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Your data matches 18 different statistics following compositions of up to 3 maps.
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Matching statistic: St001939
St001939: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 0
[1,1]
=> 0
[3]
=> 0
[2,1]
=> 1
[1,1,1]
=> 0
[4]
=> 0
[3,1]
=> 1
[2,2]
=> 2
[2,1,1]
=> 0
[1,1,1,1]
=> 0
[5]
=> 0
[4,1]
=> 1
[3,2]
=> 0
[3,1,1]
=> 0
[2,2,1]
=> 3
[2,1,1,1]
=> 0
[1,1,1,1,1]
=> 0
[6]
=> 0
[5,1]
=> 1
[4,2]
=> 0
[4,1,1]
=> 0
[3,3]
=> 0
[3,2,1]
=> 1
[3,1,1,1]
=> 0
[2,2,2]
=> 0
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 0
[1,1,1,1,1,1]
=> 0
[7]
=> 0
[6,1]
=> 1
[5,2]
=> 0
[5,1,1]
=> 0
[4,3]
=> 0
[4,2,1]
=> 1
[4,1,1,1]
=> 0
[3,3,1]
=> 1
[3,2,2]
=> 2
[3,2,1,1]
=> 0
[3,1,1,1,1]
=> 0
[2,2,2,1]
=> 1
[2,2,1,1,1]
=> 2
[2,1,1,1,1,1]
=> 0
[1,1,1,1,1,1,1]
=> 0
[8]
=> 0
[7,1]
=> 1
[6,2]
=> 0
[6,1,1]
=> 0
[5,3]
=> 0
[5,2,1]
=> 1
Description
The number of parts that are equal to their multiplicity in the integer partition.
Matching statistic: St001231
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001231: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 70%●distinct values known / distinct values provided: 57%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001231: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 70%●distinct values known / distinct values provided: 57%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1
[2]
=> []
=> ?
=> ?
=> ? ∊ {0,0}
[1,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0}
[3]
=> []
=> ?
=> ?
=> ? ∊ {0,1}
[2,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1}
[1,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[4]
=> []
=> ?
=> ?
=> ? ∊ {0,1,2}
[3,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,2}
[2,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,2}
[2,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[5]
=> []
=> ?
=> ?
=> ? ∊ {0,1,3}
[4,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,3}
[3,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,3}
[3,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[6]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,2}
[5,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,1,2}
[4,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,1,2}
[4,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,1,2}
[3,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[2,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[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]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,2}
[6,1]
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,2}
[5,2]
=> [2]
=> []
=> []
=> ? ∊ {1,1,1,2}
[5,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[4,3]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,2}
[4,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[3,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[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]
=> 2
[8]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,1,2,3}
[7,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,1,2,3}
[6,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,0,1,2,3}
[6,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[5,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,1,2,3}
[5,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[4,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,1,2,3}
[4,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[4,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[3,3,2]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[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]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[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]
=> 2
[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]
=> ? ∊ {0,0,0,1,2,3}
[9]
=> []
=> ?
=> ?
=> ? ∊ {0,1,1,1,2,3,3}
[8,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,1,1,2,3,3}
[7,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,1,1,2,3,3}
[7,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[6,3]
=> [3]
=> []
=> []
=> ? ∊ {0,1,1,1,2,3,3}
[6,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[5,4]
=> [4]
=> []
=> []
=> ? ∊ {0,1,1,1,2,3,3}
[5,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[5,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[4,4,1]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[4,3,2]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[2,1,1,1,1,1,1,1]
=> [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]
=> ? ∊ {0,1,1,1,2,3,3}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,3,3}
[10]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,1,2,2,2,3,4}
[9,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3,4}
[8,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3,4}
[7,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3,4}
[6,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3,4}
[5,5]
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3,4}
[3,1,1,1,1,1,1,1]
=> [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]
=> ? ∊ {0,0,0,0,1,2,2,2,3,4}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1,2,2,2,3,4}
[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]
=> ? ∊ {0,0,0,0,1,2,2,2,3,4}
[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]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1,2,2,2,3,4}
[11]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3,3,3}
[10,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3,3,3}
[9,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3,3,3}
[8,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3,3,3}
[7,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3,3,3}
[6,5]
=> [5]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3,3,3}
[4,1,1,1,1,1,1,1]
=> [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]
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3,3,3}
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3,3,3}
Description
The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension.
Actually the same statistics results for algebras with at most 7 simple modules when dropping the assumption that the module has projective dimension one. The author is not sure whether this holds in general.
Matching statistic: St001234
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001234: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 70%●distinct values known / distinct values provided: 57%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001234: Dyck paths ⟶ ℤResult quality: 57% ●values known / values provided: 70%●distinct values known / distinct values provided: 57%
Values
[1]
=> []
=> ?
=> ?
=> ? = 1
[2]
=> []
=> ?
=> ?
=> ? ∊ {0,0}
[1,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0}
[3]
=> []
=> ?
=> ?
=> ? ∊ {0,1}
[2,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1}
[1,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[4]
=> []
=> ?
=> ?
=> ? ∊ {0,1,2}
[3,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,2}
[2,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,2}
[2,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[5]
=> []
=> ?
=> ?
=> ? ∊ {0,1,3}
[4,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,3}
[3,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,3}
[3,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[6]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,2}
[5,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,1,2}
[4,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,1,2}
[4,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,1,2}
[3,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[2,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[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]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,2}
[6,1]
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,2}
[5,2]
=> [2]
=> []
=> []
=> ? ∊ {1,1,1,2}
[5,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[4,3]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,2}
[4,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[3,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[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]
=> 2
[8]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,1,2,3}
[7,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,1,2,3}
[6,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,0,1,2,3}
[6,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[5,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,1,2,3}
[5,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[4,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,1,2,3}
[4,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[4,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[3,3,2]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[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]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[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]
=> 2
[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]
=> ? ∊ {0,0,0,1,2,3}
[9]
=> []
=> ?
=> ?
=> ? ∊ {0,1,1,1,2,3,3}
[8,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,1,1,2,3,3}
[7,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,1,1,2,3,3}
[7,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[6,3]
=> [3]
=> []
=> []
=> ? ∊ {0,1,1,1,2,3,3}
[6,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[5,4]
=> [4]
=> []
=> []
=> ? ∊ {0,1,1,1,2,3,3}
[5,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[5,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[4,4,1]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[4,3,2]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[2,1,1,1,1,1,1,1]
=> [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]
=> ? ∊ {0,1,1,1,2,3,3}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,1,2,3,3}
[10]
=> []
=> ?
=> ?
=> ? ∊ {0,0,0,0,1,2,2,2,3,4}
[9,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3,4}
[8,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3,4}
[7,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3,4}
[6,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3,4}
[5,5]
=> [5]
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,2,3,4}
[3,1,1,1,1,1,1,1]
=> [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]
=> ? ∊ {0,0,0,0,1,2,2,2,3,4}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1,2,2,2,3,4}
[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]
=> ? ∊ {0,0,0,0,1,2,2,2,3,4}
[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]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1,2,2,2,3,4}
[11]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3,3,3}
[10,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3,3,3}
[9,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3,3,3}
[8,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3,3,3}
[7,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3,3,3}
[6,5]
=> [5]
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3,3,3}
[4,1,1,1,1,1,1,1]
=> [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]
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3,3,3}
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3,3,3}
Description
The number of indecomposable three dimensional modules with projective dimension one.
It return zero when there are no such modules.
Matching statistic: St000478
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000478: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 57%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000478: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 57%
Values
[1]
=> []
=> []
=> []
=> ? = 1
[2]
=> []
=> []
=> []
=> ? ∊ {0,0}
[1,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,0}
[3]
=> []
=> []
=> []
=> ? ∊ {0,0,1}
[2,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,0,1}
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0,1}
[4]
=> []
=> []
=> []
=> ? ∊ {0,0,0,1,2}
[3,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,0,0,1,2}
[2,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,0,0,1,2}
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0,0,1,2}
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,0,0,1,2}
[5]
=> []
=> []
=> []
=> ? ∊ {0,0,0,1,3}
[4,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,0,0,1,3}
[3,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,0,0,1,3}
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0,0,1,3}
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,0,0,1,3}
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[6]
=> []
=> []
=> []
=> ? ∊ {0,0,0,1,1,2}
[5,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,0,0,1,1,2}
[4,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,0,0,1,1,2}
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0,0,1,1,2}
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 0
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,0,0,1,1,2}
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,1,1,2}
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 0
[7]
=> []
=> []
=> []
=> ? ∊ {0,1,1,1,2,2}
[6,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,1,1,1,2,2}
[5,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,1,1,1,2,2}
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,1,1,1,2,2}
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 0
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,1,1,1,2,2}
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 0
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,1,1,2,2}
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 0
[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]
=> [4,3,2,1]
=> 0
[8]
=> []
=> []
=> []
=> ? ∊ {0,0,0,1,1,2,2,3}
[7,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,0,0,1,1,2,2,3}
[6,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,0,0,1,1,2,2,3}
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0,0,1,1,2,2,3}
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 0
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,0,0,1,1,2,2,3}
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 0
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 0
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,1,1,2,2,3}
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,1,1,2,2,3}
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> 0
[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]
=> [5,4,3,2,1]
=> ? ∊ {0,0,0,1,1,2,2,3}
[9]
=> []
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,3,3}
[8,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,3,3}
[7,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,3,3}
[7,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,3,3}
[6,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 0
[6,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,3,3}
[5,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 0
[5,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 0
[5,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,3,3}
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[4,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 0
[4,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 0
[4,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 0
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 0
[3,3,3]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,3,3}
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 0
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> 0
[3,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,3,3}
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 0
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> 0
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> 0
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 2
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> 0
[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]
=> [5,4,3,2,1,1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,3,3}
[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]
=> [5,4,3,2,1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,3,3}
[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]
=> [6,5,4,3,2,1]
=> ? ∊ {0,0,1,1,1,1,1,2,2,3,3}
[10]
=> []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,4}
[9,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,4}
[8,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,3,4}
[7,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 0
Description
Another weight of a partition according to Alladi.
According to Theorem 3.4 (Alladi 2012) in [1]
∑π∈GG1(r)w1(π)
equals the number of partitions of r whose odd parts are all distinct. GG1(r) is the set of partitions of r where consecutive entries differ by at least 2, and consecutive even entries differ by at least 4.
Matching statistic: St000929
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 43%●distinct values known / distinct values provided: 29%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 43%●distinct values known / distinct values provided: 29%
Values
[1]
=> []
=> []
=> []
=> ? = 1
[2]
=> []
=> []
=> []
=> ? ∊ {0,0}
[1,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,0}
[3]
=> []
=> []
=> []
=> ? ∊ {0,0,1}
[2,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,0,1}
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0,1}
[4]
=> []
=> []
=> []
=> ? ∊ {0,0,0,1,2}
[3,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,0,0,1,2}
[2,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,0,0,1,2}
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0,0,1,2}
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,0,0,1,2}
[5]
=> []
=> []
=> []
=> ? ∊ {0,0,0,0,3}
[4,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,0,0,0,3}
[3,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,0,0,0,3}
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0,0,0,3}
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,0,0,0,3}
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[6]
=> []
=> []
=> []
=> ? ∊ {0,0,0,0,1,2}
[5,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,0,0,0,1,2}
[4,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,0,0,0,1,2}
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0,0,0,1,2}
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 0
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,0,0,0,1,2}
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,2}
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 0
[7]
=> []
=> []
=> []
=> ? ∊ {0,1,1,1,2,2}
[6,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,1,1,1,2,2}
[5,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,1,1,1,2,2}
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,1,1,1,2,2}
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 0
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,1,1,1,2,2}
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 0
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,1,1,2,2}
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 0
[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]
=> [4,3,2,1]
=> 0
[8]
=> []
=> []
=> []
=> ? ∊ {0,0,0,0,1,2,2,3}
[7,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,0,0,0,1,2,2,3}
[6,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,0,0,0,1,2,2,3}
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0,0,0,1,2,2,3}
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 0
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,0,0,0,1,2,2,3}
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 0
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 0
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,2,2,3}
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,1,2,2,3}
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> 0
[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]
=> [5,4,3,2,1]
=> ? ∊ {0,0,0,0,1,2,2,3}
[9]
=> []
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3}
[8,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3}
[7,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3}
[7,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3}
[6,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 0
[6,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3}
[5,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 0
[5,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 0
[5,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3}
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[4,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 0
[4,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 0
[4,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 0
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 0
[3,3,3]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3}
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 0
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> 0
[3,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3}
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 0
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> 0
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> 0
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 0
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> 0
[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]
=> [5,4,3,2,1,1]
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3}
[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]
=> [5,4,3,2,1]
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3}
[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]
=> [6,5,4,3,2,1]
=> ? ∊ {0,0,1,1,1,1,2,2,2,3,3}
[10]
=> []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,4}
[9,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,4}
[8,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,4}
[7,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 0
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is 0 for partitions λ≠1n and 1 for λ=1n.
Matching statistic: St001498
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 14% ●values known / values provided: 22%●distinct values known / distinct values provided: 14%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 14% ●values known / values provided: 22%●distinct values known / distinct values provided: 14%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [1,1,0,0]
=> ? = 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> ? = 0
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,1}
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,1}
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,2}
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,3}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,3}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1,3}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1,3}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 0
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [6,2,1,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,2}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1,1,2}
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 0
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => [1,1,1,0,0,1,0,1,0,1,0,0]
=> 0
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [7,2,1,3,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [6,3,1,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [6,2,3,1,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 0
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 0
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> 0
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 0
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => [1,1,1,0,1,0,0,1,0,1,0,0]
=> 0
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,3}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [8,2,1,3,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,3}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [7,3,1,2,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,3}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [7,2,3,1,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,3}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,3}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [6,3,2,1,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,3}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,3}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,3}
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,3}
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,3}
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [5,2,3,4,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> 0
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 0
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [4,3,2,5,6,1] => [1,1,1,1,0,0,0,1,0,1,0,0]
=> 0
[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] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 0
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [1,1,1,0,1,0,1,0,1,0,0,0]
=> 0
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,4,5,2,6,1] => [1,1,1,0,1,0,1,0,0,1,0,0]
=> 0
[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] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> 0
[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] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,3}
[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] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,3}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3,3}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [9,2,1,3,4,5,6,7,8] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3,3}
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [8,3,1,2,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3,3}
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [8,2,3,1,4,5,6,7] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3,3}
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [7,4,1,2,3,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3,3}
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [7,3,2,1,4,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3,3}
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,1,5,6] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3,3}
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3,3}
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [6,4,2,1,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3,3}
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3,3}
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [6,3,2,4,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3,3}
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3,3}
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [5,6,2,1,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3,3}
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3,3}
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,3,3}
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [5,3,2,4,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> 0
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,6,7,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> 0
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => [1,1,1,1,0,1,0,1,0,0,0,0]
=> 0
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 0
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [4,5,2,3,6,1] => [1,1,1,1,0,1,0,0,0,1,0,0]
=> 0
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,1,2] => [1,1,1,1,0,0,1,0,1,0,0,0]
=> 0
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,5,2,6,1] => [1,1,1,1,0,0,1,0,0,1,0,0]
=> 0
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,5,6,7,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 0
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [1,1,1,0,1,0,1,0,1,0,0,0]
=> 0
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,4,5,2,6,7,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> 0
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 0
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [6,2,3,4,5,7,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 0
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [5,6,3,1,2,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,1,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 0
[4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [5,4,6,1,2,3] => [1,1,1,1,1,0,0,1,0,0,0,0]
=> 0
[4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,3,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> 0
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,1,2] => [1,1,1,1,1,0,0,0,1,0,0,0]
=> 0
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0]
=> 0
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St001556
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001556: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 43%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001556: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 43%
Values
[1]
=> []
=> []
=> [] => ? = 1
[2]
=> []
=> []
=> [] => ? ∊ {0,0}
[1,1]
=> [1]
=> [[1]]
=> [1] => ? ∊ {0,0}
[3]
=> []
=> []
=> [] => ? ∊ {0,1}
[2,1]
=> [1]
=> [[1]]
=> [1] => ? ∊ {0,1}
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[4]
=> []
=> []
=> [] => ? ∊ {1,2}
[3,1]
=> [1]
=> [[1]]
=> [1] => ? ∊ {1,2}
[2,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[5]
=> []
=> []
=> [] => ? ∊ {0,3}
[4,1]
=> [1]
=> [[1]]
=> [1] => ? ∊ {0,3}
[3,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[6]
=> []
=> []
=> [] => ? ∊ {0,1}
[5,1]
=> [1]
=> [[1]]
=> [1] => ? ∊ {0,1}
[4,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[3,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[2,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 2
[7]
=> []
=> []
=> [] => ? ∊ {0,1,1}
[6,1]
=> [1]
=> [[1]]
=> [1] => ? ∊ {0,1,1}
[5,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[4,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[4,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[3,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[3,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[2,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 2
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ? ∊ {0,1,1}
[8]
=> []
=> []
=> [] => ? ∊ {0,0,0,0,0,1,3}
[7,1]
=> [1]
=> [[1]]
=> [1] => ? ∊ {0,0,0,0,0,1,3}
[6,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[5,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[5,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[4,4]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[4,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[4,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[3,3,2]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 0
[3,3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 0
[3,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2
[3,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 2
[2,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ? ∊ {0,0,0,0,0,1,3}
[2,2,2,1,1]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? ∊ {0,0,0,0,0,1,3}
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => ? ∊ {0,0,0,0,0,1,3}
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ? ∊ {0,0,0,0,0,1,3}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? ∊ {0,0,0,0,0,1,3}
[9]
=> []
=> []
=> [] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,3,3}
[8,1]
=> [1]
=> [[1]]
=> [1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,3,3}
[7,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[7,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[6,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[6,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 0
[6,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[3,3,3]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,3,3}
[3,3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,3,3}
[3,3,1,1,1]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,3,3}
[3,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,3,3}
[3,2,2,1,1]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,3,3}
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,3,3}
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,3,3}
[2,2,2,2,1]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,3,3}
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,3,3}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,3,3}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,3,3}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,2,3,3}
[10]
=> []
=> []
=> [] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[9,1]
=> [1]
=> [[1]]
=> [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[4,4,2]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[4,4,1,1]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[4,3,3]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[4,3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[4,3,1,1,1]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[4,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[4,2,2,1,1]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[3,3,3,1]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[3,3,2,2]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[3,3,2,1,1]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,4}
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,3,4}
Description
The number of inversions of the third entry of a permutation.
This is, for a permutation π of length n,
#{3<k≤n∣π(3)>π(k)}.
The number of inversions of the first entry is [[St000054]] and the number of inversions of the second entry is [[St001557]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.
Matching statistic: St001629
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 57%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 57%
Values
[1]
=> [1,0,1,0]
=> [1,1] => [2] => ? = 1
[2]
=> [1,1,0,0,1,0]
=> [2,1] => [1,1] => ? ∊ {0,0}
[1,1]
=> [1,0,1,1,0,0]
=> [1,2] => [1,1] => ? ∊ {0,0}
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [1,1] => ? ∊ {0,0}
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => [3] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [1,1] => ? ∊ {0,0}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1] => ? ∊ {0,0,0,1,2}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => [1,1] => ? ∊ {0,0,0,1,2}
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [2] => ? ∊ {0,0,0,1,2}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => [1,1] => ? ∊ {0,0,0,1,2}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,1] => ? ∊ {0,0,0,1,2}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1] => [1,1] => ? ∊ {0,0,0,3}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => [1,1] => ? ∊ {0,0,0,3}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,2] => 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,1,1] => 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [2,1] => 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => [1,1] => ? ∊ {0,0,0,3}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => [1,1] => ? ∊ {0,0,0,3}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,1,2}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,1,2}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,1,2}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,1,2}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,1,2}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,1,2}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,1,2}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,1,2}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,1,2}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,6] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,1,2}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => 0
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,1,1] => 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => [1,1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => [1,1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => [1,1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5] => [1,1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [2,1] => 0
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,5] => [1,1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6] => [1,1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,7] => [1,1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,3}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,3}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,3}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,3}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,3}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,3}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,3}
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,2] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,3}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => [1,2] => 0
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [2,1] => 0
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => [1,1,1] => 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,3}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,1] => 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,2] => 0
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => [2,1] => 0
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,5] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,3}
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,3}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,3}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,5] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,3}
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,6] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,3}
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,1,1] => [1,2] => 0
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => [1,1,1] => 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => [1,3] => 0
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => [1,1,2] => 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => [2,1,1] => 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => [3,1] => 0
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => [2,1] => 0
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [4,1,1] => [1,2] => 0
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,1] => [1,1,1] => 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => [5] => 1
[3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,4] => [2,1] => 0
[6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,1,1] => [1,2] => 0
[6,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,5,1] => [1,1,1] => 1
[5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [4,1,1] => [1,2] => 0
[5,4,1,1]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,1,1] => [1,2] => 0
[5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,2,1] => [1,1,1] => 1
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,1] => [1,1,1] => 1
[5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1] => [1,1,1] => 1
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,4,1] => [1,1,1] => 1
[4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,2] => [1,1,1] => 1
[4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,3,2] => [1,1,1] => 1
[4,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,4] => [2,1] => 0
[3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3] => [1,1,1] => 1
[3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,3] => [1,1,1] => 1
[3,3,2,2,1]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,1,4] => [2,1] => 0
[2,2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,5] => [2,1] => 0
[6,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [5,1,1] => [1,2] => 0
[6,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,5,1] => [1,1,1] => 1
[5,4,3]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,1,1,1] => [1,3] => 0
[5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [4,1,1] => [1,2] => 0
[5,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,1,1] => [1,1,2] => 1
[5,3,3,1]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [3,2,1] => [1,1,1] => 1
[5,3,2,2]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,3,1] => [1,1,1] => 1
[5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,4,1] => [1,1,1] => 1
[5,2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,3,1] => [2,1,1] => 1
[4,4,3,1]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [3,1,2] => [1,1,1] => 1
Description
The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
Matching statistic: St000749
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 43%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 43%
Values
[1]
=> [1,0,1,0]
=> [[1,1],[]]
=> []
=> ? = 1
[2]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 0
[1,1]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? = 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 0
[2,1]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {0,1}
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,1}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {1,2}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,2}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> [2]
=> 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {0,1,3}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {0,1,3}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,1,3}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3]]
=> [3]
=> 0
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> [3]
=> 0
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 0
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3]]
=> [3]
=> 0
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,3],[2]]
=> [2]
=> 0
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1]]
=> [2,1]
=> 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> [2]
=> 0
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 0
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 0
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 0
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 0
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ? ∊ {1,1,1,2,2}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> [1]
=> 0
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,2}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ? ∊ {1,1,1,2,2}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3,1],[]]
=> []
=> ? ∊ {1,1,1,2,2}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[4,4,4,4,1],[]]
=> []
=> ? ∊ {1,1,1,2,2}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5],[4]]
=> [4]
=> 0
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[5,5,5,5],[4]]
=> [4]
=> 0
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3]]
=> [3]
=> 0
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3,1]]
=> [3,1]
=> 0
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[4,4,3],[3]]
=> [3]
=> 0
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,2],[1]]
=> [1]
=> 0
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4],[3,1]]
=> [3,1]
=> 0
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[4,3,3],[2]]
=> [2]
=> 0
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 0
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ? ∊ {0,0,1,1,2,3}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 0
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 0
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,2,3}
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ? ∊ {0,0,1,1,2,3}
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [[4,4,3,1],[]]
=> []
=> ? ∊ {0,0,1,1,2,3}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2],[1,1]]
=> [1,1]
=> 0
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [[4,4,1],[1]]
=> [1]
=> 0
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [[4,4,4,1],[1]]
=> [1]
=> 0
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[5,5,5,1],[]]
=> []
=> ? ∊ {0,0,1,1,2,3}
[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]
=> [[5,5,5,5,1],[]]
=> []
=> ? ∊ {0,0,1,1,2,3}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5,5],[4]]
=> [4]
=> 0
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4,4],[3]]
=> [3]
=> 0
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3,1]]
=> [3,1]
=> 0
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,3],[3]]
=> [3]
=> 0
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [[3,3,3,3,3],[2,1]]
=> [2,1]
=> 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [[5,5,5],[4]]
=> [4]
=> 0
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,2],[2]]
=> [2]
=> 0
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2]]
=> [2,2]
=> 2
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,3,3}
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,3,3}
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [[5,5,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,3,3}
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,3,3}
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [[3,3,3,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,3,3}
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [[4,4,4,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,3,3}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [[4,4,4,4,4,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,3,3}
[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]
=> [[5,5,5,5,5,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,3,3}
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[4,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,3,4}
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,3,4}
[4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,3,4}
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [[5,5,4,1],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,3,4}
[3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,3,4}
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [[4,4,2,1],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,3,4}
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [[3,3,3,3,3,1],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,3,4}
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [[5,5,5,4,1],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,3,4}
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [[6,6,6,6,1],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,3,4}
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [[6,6,6,6,6,1],[]]
=> []
=> ? ∊ {0,0,0,1,2,2,2,2,3,4}
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5,4],[4]]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [[6,6,6,6],[5]]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4,3],[3]]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [[4,4,4,4,4],[3,2]]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
Description
The 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.
For example, restricting S(6,3) to S8 yields S(5,3)⊕S(6,2) of degrees (number of standard Young tableaux) 28 and 20, none of which are odd. Restricting to S7 yields S(4,3)⊕2S(5,2)⊕S(6,1) of degrees 14, 14 and 6. However, restricting to S6 yields
S(3,3)⊕3S(4,2)⊕3S(5,1)⊕S6 of degrees 5,9,5 and 1. Therefore, the statistic on the partition (6,3) gives 3.
This is related to 2-saturations of Welter's game, see [1, Corollary 1.2].
Matching statistic: St001586
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001586: Integer partitions ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 57%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001586: Integer partitions ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 57%
Values
[1]
=> [1,0,1,0]
=> [[1,1],[]]
=> []
=> ? = 1
[2]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 0
[1,1]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? = 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 0
[2,1]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {0,1}
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,1}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {1,2}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,2}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> [2]
=> 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 0
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {0,1,3}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {0,1,3}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {0,1,3}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3]]
=> [3]
=> 0
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> [3]
=> 0
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> 0
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4,1],[]]
=> []
=> ? ∊ {0,0,1,2}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3]]
=> [3]
=> 0
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,3],[2]]
=> [2]
=> 0
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1]]
=> [2,1]
=> 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> [2]
=> 0
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 0
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 0
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> 0
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> 0
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ? ∊ {1,1,1,2,2}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> [1]
=> 0
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,2}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ? ∊ {1,1,1,2,2}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3,1],[]]
=> []
=> ? ∊ {1,1,1,2,2}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [[4,4,4,4,1],[]]
=> []
=> ? ∊ {1,1,1,2,2}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5],[4]]
=> [4]
=> 0
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[5,5,5,5],[4]]
=> [4]
=> 0
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3]]
=> [3]
=> 0
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3,1]]
=> [3,1]
=> 0
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[4,4,3],[3]]
=> [3]
=> 0
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,2],[1]]
=> [1]
=> 0
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4],[3,1]]
=> [3,1]
=> 0
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[4,3,3],[2]]
=> [2]
=> 0
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 0
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> 0
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ? ∊ {0,1,1,2,2,3}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 0
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 0
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ? ∊ {0,1,1,2,2,3}
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ? ∊ {0,1,1,2,2,3}
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [[4,4,3,1],[]]
=> []
=> ? ∊ {0,1,1,2,2,3}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2],[1,1]]
=> [1,1]
=> 0
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [[4,4,1],[1]]
=> [1]
=> 0
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [[4,4,4,1],[1]]
=> [1]
=> 0
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [[5,5,5,1],[]]
=> []
=> ? ∊ {0,1,1,2,2,3}
[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]
=> [[5,5,5,5,1],[]]
=> []
=> ? ∊ {0,1,1,2,2,3}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5,5],[4]]
=> [4]
=> 0
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4,4],[3]]
=> [3]
=> 0
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3,1]]
=> [3,1]
=> 0
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,3],[3]]
=> [3]
=> 0
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [[3,3,3,3,3],[2,1]]
=> [2,1]
=> 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [[5,5,5],[4]]
=> [4]
=> 0
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,2],[2]]
=> [2]
=> 0
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2]]
=> [2,2]
=> 0
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [[3,2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,3,3}
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,3,3}
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [[5,5,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,3,3}
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,3,3}
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [[3,3,3,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,3,3}
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [[4,4,4,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,3,3}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [[4,4,4,4,4,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,3,3}
[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]
=> [[5,5,5,5,5,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,3,3}
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [[4,3,3,1],[]]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,3,4}
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,3,4}
[4,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [[4,2,1],[]]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,3,4}
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [[5,5,4,1],[]]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,3,4}
[3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1,1],[]]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,3,4}
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [[4,4,2,1],[]]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,3,4}
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [[3,3,3,3,3,1],[]]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,3,4}
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [[5,5,5,4,1],[]]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,3,4}
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [[6,6,6,6,1],[]]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,3,4}
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [[6,6,6,6,6,1],[]]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,3,4}
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5,4],[4]]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [[6,6,6,6],[5]]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4,3],[3]]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [[4,4,4,4,4],[3,2]]
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3,3,3}
Description
The number of odd parts smaller than the largest even part in an integer partition.
The following 8 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001651The Frankl number of a lattice. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001615The number of join prime elements of a lattice. St001632The number of indecomposable injective modules I with dimExt1(I,A)=1 for the incidence algebra A of a poset. St000455The second largest eigenvalue of a graph if it is integral.
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