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Your data matches 8 different statistics following compositions of up to 3 maps.
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Matching statistic: St000928
St000928: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> 0
[1,1]
=> -1
[3]
=> 1
[2,1]
=> 0
[1,1,1]
=> 1
[4]
=> 0
[3,1]
=> -1
[2,2]
=> -1
[2,1,1]
=> 1
[1,1,1,1]
=> -1
[5]
=> 1
[4,1]
=> -1
[3,2]
=> 1
[3,1,1]
=> 1
[2,2,1]
=> 0
[2,1,1,1]
=> -1
[1,1,1,1,1]
=> 0
[6]
=> 1
[5,1]
=> -1
[4,2]
=> 1
[4,1,1]
=> 0
[3,3]
=> -1
[3,2,1]
=> 0
[3,1,1,1]
=> -1
[2,2,2]
=> -1
[2,2,1,1]
=> 0
[2,1,1,1,1]
=> 2
[1,1,1,1,1,1]
=> 0
[7]
=> 1
[6,1]
=> -2
[5,2]
=> 0
[5,1,1]
=> 2
[4,3]
=> 0
[4,2,1]
=> 0
[4,1,1,1]
=> -1
[3,3,1]
=> -1
[3,2,2]
=> 0
[3,2,1,1]
=> 1
[3,1,1,1,1]
=> 0
[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]
=> 1
[7,1]
=> -1
[6,2]
=> 0
[6,1,1]
=> 2
[5,3]
=> 1
[5,2,1]
=> -1
[5,1,1,1]
=> -1
Description
The sum of the coefficients of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1].
Matching statistic: St001604
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 22% ●values known / values provided: 23%●distinct values known / distinct values provided: 22%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 22% ●values known / values provided: 23%●distinct values known / distinct values provided: 22%
Values
[2]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> ? ∊ {-1,0}
[1,1]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {-1,0}
[3]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> ? ∊ {0,1,1}
[2,1]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {0,1,1}
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {0,1,1}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> ? ∊ {-1,-1,-1,0,1}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> [2]
=> ? ∊ {-1,-1,-1,0,1}
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> ? ∊ {-1,-1,-1,0,1}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> []
=> ? ∊ {-1,-1,-1,0,1}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {-1,-1,-1,0,1}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> [2]
=> ? ∊ {-1,-1,0,0,1,1,1}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> [1]
=> ? ∊ {-1,-1,0,0,1,1,1}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> ? ∊ {-1,-1,0,0,1,1,1}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> ? ∊ {-1,-1,0,0,1,1,1}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {-1,-1,0,0,1,1,1}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> []
=> ? ∊ {-1,-1,0,0,1,1,1}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {-1,-1,0,0,1,1,1}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3]]
=> [3]
=> 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> [3]
=> 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> [2]
=> ? ∊ {-1,-1,-1,-1,0,0,0,2}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 0
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> ? ∊ {-1,-1,-1,-1,0,0,0,2}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {-1,-1,-1,-1,0,0,0,2}
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> []
=> ? ∊ {-1,-1,-1,-1,0,0,0,2}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> ? ∊ {-1,-1,-1,-1,0,0,0,2}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> [1]
=> ? ∊ {-1,-1,-1,-1,0,0,0,2}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> []
=> ? ∊ {-1,-1,-1,-1,0,0,0,2}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4,1],[]]
=> []
=> ? ∊ {-1,-1,-1,-1,0,0,0,2}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3]]
=> [3]
=> 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,3],[2]]
=> [2]
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,2}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1]]
=> [2,1]
=> 0
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> [2]
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,2}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,2}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> [3]
=> 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,2}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> [2]
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,2}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,2}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,2}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> [1]
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,2}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,2}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,2}
[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],[]]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,2}
[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],[]]
=> []
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,2}
[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]
=> 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[5,5,5,5],[4]]
=> [4]
=> 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3]]
=> [3]
=> 1
[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]
=> 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,2],[1]]
=> [1]
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,2,2}
[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]
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,2,2}
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 0
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [[3,3,1],[2]]
=> [2]
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,2,2}
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[4,3,1],[]]
=> []
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,2,2}
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,2,2}
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,2,2}
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[3,1,1],[]]
=> []
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,2,2}
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [[2,2,2,2,1],[]]
=> []
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,2,2}
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [[4,4,3,1],[]]
=> []
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,2,2}
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[4,4,2],[1,1]]
=> [1,1]
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,2,2}
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [[4,4,1],[1]]
=> [1]
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,2,2}
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [[4,4,4,1],[1]]
=> [1]
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,2,2}
[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],[]]
=> []
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,1,1,1,2,2}
[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]
=> 1
[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]
=> 1
[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]
=> 1
[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]
=> 0
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [[5,5,5],[4]]
=> [4]
=> 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2]]
=> [2,2]
=> 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1]]
=> [2,1]
=> 0
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1,1]]
=> [2,1,1]
=> 0
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [[3,3,3,2],[1,1,1]]
=> [1,1,1]
=> 0
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [[6,6,6,6,6,6],[5]]
=> [5]
=> 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [[6,6,6,6,6],[5]]
=> [5]
=> 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,4],[4]]
=> [4]
=> 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [[5,5,5,5,5],[4,1]]
=> [4,1]
=> 0
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [[5,5,5,4],[4]]
=> [4]
=> 1
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [[5,5,5,5],[4,1]]
=> [4,1]
=> 0
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [[4,4,3,3],[3]]
=> [3]
=> 1
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [[4,4,4,4],[3,2]]
=> [3,2]
=> 1
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3,1]]
=> [3,1]
=> 0
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4,2],[3]]
=> [3]
=> 1
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3,1,1]]
=> [3,1,1]
=> 0
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [[4,4,2],[3]]
=> [3]
=> 1
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [[4,4,3],[3,1]]
=> [3,1]
=> 0
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [[4,4,4],[3,2]]
=> [3,2]
=> 1
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [[4,3,3],[2,1]]
=> [2,1]
=> 0
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [[4,4,3],[2,2]]
=> [2,2]
=> 1
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> [2,1]
=> 0
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [[4,4,4,2],[1,1,1]]
=> [1,1,1]
=> 0
[6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [[3,3,3,3,3],[2,2]]
=> [2,2]
=> 1
[6,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [[3,3,3,3,3],[2,1,1]]
=> [2,1,1]
=> 0
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [[5,5,5],[4,1]]
=> [4,1]
=> 0
[5,4,2]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [[3,3,3,2],[2,2]]
=> [2,2]
=> 1
[5,4,1,1]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [[3,3,3,3],[2,2,1]]
=> [2,2,1]
=> 1
[5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1]]
=> [2,1]
=> 0
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [[5,5],[4]]
=> [4]
=> 1
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001629
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 22%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 20% ●values known / values provided: 20%●distinct values known / distinct values provided: 22%
Values
[2]
=> 0 => [1] => [1] => ? ∊ {-1,0}
[1,1]
=> 11 => [2] => [1] => ? ∊ {-1,0}
[3]
=> 1 => [1] => [1] => ? ∊ {0,1,1}
[2,1]
=> 01 => [1,1] => [2] => ? ∊ {0,1,1}
[1,1,1]
=> 111 => [3] => [1] => ? ∊ {0,1,1}
[4]
=> 0 => [1] => [1] => ? ∊ {-1,-1,-1,0,1}
[3,1]
=> 11 => [2] => [1] => ? ∊ {-1,-1,-1,0,1}
[2,2]
=> 00 => [2] => [1] => ? ∊ {-1,-1,-1,0,1}
[2,1,1]
=> 011 => [1,2] => [1,1] => ? ∊ {-1,-1,-1,0,1}
[1,1,1,1]
=> 1111 => [4] => [1] => ? ∊ {-1,-1,-1,0,1}
[5]
=> 1 => [1] => [1] => ? ∊ {-1,-1,0,0,1,1,1}
[4,1]
=> 01 => [1,1] => [2] => ? ∊ {-1,-1,0,0,1,1,1}
[3,2]
=> 10 => [1,1] => [2] => ? ∊ {-1,-1,0,0,1,1,1}
[3,1,1]
=> 111 => [3] => [1] => ? ∊ {-1,-1,0,0,1,1,1}
[2,2,1]
=> 001 => [2,1] => [1,1] => ? ∊ {-1,-1,0,0,1,1,1}
[2,1,1,1]
=> 0111 => [1,3] => [1,1] => ? ∊ {-1,-1,0,0,1,1,1}
[1,1,1,1,1]
=> 11111 => [5] => [1] => ? ∊ {-1,-1,0,0,1,1,1}
[6]
=> 0 => [1] => [1] => ? ∊ {-1,-1,-1,-1,0,0,0,0,1,2}
[5,1]
=> 11 => [2] => [1] => ? ∊ {-1,-1,-1,-1,0,0,0,0,1,2}
[4,2]
=> 00 => [2] => [1] => ? ∊ {-1,-1,-1,-1,0,0,0,0,1,2}
[4,1,1]
=> 011 => [1,2] => [1,1] => ? ∊ {-1,-1,-1,-1,0,0,0,0,1,2}
[3,3]
=> 11 => [2] => [1] => ? ∊ {-1,-1,-1,-1,0,0,0,0,1,2}
[3,2,1]
=> 101 => [1,1,1] => [3] => 1
[3,1,1,1]
=> 1111 => [4] => [1] => ? ∊ {-1,-1,-1,-1,0,0,0,0,1,2}
[2,2,2]
=> 000 => [3] => [1] => ? ∊ {-1,-1,-1,-1,0,0,0,0,1,2}
[2,2,1,1]
=> 0011 => [2,2] => [2] => ? ∊ {-1,-1,-1,-1,0,0,0,0,1,2}
[2,1,1,1,1]
=> 01111 => [1,4] => [1,1] => ? ∊ {-1,-1,-1,-1,0,0,0,0,1,2}
[1,1,1,1,1,1]
=> 111111 => [6] => [1] => ? ∊ {-1,-1,-1,-1,0,0,0,0,1,2}
[7]
=> 1 => [1] => [1] => ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,2}
[6,1]
=> 01 => [1,1] => [2] => ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,2}
[5,2]
=> 10 => [1,1] => [2] => ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,2}
[5,1,1]
=> 111 => [3] => [1] => ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,2}
[4,3]
=> 01 => [1,1] => [2] => ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,2}
[4,2,1]
=> 001 => [2,1] => [1,1] => ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,2}
[4,1,1,1]
=> 0111 => [1,3] => [1,1] => ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,2}
[3,3,1]
=> 111 => [3] => [1] => ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,2}
[3,2,2]
=> 100 => [1,2] => [1,1] => ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,2}
[3,2,1,1]
=> 1011 => [1,1,2] => [2,1] => 0
[3,1,1,1,1]
=> 11111 => [5] => [1] => ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,2}
[2,2,2,1]
=> 0001 => [3,1] => [1,1] => ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,2}
[2,2,1,1,1]
=> 00111 => [2,3] => [1,1] => ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,2}
[2,1,1,1,1,1]
=> 011111 => [1,5] => [1,1] => ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,2}
[1,1,1,1,1,1,1]
=> 1111111 => [7] => [1] => ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,1,2}
[8]
=> 0 => [1] => [1] => ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,1,1,1,1,2,2}
[7,1]
=> 11 => [2] => [1] => ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,1,1,1,1,2,2}
[6,2]
=> 00 => [2] => [1] => ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,1,1,1,1,2,2}
[6,1,1]
=> 011 => [1,2] => [1,1] => ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,1,1,1,1,2,2}
[5,3]
=> 11 => [2] => [1] => ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,1,1,1,1,2,2}
[5,2,1]
=> 101 => [1,1,1] => [3] => 1
[5,1,1,1]
=> 1111 => [4] => [1] => ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,1,1,1,1,2,2}
[4,4]
=> 00 => [2] => [1] => ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,1,1,1,1,2,2}
[4,3,1]
=> 011 => [1,2] => [1,1] => ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,1,1,1,1,2,2}
[4,2,2]
=> 000 => [3] => [1] => ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,1,1,1,1,1,1,2,2}
[3,2,2,1]
=> 1001 => [1,2,1] => [1,1,1] => 1
[3,2,1,1,1]
=> 10111 => [1,1,3] => [2,1] => 0
[5,2,1,1]
=> 1011 => [1,1,2] => [2,1] => 0
[4,3,2]
=> 010 => [1,1,1] => [3] => 1
[3,3,2,1]
=> 1101 => [2,1,1] => [1,2] => 0
[3,2,2,1,1]
=> 10011 => [1,2,2] => [1,2] => 0
[3,2,1,1,1,1]
=> 101111 => [1,1,4] => [2,1] => 0
[7,2,1]
=> 101 => [1,1,1] => [3] => 1
[5,4,1]
=> 101 => [1,1,1] => [3] => 1
[5,2,2,1]
=> 1001 => [1,2,1] => [1,1,1] => 1
[5,2,1,1,1]
=> 10111 => [1,1,3] => [2,1] => 0
[4,3,2,1]
=> 0101 => [1,1,1,1] => [4] => 1
[3,3,2,1,1]
=> 11011 => [2,1,2] => [1,1,1] => 1
[3,2,2,2,1]
=> 10001 => [1,3,1] => [1,1,1] => 1
[3,2,2,1,1,1]
=> 100111 => [1,2,3] => [1,1,1] => 1
[3,2,1,1,1,1,1]
=> 1011111 => [1,1,5] => [2,1] => 0
[7,2,1,1]
=> 1011 => [1,1,2] => [2,1] => 0
[6,3,2]
=> 010 => [1,1,1] => [3] => 1
[5,4,1,1]
=> 1011 => [1,1,2] => [2,1] => 0
[5,3,2,1]
=> 1101 => [2,1,1] => [1,2] => 0
[5,2,2,1,1]
=> 10011 => [1,2,2] => [1,2] => 0
[5,2,1,1,1,1]
=> 101111 => [1,1,4] => [2,1] => 0
[4,3,2,2]
=> 0100 => [1,1,2] => [2,1] => 0
[4,3,2,1,1]
=> 01011 => [1,1,1,2] => [3,1] => 0
[3,3,2,2,1]
=> 11001 => [2,2,1] => [2,1] => 0
[3,3,2,1,1,1]
=> 110111 => [2,1,3] => [1,1,1] => 1
[3,2,2,2,1,1]
=> 100011 => [1,3,2] => [1,1,1] => 1
[3,2,2,1,1,1,1]
=> 1001111 => [1,2,4] => [1,1,1] => 1
[3,2,1,1,1,1,1,1]
=> 10111111 => [1,1,6] => [2,1] => 0
[9,2,1]
=> 101 => [1,1,1] => [3] => 1
[7,4,1]
=> 101 => [1,1,1] => [3] => 1
[7,2,2,1]
=> 1001 => [1,2,1] => [1,1,1] => 1
[7,2,1,1,1]
=> 10111 => [1,1,3] => [2,1] => 0
[6,3,2,1]
=> 0101 => [1,1,1,1] => [4] => 1
[5,4,3]
=> 101 => [1,1,1] => [3] => 1
[5,4,2,1]
=> 1001 => [1,2,1] => [1,1,1] => 1
[5,4,1,1,1]
=> 10111 => [1,1,3] => [2,1] => 0
[5,3,2,1,1]
=> 11011 => [2,1,2] => [1,1,1] => 1
[5,2,2,2,1]
=> 10001 => [1,3,1] => [1,1,1] => 1
[5,2,2,1,1,1]
=> 100111 => [1,2,3] => [1,1,1] => 1
[5,2,1,1,1,1,1]
=> 1011111 => [1,1,5] => [2,1] => 0
[4,3,3,2]
=> 0110 => [1,2,1] => [1,1,1] => 1
[4,3,2,2,1]
=> 01001 => [1,1,2,1] => [2,1,1] => 1
[4,3,2,1,1,1]
=> 010111 => [1,1,1,3] => [3,1] => 0
[3,3,3,2,1]
=> 11101 => [3,1,1] => [1,2] => 0
[3,3,2,2,1,1]
=> 110011 => [2,2,2] => [3] => 1
[3,3,2,1,1,1,1]
=> 1101111 => [2,1,4] => [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: St000454
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 44%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 44%
Values
[2]
=> [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,1]
=> [1,1,0,0]
=> [1,2] => ([],2)
=> 0 = -1 + 1
[3]
=> [1,0,1,0,1,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ? ∊ {1,1} + 1
[2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> 1 = 0 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {1,1} + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {-1,-1,0} + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {-1,-1,0} + 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 0 = -1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {-1,-1,0} + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {-1,-1,0,1,1} + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 1 = 0 + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {-1,-1,0,1,1} + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {-1,-1,0,1,1} + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {-1,-1,0,1,1} + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {-1,-1,0,1,1} + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {-1,-1,-1,0,0,0,0,2} + 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {-1,-1,-1,0,0,0,0,2} + 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {-1,-1,-1,0,0,0,0,2} + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {-1,-1,-1,0,0,0,0,2} + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {-1,-1,-1,0,0,0,0,2} + 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {-1,-1,-1,0,0,0,0,2} + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> 0 = -1 + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {-1,-1,-1,0,0,0,0,2} + 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {-1,-1,-1,0,0,0,0,2} + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,2} + 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,1,7] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,2} + 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,2} + 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,7,1,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,2} + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,2} + 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,6,4,5] => ([(0,5),(1,5),(2,4),(3,4)],6)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,2} + 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,6,7,1,5] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,2} + 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,2} + 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 1 = 0 + 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,1,5,6,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 1 + 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,1,4] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,2} + 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,2} + 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,5,6,2,3] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,2} + 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,7,1,3] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,2} + 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,7,1,2] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,1,1,2} + 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,2,2} + 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,7,1,8] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,2,2} + 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,5,1,6,7] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,5,6,8,1,7] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,2,2} + 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [2,3,4,1,7,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5)],7)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,2,2} + 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,4,5,7,8,1,6] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,2,2} + 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,2,2} + 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,2,2} + 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,2,2} + 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [2,3,1,6,7,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)
=> 2 = 1 + 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,6,7,8,1,5] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,2,2} + 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,2,2} + 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,1,5,6,2,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,2,2} + 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,2,2} + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [2,1,5,6,7,3,4] => ([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,2,2} + 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,5,6,7,8,1,4] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,2,2} + 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,2,5,6,3,4] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 2 = 1 + 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,4,5,6,7,2,3] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,2,2} + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,7,8,1,3] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,2,2} + 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]
=> [3,4,5,6,7,8,1,2] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,2,2} + 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,9,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? ∊ {-3,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2} + 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,6,7,8,1,9] => ([(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? ∊ {-3,-2,-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2} + 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([],5)
=> 0 = -1 + 1
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [2,1,3,6,7,4,5] => ([(1,2),(3,5),(3,6),(4,5),(4,6)],7)
=> 2 = 1 + 1
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> 3 = 2 + 1
[5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,3,6,1,4,5,7] => ([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 2 = 1 + 1
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => ([(4,5)],6)
=> 1 = 0 + 1
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,2,3,4,5] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,2,3,6,7,4,5] => ([(3,5),(3,6),(4,5),(4,6)],7)
=> 2 = 1 + 1
[5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [4,5,6,1,2,3,7] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> 3 = 2 + 1
[4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,1,7,3,4,5,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> 2 = 1 + 1
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([],6)
=> 0 = -1 + 1
Description
The largest eigenvalue of a graph if it is integral.
If a graph is d-regular, then its largest eigenvalue equals d. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001630
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 22%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 22%
Values
[2]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> ([],1)
=> ? ∊ {-1,0}
[1,1]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> ([],1)
=> ? ∊ {-1,0}
[3]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> ([],1)
=> ? ∊ {0,1,1}
[2,1]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1}
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> ([],1)
=> ? ∊ {0,1,1}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ([],1)
=> ? ∊ {-1,-1,-1,0,1}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> ([],1)
=> ? ∊ {-1,-1,-1,0,1}
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,0,1}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,-1,0,1}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,-1,0,1}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> ([],1)
=> ? ∊ {-1,-1,0,0,1,1,1}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ([],1)
=> ? ∊ {-1,-1,0,0,1,1,1}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {-1,-1,0,0,1,1,1}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,0,0,1,1,1}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,0,0,1,1,1}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,0,0,1,1,1}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,0,0,1,1,1}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,3],[2]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ([(0,1)],2)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3,1],[]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[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)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[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]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,1,1,1,2,2}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[5,5,5,5],[4]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,1,1,1,2,2}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,1,1,1,2,2}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3,1]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,1,1,1,2,2}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[4,4,3],[3]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,1,1,1,2,2}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,2],[1]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,1,1,1,2,2}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4],[3,1]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,1,1,1,2,2}
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [[4,4,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> 1
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [[4,3,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [[4,3,3],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [[4,3,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[5,3,3,1]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
[5,3,2,2]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[4,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[4,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001876
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 22%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 22%
Values
[2]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> ([],1)
=> ? ∊ {-1,0}
[1,1]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> ([],1)
=> ? ∊ {-1,0}
[3]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> ([],1)
=> ? ∊ {0,1,1}
[2,1]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1}
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> ([],1)
=> ? ∊ {0,1,1}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ([],1)
=> ? ∊ {-1,-1,-1,0,1}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> ([],1)
=> ? ∊ {-1,-1,-1,0,1}
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,0,1}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,-1,0,1}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,-1,0,1}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> ([],1)
=> ? ∊ {-1,-1,0,0,1,1,1}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ([],1)
=> ? ∊ {-1,-1,0,0,1,1,1}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {-1,-1,0,0,1,1,1}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,0,0,1,1,1}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,0,0,1,1,1}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,0,0,1,1,1}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,0,0,1,1,1}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,3],[2]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ([(0,1)],2)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3,1],[]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[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)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[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]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,1,1,1,1,1,1,2,2}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[5,5,5,5],[4]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,1,1,1,1,1,1,2,2}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,1,1,1,1,1,1,2,2}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3,1]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,1,1,1,1,1,1,2,2}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[4,4,3],[3]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,1,1,1,1,1,1,2,2}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,2],[1]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,1,1,1,1,1,1,2,2}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4],[3,1]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,1,1,1,1,1,1,2,2}
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> 0
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> 0
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [[4,4,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> 0
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [[4,3,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [[4,3,3],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [[4,3,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> 0
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
[5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> 0
[5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> 0
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
[4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
[4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
[3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[5,3,3,1]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
[5,3,2,2]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[4,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St001877
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 22%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 22%
Values
[2]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> ([],1)
=> ? ∊ {-1,0}
[1,1]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> ([],1)
=> ? ∊ {-1,0}
[3]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> ([],1)
=> ? ∊ {0,1,1}
[2,1]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1}
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> ([],1)
=> ? ∊ {0,1,1}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ([],1)
=> ? ∊ {-1,-1,-1,0,1}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> ([],1)
=> ? ∊ {-1,-1,-1,0,1}
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,0,1}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,-1,0,1}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,-1,0,1}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> ([],1)
=> ? ∊ {-1,-1,0,0,1,1,1}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ([],1)
=> ? ∊ {-1,-1,0,0,1,1,1}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {-1,-1,0,0,1,1,1}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,0,0,1,1,1}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,0,0,1,1,1}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,0,0,1,1,1}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,0,0,1,1,1}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,3],[2]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ([(0,1)],2)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3,1],[]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[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)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[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]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,1,1,1,1,1,1,2,2}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[5,5,5,5],[4]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,1,1,1,1,1,1,2,2}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,1,1,1,1,1,1,2,2}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3,1]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,1,1,1,1,1,1,2,2}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[4,4,3],[3]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,1,1,1,1,1,1,2,2}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,2],[1]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,1,1,1,1,1,1,2,2}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4],[3,1]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,1,1,1,1,1,1,1,1,2,2}
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> 0
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> 0
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [[4,4,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> 0
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [[4,3,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [[4,3,3],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [[4,3,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> 0
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
[5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> 0
[5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> 0
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> 0
[4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
[4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 0
[3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[5,3,3,1]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 0
[5,3,2,2]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[4,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 0
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St001878
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 22%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 22%
Values
[2]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> ([],1)
=> ? ∊ {-1,0}
[1,1]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> ([],1)
=> ? ∊ {-1,0}
[3]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> ([],1)
=> ? ∊ {0,1,1}
[2,1]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> ([],1)
=> ? ∊ {0,1,1}
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> ([],1)
=> ? ∊ {0,1,1}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> ([],1)
=> ? ∊ {-1,-1,-1,0,1}
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [[3,3],[2]]
=> ([],1)
=> ? ∊ {-1,-1,-1,0,1}
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,0,1}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[3,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,-1,0,1}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,-1,0,1}
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[3,3,3,3],[2]]
=> ([],1)
=> ? ∊ {-1,-1,0,0,1,1,1}
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[2,2,2,2],[1]]
=> ([],1)
=> ? ∊ {-1,-1,0,0,1,1,1}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {-1,-1,0,0,1,1,1}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,0,0,1,1,1}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,0,0,1,1,1}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[2,2,2,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,0,0,1,1,1}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,0,0,1,1,1}
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [[4,4,4],[3]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2],[2]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3],[2,1]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[3,2,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[3,3,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[4,4,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4,1],[]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,0,0,0,0,1,1,2}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [[4,4,4,4,4],[3]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [[3,3,3,3,3],[2]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [[3,3,3,3],[2,1]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [[3,3,3,2],[2]]
=> ([(0,1)],2)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [[4,4],[3]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [[4,3],[2]]
=> ([(0,1)],2)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [[4,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[4,1],[]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [[3,3,3,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [[3,3,2,1],[]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [[3,3,3,3,1],[]]
=> ([],1)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[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)
=> ? ∊ {-2,-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,2}
[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]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,1,1,1,2,2}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [[5,5,5,5],[4]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,1,1,1,2,2}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,1,1,1,2,2}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [[4,4,4,4],[3,1]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,1,1,1,2,2}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [[4,4,3],[3]]
=> ([(0,1)],2)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,1,1,1,2,2}
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [[2,2,2,2,2],[1]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,1,1,1,2,2}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[4,4,4],[3,1]]
=> ([],1)
=> ? ∊ {-1,-1,-1,-1,-1,-1,-1,-1,-1,0,0,0,1,1,1,1,1,1,2,2}
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[3,3,2,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [[3,3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [[4,4,4,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [[4,4,3],[3,1]]
=> ([(0,2),(2,1)],3)
=> 1
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [[4,3,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [[4,3,3],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> [[4,3,2],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [[4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [[4,4,2],[2,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [[4,4,3,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,3,3]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [[3,3,2,2],[2,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[5,3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> [[3,3,2,1],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[5,2,2,2]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[3,3,3,2],[2,1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[5,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> [[3,3,3,1],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[4,4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[3,3,3,1,1]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2,1],[1,1]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[5,3,3,1]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
[5,3,2,2]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[4,4,2,2]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[4,4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[4,3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
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