [3]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,4,8}
[5,2,1]
=> [2,1]
=> [1]
=> [[1],[]]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[4,4]
=> [4]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,4,8}
[4,3,1]
=> [3,1]
=> [1]
=> [[1],[]]
=> 1
[4,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 1
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[3,3,2]
=> [3,2]
=> [2]
=> [[2],[]]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[2,1],[]]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [[2,2],[]]
=> 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? ∊ {1,2,2,2,4,8}
[9]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,3,3,9}
[8,1]
=> [1]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,1,3,3,9}
[7,2]
=> [2]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,1,3,3,9}
[7,1,1]
=> [1,1]
=> [1]
=> [[1],[]]
=> 1
[6,3]
=> [3]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,1,3,3,9}
[6,2,1]
=> [2,1]
=> [1]
=> [[1],[]]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[5,4]
=> [4]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,1,3,3,9}
[5,3,1]
=> [3,1]
=> [1]
=> [[1],[]]
=> 1
[5,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 1
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[4,4,1]
=> [4,1]
=> [1]
=> [[1],[]]
=> 1
[4,3,2]
=> [3,2]
=> [2]
=> [[2],[]]
=> 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? ∊ {1,1,1,1,3,3,9}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? ∊ {1,1,1,1,3,3,9}
[10]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,5,10}
[9,1]
=> [1]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,5,10}
[8,2]
=> [2]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,5,10}
[7,3]
=> [3]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,5,10}
[6,4]
=> [4]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,5,10}
[5,5]
=> [5]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,5,10}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,5,10}
[2,2,2,2,2]
=> [2,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,5,10}
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,5,10}
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,5,10}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,5,10}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,5,10}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1,1],[]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,5,10}
[11]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,11}
[10,1]
=> [1]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,11}
[9,2]
=> [2]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,11}
[8,3]
=> [3]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,11}
[7,4]
=> [4]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,11}
Description
The number of inner corners of a skew partition.
Matching statistic: St001490
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 8% ●values known / values provided: 65%●distinct values known / distinct values provided: 8%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 8% ●values known / values provided: 65%●distinct values known / distinct values provided: 8%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1
[2]
=> []
=> ?
=> ?
=> ? ∊ {1,2}
[1,1]
=> [1]
=> []
=> [[],[]]
=> ? ∊ {1,2}
[3]
=> []
=> ?
=> ?
=> ? ∊ {1,3}
[2,1]
=> [1]
=> []
=> [[],[]]
=> ? ∊ {1,3}
[1,1,1]
=> [1,1]
=> [1]
=> [[1],[]]
=> 1
[4]
=> []
=> ?
=> ?
=> ? ∊ {1,2,4}
[3,1]
=> [1]
=> []
=> [[],[]]
=> ? ∊ {1,2,4}
[2,2]
=> [2]
=> []
=> [[],[]]
=> ? ∊ {1,2,4}
[2,1,1]
=> [1,1]
=> [1]
=> [[1],[]]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[5]
=> []
=> ?
=> ?
=> ? ∊ {1,1,5}
[4,1]
=> [1]
=> []
=> [[],[]]
=> ? ∊ {1,1,5}
[3,2]
=> [2]
=> []
=> [[],[]]
=> ? ∊ {1,1,5}
[3,1,1]
=> [1,1]
=> [1]
=> [[1],[]]
=> 1
[2,2,1]
=> [2,1]
=> [1]
=> [[1],[]]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[6]
=> []
=> ?
=> ?
=> ? ∊ {2,2,3,6}
[5,1]
=> [1]
=> []
=> [[],[]]
=> ? ∊ {2,2,3,6}
[4,2]
=> [2]
=> []
=> [[],[]]
=> ? ∊ {2,2,3,6}
[4,1,1]
=> [1,1]
=> [1]
=> [[1],[]]
=> 1
[3,3]
=> [3]
=> []
=> [[],[]]
=> ? ∊ {2,2,3,6}
[3,2,1]
=> [2,1]
=> [1]
=> [[1],[]]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[2,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[7]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,7}
[6,1]
=> [1]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,7}
[5,2]
=> [2]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,7}
[5,1,1]
=> [1,1]
=> [1]
=> [[1],[]]
=> 1
[4,3]
=> [3]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,7}
[4,2,1]
=> [2,1]
=> [1]
=> [[1],[]]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[3,3,1]
=> [3,1]
=> [1]
=> [[1],[]]
=> 1
[3,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[2,1],[]]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1
[8]
=> []
=> ?
=> ?
=> ? ∊ {1,2,2,2,4,8}
[7,1]
=> [1]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,4,8}
[6,2]
=> [2]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,4,8}
[6,1,1]
=> [1,1]
=> [1]
=> [[1],[]]
=> 1
[5,3]
=> [3]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,4,8}
[5,2,1]
=> [2,1]
=> [1]
=> [[1],[]]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[4,4]
=> [4]
=> []
=> [[],[]]
=> ? ∊ {1,2,2,2,4,8}
[4,3,1]
=> [3,1]
=> [1]
=> [[1],[]]
=> 1
[4,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 1
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[3,3,2]
=> [3,2]
=> [2]
=> [[2],[]]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[2,1],[]]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [[2,2],[]]
=> 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? ∊ {1,2,2,2,4,8}
[9]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,3,3,9}
[8,1]
=> [1]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,1,3,3,9}
[7,2]
=> [2]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,1,3,3,9}
[7,1,1]
=> [1,1]
=> [1]
=> [[1],[]]
=> 1
[6,3]
=> [3]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,1,3,3,9}
[6,2,1]
=> [2,1]
=> [1]
=> [[1],[]]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[5,4]
=> [4]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,1,3,3,9}
[5,3,1]
=> [3,1]
=> [1]
=> [[1],[]]
=> 1
[5,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 1
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1
[4,4,1]
=> [4,1]
=> [1]
=> [[1],[]]
=> 1
[4,3,2]
=> [3,2]
=> [2]
=> [[2],[]]
=> 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? ∊ {1,1,1,1,3,3,9}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? ∊ {1,1,1,1,3,3,9}
[10]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,5,10}
[9,1]
=> [1]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,5,10}
[8,2]
=> [2]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,5,10}
[7,3]
=> [3]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,5,10}
[6,4]
=> [4]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,5,10}
[5,5]
=> [5]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,5,10}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,5,10}
[2,2,2,2,2]
=> [2,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,5,10}
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,5,10}
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,5,10}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,5,10}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,5,10}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1,1],[]]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,2,5,10}
[11]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,11}
[10,1]
=> [1]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,11}
[9,2]
=> [2]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,11}
[8,3]
=> [3]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,11}
[7,4]
=> [4]
=> []
=> [[],[]]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,11}
Description
The number of connected components of a skew partition.
Matching statistic: St000781
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 63%●distinct values known / distinct values provided: 8%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 63%●distinct values known / distinct values provided: 8%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1
[2]
=> []
=> ?
=> ?
=> ? ∊ {1,2}
[1,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,2}
[3]
=> []
=> ?
=> ?
=> ? ∊ {1,1,3}
[2,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,1,3}
[1,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,3}
[4]
=> []
=> ?
=> ?
=> ? ∊ {1,1,2,4}
[3,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,1,2,4}
[2,2]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,4}
[2,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,4}
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[5]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,5}
[4,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,1,1,1,5}
[3,2]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,1,1,5}
[3,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,5}
[2,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,5}
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[6]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,2,2,3,6}
[5,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,3,6}
[4,2]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,3,6}
[4,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,1,2,2,3,6}
[3,3]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,3,6}
[3,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,2,2,3,6}
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[2,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,2,2,3,6}
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[7]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,7}
[6,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,7}
[5,2]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,7}
[5,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,7}
[4,3]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,7}
[4,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,7}
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[3,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,7}
[3,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,7}
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[8]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,4,8}
[7,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,4,8}
[6,2]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,4,8}
[6,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,4,8}
[5,3]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,4,8}
[5,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,4,8}
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[4,4]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,4,8}
[4,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,4,8}
[4,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,4,8}
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,3,2]
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,4,8}
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[9]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,9}
[8,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,9}
[7,2]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,9}
[7,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,9}
[6,3]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,9}
[6,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,9}
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[5,4]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,9}
[5,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,9}
[5,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,9}
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,4,1]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,9}
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[3,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1
[7,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[6,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[5,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[5,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
Description
The number of proper colouring schemes of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
Matching statistic: St001901
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 63%●distinct values known / distinct values provided: 8%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 8% ●values known / values provided: 63%●distinct values known / distinct values provided: 8%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1
[2]
=> []
=> ?
=> ?
=> ? ∊ {1,2}
[1,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,2}
[3]
=> []
=> ?
=> ?
=> ? ∊ {1,1,3}
[2,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,1,3}
[1,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,3}
[4]
=> []
=> ?
=> ?
=> ? ∊ {1,1,2,4}
[3,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,1,2,4}
[2,2]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,2,4}
[2,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,2,4}
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[5]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,5}
[4,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,1,1,1,5}
[3,2]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,1,1,5}
[3,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,5}
[2,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,5}
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[6]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,2,2,3,6}
[5,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,3,6}
[4,2]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,3,6}
[4,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,1,2,2,3,6}
[3,3]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,3,6}
[3,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,2,2,3,6}
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[2,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,2,2,3,6}
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[7]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,7}
[6,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,7}
[5,2]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,7}
[5,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,7}
[4,3]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,7}
[4,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,7}
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[3,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,7}
[3,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,7}
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[8]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,4,8}
[7,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,4,8}
[6,2]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,4,8}
[6,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,4,8}
[5,3]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,4,8}
[5,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,4,8}
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[4,4]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,2,2,2,4,8}
[4,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,4,8}
[4,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,4,8}
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,3,2]
=> [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,4,8}
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[9]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,9}
[8,1]
=> [1]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,9}
[7,2]
=> [2]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,9}
[7,1,1]
=> [1,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,9}
[6,3]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,9}
[6,2,1]
=> [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,9}
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[5,4]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,9}
[5,3,1]
=> [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,9}
[5,2,2]
=> [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,9}
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,4,1]
=> [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,9}
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> [1]
=> 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,2,2,2]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[3,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1
[7,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1]
=> 1
[6,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1]
=> 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[5,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1]
=> 1
[5,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1]
=> 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
Description
The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition.
The following 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000260The radius of a connected graph. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St000782The indicator function of whether a given perfect matching is an L & P matching. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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