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Your data matches 10 different statistics following compositions of up to 3 maps.
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Matching statistic: St000460
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Mp00040: Integer compositions —to partition⟶ Integer partitions
St000460: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000460: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 1
[1,1] => [1,1]
=> 2
[2] => [2]
=> 2
[1,1,1] => [1,1,1]
=> 3
[1,2] => [2,1]
=> 3
[2,1] => [2,1]
=> 3
[3] => [3]
=> 3
[1,1,1,1] => [1,1,1,1]
=> 4
[1,1,2] => [2,1,1]
=> 4
[1,2,1] => [2,1,1]
=> 4
[1,3] => [3,1]
=> 4
[2,1,1] => [2,1,1]
=> 4
[2,2] => [2,2]
=> 1
[3,1] => [3,1]
=> 4
[4] => [4]
=> 4
[1,1,1,1,1] => [1,1,1,1,1]
=> 5
[1,1,1,2] => [2,1,1,1]
=> 5
[1,1,2,1] => [2,1,1,1]
=> 5
[1,1,3] => [3,1,1]
=> 5
[1,2,1,1] => [2,1,1,1]
=> 5
[1,2,2] => [2,2,1]
=> 1
[1,3,1] => [3,1,1]
=> 5
[1,4] => [4,1]
=> 5
[2,1,1,1] => [2,1,1,1]
=> 5
[2,1,2] => [2,2,1]
=> 1
[2,2,1] => [2,2,1]
=> 1
[2,3] => [3,2]
=> 1
[3,1,1] => [3,1,1]
=> 5
[3,2] => [3,2]
=> 1
[4,1] => [4,1]
=> 5
[5] => [5]
=> 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 6
[1,1,1,1,2] => [2,1,1,1,1]
=> 6
[1,1,1,2,1] => [2,1,1,1,1]
=> 6
[1,1,1,3] => [3,1,1,1]
=> 6
[1,1,2,1,1] => [2,1,1,1,1]
=> 6
[1,1,2,2] => [2,2,1,1]
=> 1
[1,1,3,1] => [3,1,1,1]
=> 6
[1,1,4] => [4,1,1]
=> 6
[1,2,1,1,1] => [2,1,1,1,1]
=> 6
[1,2,1,2] => [2,2,1,1]
=> 1
[1,2,2,1] => [2,2,1,1]
=> 1
[1,2,3] => [3,2,1]
=> 1
[1,3,1,1] => [3,1,1,1]
=> 6
[1,3,2] => [3,2,1]
=> 1
[1,4,1] => [4,1,1]
=> 6
[1,5] => [5,1]
=> 6
[2,1,1,1,1] => [2,1,1,1,1]
=> 6
[2,1,1,2] => [2,2,1,1]
=> 1
[2,1,2,1] => [2,2,1,1]
=> 1
Description
The hook length of the last cell along the main diagonal of an integer partition.
Matching statistic: St001432
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 30%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 30%
Values
[1] => [[1],[]]
=> []
=> ?
=> ? = 1
[1,1] => [[1,1],[]]
=> []
=> ?
=> ? ∊ {2,2}
[2] => [[2],[]]
=> []
=> ?
=> ? ∊ {2,2}
[1,1,1] => [[1,1,1],[]]
=> []
=> ?
=> ? ∊ {3,3,3,3}
[1,2] => [[2,1],[]]
=> []
=> ?
=> ? ∊ {3,3,3,3}
[2,1] => [[2,2],[1]]
=> [1]
=> []
=> ? ∊ {3,3,3,3}
[3] => [[3],[]]
=> []
=> ?
=> ? ∊ {3,3,3,3}
[1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {4,4,4,4,4,4,4}
[1,1,2] => [[2,1,1],[]]
=> []
=> ?
=> ? ∊ {4,4,4,4,4,4,4}
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> []
=> ? ∊ {4,4,4,4,4,4,4}
[1,3] => [[3,1],[]]
=> []
=> ?
=> ? ∊ {4,4,4,4,4,4,4}
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[2,2] => [[3,2],[1]]
=> [1]
=> []
=> ? ∊ {4,4,4,4,4,4,4}
[3,1] => [[3,3],[2]]
=> [2]
=> []
=> ? ∊ {4,4,4,4,4,4,4}
[4] => [[4],[]]
=> []
=> ?
=> ? ∊ {4,4,4,4,4,4,4}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5}
[1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ?
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5}
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> []
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5}
[1,1,3] => [[3,1,1],[]]
=> []
=> ?
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> []
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5}
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> []
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5}
[1,4] => [[4,1],[]]
=> []
=> ?
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1]
=> 1
[2,3] => [[4,2],[1]]
=> [1]
=> []
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5}
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [2]
=> 1
[3,2] => [[4,3],[2]]
=> [2]
=> []
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5}
[4,1] => [[4,4],[3]]
=> [3]
=> []
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5}
[5] => [[5],[]]
=> []
=> ?
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> []
=> ? ∊ {6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ?
=> ? ∊ {6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> []
=> ? ∊ {6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> []
=> ? ∊ {6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,4] => [[4,1,1],[]]
=> []
=> ?
=> ? ∊ {6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> []
=> ? ∊ {6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> []
=> ? ∊ {6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> []
=> ? ∊ {6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,5] => [[5,1],[]]
=> []
=> ?
=> ? ∊ {6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 1
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 2
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1]
=> 1
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1]
=> 1
[2,4] => [[5,2],[1]]
=> [1]
=> []
=> ? ∊ {6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 2
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [2]
=> 1
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [2]
=> 1
[3,3] => [[5,3],[2]]
=> [2]
=> []
=> ? ∊ {6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [3]
=> 1
[4,2] => [[5,4],[3]]
=> [3]
=> []
=> ? ∊ {6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[5,1] => [[5,5],[4]]
=> [4]
=> []
=> ? ∊ {6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[6] => [[6],[]]
=> []
=> ?
=> ? ∊ {6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> []
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,1,1,3] => [[3,1,1,1,1],[]]
=> []
=> ?
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> []
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> []
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,1,4] => [[4,1,1,1],[]]
=> []
=> ?
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> []
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[1,1,3,2] => [[4,3,1,1],[2]]
=> [2]
=> []
=> ? ∊ {7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 1
[1,2,1,3] => [[4,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 2
[1,2,2,2] => [[4,3,2,1],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,2,3,1] => [[4,4,2,1],[3,1]]
=> [3,1]
=> [1]
=> 1
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 2
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[1,3,2,1] => [[4,4,3,1],[3,2]]
=> [3,2]
=> [2]
=> 1
[1,4,1,1] => [[4,4,4,1],[3,3]]
=> [3,3]
=> [3]
=> 1
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,3] => [[4,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> [2,2,1,1]
=> [2,1,1]
=> 2
[2,1,2,2] => [[4,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 1
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> [3,1,1]
=> [1,1]
=> 1
[2,1,4] => [[5,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> [2,2,2,1]
=> [2,2,1]
=> 2
[2,2,1,2] => [[4,3,3,2],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 2
[2,2,2,1] => [[4,4,3,2],[3,2,1]]
=> [3,2,1]
=> [2,1]
=> 2
[2,2,3] => [[5,3,2],[2,1]]
=> [2,1]
=> [1]
=> 1
Description
The order dimension of the partition.
Given a partition $\lambda$, let $I(\lambda)$ be the principal order ideal in the Young lattice generated by $\lambda$. The order dimension of a partition is defined as the order dimension of the poset $I(\lambda)$.
Matching statistic: St001630
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 20%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 20%
Values
[1] => [1] => [[1],[]]
=> ([],1)
=> ? = 1
[1,1] => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {2,2}
[2] => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {2,2}
[1,1,1] => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {3,3,3,3}
[1,2] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {3,3,3,3}
[2,1] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {3,3,3,3}
[3] => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {3,3,3,3}
[1,1,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {1,4,4,4,4,4,4,4}
[1,1,2] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {1,4,4,4,4,4,4,4}
[1,2,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {1,4,4,4,4,4,4,4}
[1,3] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {1,4,4,4,4,4,4,4}
[2,1,1] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {1,4,4,4,4,4,4,4}
[2,2] => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {1,4,4,4,4,4,4,4}
[3,1] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {1,4,4,4,4,4,4,4}
[4] => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {1,4,4,4,4,4,4,4}
[1,1,1,1,1] => [5] => [[5],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[1,1,1,2] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[1,1,3] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[1,2,2] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[1,4] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[2,1,1,1] => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[2,2,1] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[2,3] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[3,1,1] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[3,2] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[4,1] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[5] => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[1,1,1,1,1,1] => [6] => [[6],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,1,3] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,2,2] => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,4] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,5] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,2,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,2,2,1,1] => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[1,2,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,2,1,1,1] => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,1,1,2] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,2,1,1,1] => [4,1,3] => [[6,4,4],[3,3]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,2,2,1] => [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,2,1,1,2] => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,2,2,1,1] => [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[1,1,1,2,2,2] => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,3,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,1,1,1,2] => [2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[1,1,2,1,1,2,1] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,1,2,1,1,3] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,2,2,1,1,1] => [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[1,1,2,2,1,2] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,2,2,1] => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,2,2,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,3,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,3,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,1,1,1,2,1] => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,1,1,2,1,1] => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,2,1,1,2,2] => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,1,1,3,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,2,1,1,1,1] => [1,2,4] => [[5,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,2,1,1,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,2,2,2,1,1] => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,3,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,1,1,2,1,1] => [1,3,1,2] => [[4,3,3,1],[2,2]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[2,1,1,1,2,2] => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1,2,1,1,1] => [1,2,1,3] => [[4,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1,2,2,1] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[2,1,1,3,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,2,2,1,1] => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,1,1,1,2] => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,2,1,1,2,1] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,1,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,2,1,1,1] => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,2,1,1,2] => [4,1,2,1] => [[5,5,4,4],[4,3,3]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,2,2,1,1] => [4,2,2] => [[6,5,4],[4,3]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St001878
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 20%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 20%
Values
[1] => [1] => [[1],[]]
=> ([],1)
=> ? = 1
[1,1] => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {2,2}
[2] => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {2,2}
[1,1,1] => [3] => [[3],[]]
=> ([],1)
=> ? ∊ {3,3,3,3}
[1,2] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {3,3,3,3}
[2,1] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {3,3,3,3}
[3] => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {3,3,3,3}
[1,1,1,1] => [4] => [[4],[]]
=> ([],1)
=> ? ∊ {1,4,4,4,4,4,4,4}
[1,1,2] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {1,4,4,4,4,4,4,4}
[1,2,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {1,4,4,4,4,4,4,4}
[1,3] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {1,4,4,4,4,4,4,4}
[2,1,1] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {1,4,4,4,4,4,4,4}
[2,2] => [2] => [[2],[]]
=> ([],1)
=> ? ∊ {1,4,4,4,4,4,4,4}
[3,1] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {1,4,4,4,4,4,4,4}
[4] => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {1,4,4,4,4,4,4,4}
[1,1,1,1,1] => [5] => [[5],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[1,1,1,2] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[1,1,2,1] => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[1,1,3] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[1,2,1,1] => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[1,2,2] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[1,3,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[1,4] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[2,1,1,1] => [1,3] => [[3,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[2,1,2] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[2,2,1] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[2,3] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[3,1,1] => [1,2] => [[2,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[3,2] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[4,1] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[5] => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[1,1,1,1,1,1] => [6] => [[6],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,1,1,2] => [4,1] => [[4,4],[3]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,1,2,1] => [3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,1,3] => [3,1] => [[3,3],[2]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,2,1,1] => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,2,2] => [2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,4] => [2,1] => [[2,2],[1]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,1,1,1] => [1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,1,2] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,2,1] => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,3] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,3,1,1] => [1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,3,2] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,4,1] => [1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,5] => [1,1] => [[1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[2,1,1,1,1] => [1,4] => [[4,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[2,1,1,2] => [1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[2,1,2,1] => [1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,2,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,2,2,1] => [3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,2,2,1,1] => [2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[1,2,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,2,1,1,1] => [1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,1,1,2] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,2,1,1,1] => [4,1,3] => [[6,4,4],[3,3]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,2,2,1] => [4,2,1] => [[5,5,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,2,1,1,2] => [3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,2,2,1,1] => [3,2,2] => [[5,4,3],[3,2]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[1,1,1,2,2,2] => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,3,1,1,1] => [3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,1,1,1,2] => [2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[1,1,2,1,1,2,1] => [2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,1,2,1,1,3] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,2,2,1,1,1] => [2,2,3] => [[5,3,2],[2,1]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[1,1,2,2,1,2] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,2,2,2,1] => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,2,2,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,3,1,1,2] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,3,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,1,1,1,2,1] => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,1,1,2,1,1] => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,2,1,1,2,2] => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,1,1,3,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,2,1,1,1,1] => [1,2,4] => [[5,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,2,2,1,1,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,2,2,2,1,1] => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,3,1,1,2,1] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,3,3,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,1,1,1,2,1,1] => [1,3,1,2] => [[4,3,3,1],[2,2]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 2
[2,1,1,1,2,2] => [1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1,2,1,1,1] => [1,2,1,3] => [[4,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,1,2,2,1] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[2,1,1,3,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,1,2,2,1,1] => [1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,1,1,1,2] => [2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,2,1,1,2,1] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,1,1,3] => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> 1
[2,2,2,1,1,1] => [3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[3,1,1,2,1,1] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,1,1,2,2] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[3,2,2,1,1] => [1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> 1
[1,1,1,1,2,1,1,2] => [4,1,2,1] => [[5,5,4,4],[4,3,3]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,1,1,1,2,2,1,1] => [4,2,2] => [[6,5,4],[4,3]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001877
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 10%
Mp00192: Skew partitions —dominating sublattice⟶ Lattices
Mp00197: Lattices —lattice of congruences⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 10%
Values
[1] => [[1],[]]
=> ([],1)
=> ([],1)
=> ? = 1
[1,1] => [[1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {2,2}
[2] => [[2],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {2,2}
[1,1,1] => [[1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {3,3,3,3}
[1,2] => [[2,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {3,3,3,3}
[2,1] => [[2,2],[1]]
=> ([],1)
=> ([],1)
=> ? ∊ {3,3,3,3}
[3] => [[3],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {3,3,3,3}
[1,1,1,1] => [[1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,4,4,4,4,4,4,4}
[1,1,2] => [[2,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,4,4,4,4,4,4,4}
[1,2,1] => [[2,2,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,4,4,4,4,4,4,4}
[1,3] => [[3,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,4,4,4,4,4,4,4}
[2,1,1] => [[2,2,2],[1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,4,4,4,4,4,4,4}
[2,2] => [[3,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,4,4,4,4,4,4,4}
[3,1] => [[3,3],[2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,4,4,4,4,4,4,4}
[4] => [[4],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,4,4,4,4,4,4,4}
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[1,1,1,2] => [[2,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[1,1,3] => [[3,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[1,2,2] => [[3,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,1] => [[3,3,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[1,4] => [[4,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[2,1,2] => [[3,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[2,2,1] => [[3,3,2],[2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,3] => [[4,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[3,1,1] => [[3,3,3],[2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[3,2] => [[4,3],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[4,1] => [[4,4],[3]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[5] => [[5],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,5,5,5,5,5,5,5,5,5,5,5}
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,1,3] => [[3,1,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,4] => [[4,1,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,2,3] => [[4,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,3,2] => [[4,3,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,4,1] => [[4,4,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,5] => [[5,1],[]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,3] => [[4,2,2],[1,1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,2,2] => [[4,3,2],[2,1]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? ∊ {1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[2,3,1] => [[4,4,2],[3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,4] => [[5,2],[1]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[3,1,2] => [[4,3,3],[2,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[3,2,1] => [[4,4,3],[3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,3] => [[5,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,1,1] => [[4,4,4],[3,3]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[4,2] => [[5,4],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[5,1] => [[5,5],[4]]
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6}
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,3,2] => [[4,3,1,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,2,1,3] => [[4,2,2,1],[1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,2,4] => [[5,2,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,1,2] => [[4,3,3,1],[2,2]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,4,2] => [[5,4,1],[3]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,3,1] => [[4,4,2,2],[3,1,1]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,3,1,1] => [[4,4,4,2],[3,3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,4,1] => [[5,5,2],[4,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,1,2,1] => [[4,4,3,3],[3,2,2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,1,3] => [[5,3,3],[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,2,1,1] => [[4,4,4,3],[3,3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,4] => [[6,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,2,1] => [[5,5,4],[4,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,3] => [[6,4],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,1,1,2,1,1] => [[2,2,2,1,1,1,1],[1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,1,1,2,2] => [[3,2,1,1,1,1],[1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,1,3,2] => [[4,3,1,1,1],[2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,2,1,1,1,1] => [[2,2,2,2,2,1,1],[1,1,1,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,2,1,1,1,2] => [[3,2,2,2,2,1],[1,1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3,1,1,2] => [[4,3,3,3,1],[2,2,2]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,4,1,2] => [[5,4,4,1],[3,3]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[2,1,1,1,2,1] => [[3,3,2,2,2,2],[2,1,1,1,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,1,3,1] => [[4,4,2,2,2],[3,1,1,1]]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[2,2,1,1,1,1] => [[3,3,3,3,3,2],[2,2,2,2,1]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,3,1,1,1] => [[4,4,4,4,2],[3,3,3,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,4,1,1] => [[5,5,5,2],[4,4,1]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,1,1,2,1] => [[4,4,3,3,3],[3,2,2,2]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,1,1,3] => [[5,3,3,3],[2,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[4,1,2,1] => [[5,5,4,4],[4,3,3]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
Description
Number of indecomposable injective modules with projective dimension 2.
Matching statistic: St001880
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 50%
Values
[1] => [1,0]
=> [.,.]
=> ([],1)
=> ? = 1
[1,1] => [1,0,1,0]
=> [.,[.,.]]
=> ([(0,1)],2)
=> ? ∊ {2,2}
[2] => [1,1,0,0]
=> [[.,.],.]
=> ([(0,1)],2)
=> ? ∊ {2,2}
[1,1,1] => [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 3
[1,2] => [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 3
[2,1] => [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 3
[3] => [1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,3] => [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {1,4,4}
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,2] => [1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[3,1] => [1,1,1,0,0,0,1,0]
=> [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,4,4}
[4] => [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {1,4,4}
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? ∊ {1,1,1,1,1,5,5,5}
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {1,1,1,1,1,5,5,5}
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {1,1,1,1,1,5,5,5}
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? ∊ {1,1,1,1,1,5,5,5}
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,5,5,5}
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,5,5,5}
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,5,5,5}
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,5,5,5}
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[[.,.],[.,.]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6}
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [.,[.,[[.,.],[.,[.,.]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6}
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[[[.,.],.],[.,.]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6}
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [.,[[.,[[.,.],[.,.]]],.]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6}
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [.,[[.,.],[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6}
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [.,[[.,.],[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6}
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [.,[[[.,.],.],[.,[.,.]]]]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6}
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[[[.,.],[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6}
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [[.,[.,[[.,.],[.,.]]]],.]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6}
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [[.,[[.,[.,[.,.]]],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [[.,[[.,[[.,.],.]],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[.,[[.,.],[.,[.,.]]]],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6}
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[.,[[[.,.],.],[.,.]]],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6}
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [[.,.],[.,[.,[.,[.,.]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6}
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [[.,.],[.,[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6}
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [[.,.],[.,[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6}
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6}
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[[.,.],.],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6}
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[[.,.],.],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6}
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[.,.],[.,[.,.]]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6}
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[[.,.],[.,.]],[[.,.],.]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6}
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,[.,[.,[.,.]]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [.,[.,[.,[.,[.,[[.,.],.]]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [.,[.,[.,[.,[[.,[.,.]],.]]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [.,[.,[.,[.,[[.,.],[.,.]]]]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [.,[.,[.,[[.,[.,[.,.]]],.]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [.,[.,[.,[[.,[[.,.],.]],.]]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [.,[.,[.,[[.,.],[.,[.,.]]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7}
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [.,[.,[.,[[[.,.],.],[.,.]]]]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7}
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [.,[.,[[.,[.,[.,[.,.]]]],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [.,[.,[[.,[.,[[.,.],.]]],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [.,[.,[[.,[[.,[.,.]],.]],.]]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [.,[.,[[.,[[.,.],[.,.]]],.]]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7}
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [.,[.,[[.,.],[.,[.,[.,.]]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7}
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [.,[.,[[.,.],[.,[[.,.],.]]]]]
=> ([(0,6),(1,4),(3,6),(4,3),(5,2),(6,5)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7}
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [.,[.,[[[.,.],.],[.,[.,.]]]]]
=> ([(0,4),(1,3),(3,6),(4,6),(5,2),(6,5)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7}
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [.,[.,[[[.,.],[.,.]],[.,.]]]]
=> ([(0,6),(1,5),(2,5),(3,4),(5,6),(6,3)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7}
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [.,[[.,[.,[.,[.,[.,.]]]]],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [.,[[.,[.,[.,[[.,.],.]]]],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [.,[[.,[.,[[.,[.,.]],.]]],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [.,[[.,[.,[[.,.],[.,.]]]],.]]
=> ([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7}
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [.,[[.,[[.,[.,[.,.]]],.]],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [.,[[.,[[.,[[.,.],.]],.]],.]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [.,[[.,[[.,.],[.,[.,.]]]],.]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7}
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [.,[[.,[[[.,.],.],[.,.]]],.]]
=> ([(0,6),(1,3),(3,6),(4,2),(5,4),(6,5)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7}
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [.,[[.,.],[.,[.,[.,[.,.]]]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7}
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [.,[[.,.],[.,[.,[[.,.],.]]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7}
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [.,[[.,.],[.,[[.,[.,.]],.]]]]
=> ([(0,6),(1,5),(2,6),(4,2),(5,4),(6,3)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7}
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,[[.,.],[.,.]]]]]
=> ([(0,6),(1,5),(2,5),(4,6),(5,4),(6,3)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7}
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [.,[[[.,.],.],[.,[.,[.,.]]]]]
=> ([(0,4),(1,5),(2,6),(4,6),(5,2),(6,3)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7}
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[.,[.,[.,[.,[.,[.,.]]]]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [[.,[.,[.,[.,[[.,.],.]]]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [[.,[.,[.,[[.,[.,.]],.]]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [[.,[.,[[.,[.,[.,.]]],.]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [[.,[.,[[.,[[.,.],.]],.]]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [[.,[[.,[.,[.,[.,.]]]],.]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [[.,[[.,[.,[[.,.],.]]],.]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [[.,[[.,[[.,[.,.]],.]],.]],.]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001583
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 60%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 60%
Values
[1] => [1]
=> [1,0,1,0]
=> [1,2] => 1
[1,1] => [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 2
[2] => [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 2
[1,1,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[1,2] => [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 3
[1,1,1,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ? ∊ {1,4}
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4
[2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 4
[2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 4
[4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => ? ∊ {1,4}
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => ? ∊ {1,1,1,1,1,5,5,5}
[1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ? ∊ {1,1,1,1,1,5,5,5}
[1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ? ∊ {1,1,1,1,1,5,5,5}
[1,1,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5
[1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ? ∊ {1,1,1,1,1,5,5,5}
[1,2,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 5
[1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5
[1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ? ∊ {1,1,1,1,1,5,5,5}
[2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ? ∊ {1,1,1,1,1,5,5,5}
[2,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 5
[2,2,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 5
[2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5
[3,1,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 5
[3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 5
[4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => ? ∊ {1,1,1,1,1,5,5,5}
[5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => ? ∊ {1,1,1,1,1,5,5,5}
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6}
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6}
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6}
[1,1,1,3] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6}
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6}
[1,1,2,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6}
[1,1,3,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6}
[1,1,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6}
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6}
[1,2,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6}
[1,2,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6}
[1,2,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 6
[1,3,1,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6}
[1,3,2] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 6
[1,4,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6}
[1,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6}
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,6,4,5,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6}
[2,1,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6}
[2,1,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6}
[2,1,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 6
[2,2,1,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6}
[2,2,2] => [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6}
[2,3,1] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 6
[2,4] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6}
[3,1,1,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6}
[3,1,2] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 6
[3,2,1] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 6
[3,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6}
[4,1,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6}
[4,2] => [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6}
[5,1] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,3,4,2,1,6] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6}
[6] => [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6}
[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,1,1,1,2] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,7,6,4,5,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,1,1,2,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,7,6,4,5,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,1,1,3] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,6,4,3,5,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,1,2,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,7,6,4,5,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,1,2,2] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,6,3,5,4,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,1,3,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,6,4,3,5,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,1,4] => [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,2,1,1,1] => [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,7,6,4,5,3,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,2,1,2] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,6,3,5,4,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,2,2,1] => [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,6,3,5,4,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,2,3] => [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,3,1,1] => [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,6,4,3,5,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,3,2] => [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
Description
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
Matching statistic: St001637
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001637: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 40%
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001637: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 40%
Values
[1] => [1,0]
=> ([],1)
=> ? = 1 - 1
[1,1] => [1,0,1,0]
=> ([(0,1)],2)
=> 1 = 2 - 1
[2] => [1,1,0,0]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,1] => [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,2] => [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,1] => [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3] => [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,3] => [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3 = 4 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[2,2] => [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,1] => [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 4 - 1
[4] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? ∊ {1,1,1,1,1,5,5,5} - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? ∊ {1,1,1,1,1,5,5,5} - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? ∊ {1,1,1,1,1,5,5,5} - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? ∊ {1,1,1,1,1,5,5,5} - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? ∊ {1,1,1,1,1,5,5,5} - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? ∊ {1,1,1,1,1,5,5,5} - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? ∊ {1,1,1,1,1,5,5,5} - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,5,5,5} - 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,4),(4,6),(5,3),(6,1),(6,2)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4),(7,8)],9)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
Description
The number of (upper) dissectors of a poset.
Matching statistic: St001668
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001668: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 40%
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001668: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 40%
Values
[1] => [1,0]
=> ([],1)
=> ? = 1 - 1
[1,1] => [1,0,1,0]
=> ([(0,1)],2)
=> 1 = 2 - 1
[2] => [1,1,0,0]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,1,1] => [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,2] => [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[2,1] => [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[3] => [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 3 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,3] => [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 3 = 4 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[2,2] => [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[3,1] => [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 3 = 4 - 1
[4] => [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? ∊ {1,1,1,1,1,5,5,5} - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? ∊ {1,1,1,1,1,5,5,5} - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? ∊ {1,1,1,1,1,5,5,5} - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? ∊ {1,1,1,1,1,5,5,5} - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? ∊ {1,1,1,1,1,5,5,5} - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? ∊ {1,1,1,1,1,5,5,5} - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? ∊ {1,1,1,1,1,5,5,5} - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? ∊ {1,1,1,1,1,5,5,5} - 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6} - 1
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,4),(4,6),(5,3),(6,1),(6,2)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4),(7,8)],9)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7} - 1
Description
The number of points of the poset minus the width of the poset.
Matching statistic: St001232
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 60%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 60%
Values
[1] => 1 => [1,1] => [1,0,1,0]
=> 1
[1,1] => 11 => [1,1,1] => [1,0,1,0,1,0]
=> ? = 2
[2] => 10 => [1,2] => [1,0,1,1,0,0]
=> 2
[1,1,1] => 111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> ? ∊ {3,3}
[1,2] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> ? ∊ {3,3}
[2,1] => 101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 3
[3] => 100 => [1,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,1,1,1] => 1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,4,4,4,4}
[1,1,2] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,4,4,4,4}
[1,2,1] => 1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,4,4,4,4}
[1,3] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,4,4,4,4}
[2,1,1] => 1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {1,4,4,4,4}
[2,2] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[3,1] => 1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 4
[4] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 4
[1,1,1,1,1] => 11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5}
[1,1,1,2] => 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5}
[1,1,2,1] => 11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5}
[1,1,3] => 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5}
[1,2,1,1] => 11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5}
[1,2,2] => 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5}
[1,3,1] => 11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5}
[1,4] => 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5}
[2,1,1,1] => 10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5}
[2,1,2] => 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5}
[2,2,1] => 10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[2,3] => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 5
[3,1,1] => 10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,5,5,5,5,5,5}
[3,2] => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
[4,1] => 10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
[5] => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,1,1,1,1] => 111111 => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6}
[1,1,1,1,2] => 111110 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6}
[1,1,1,2,1] => 111101 => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6}
[1,1,1,3] => 111100 => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6}
[1,1,2,1,1] => 111011 => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6}
[1,1,2,2] => 111010 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6}
[1,1,3,1] => 111001 => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6}
[1,1,4] => 111000 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6}
[1,2,1,1,1] => 110111 => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6}
[1,2,1,2] => 110110 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6}
[1,2,2,1] => 110101 => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6}
[1,2,3] => 110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6}
[1,3,1,1] => 110011 => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6}
[1,3,2] => 110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6}
[1,4,1] => 110001 => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6}
[1,5] => 110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6}
[2,1,1,1,1] => 101111 => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6}
[2,1,1,2] => 101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6}
[2,1,2,1] => 101101 => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6}
[2,1,3] => 101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6}
[2,2,1,1] => 101011 => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6}
[2,2,2] => 101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 6
[2,3,1] => 101001 => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> 6
[2,4] => 101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> 6
[3,1,1,1] => 100111 => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6}
[3,1,2] => 100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6}
[3,2,1] => 100101 => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> 6
[3,3] => 100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> 6
[4,1,1] => 100011 => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6,6,6,6,6,6,6,6}
[4,2] => 100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> 6
[5,1] => 100001 => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[6] => 100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[1,1,1,1,1,1,1] => 1111111 => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,1,1,1,2] => 1111110 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,1,1,2,1] => 1111101 => [1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,1,1,3] => 1111100 => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,1,2,1,1] => 1111011 => [1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,1,2,2] => 1111010 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
[1,1,1,3,1] => 1111001 => [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7}
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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