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Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St000579
St000579: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> 0
{{1},{2}}
=> 1
{{1,2,3}}
=> 0
{{1,2},{3}}
=> 2
{{1,3},{2}}
=> 2
{{1},{2,3}}
=> 1
{{1},{2},{3}}
=> 3
{{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> 3
{{1,2,4},{3}}
=> 3
{{1,2},{3,4}}
=> 2
{{1,2},{3},{4}}
=> 5
{{1,3,4},{2}}
=> 2
{{1,3},{2,4}}
=> 3
{{1,3},{2},{4}}
=> 5
{{1,4},{2,3}}
=> 3
{{1},{2,3,4}}
=> 1
{{1},{2,3},{4}}
=> 4
{{1,4},{2},{3}}
=> 5
{{1},{2,4},{3}}
=> 4
{{1},{2},{3,4}}
=> 3
{{1},{2},{3},{4}}
=> 6
{{1,2,3,4,5}}
=> 0
{{1,2,3,4},{5}}
=> 4
{{1,2,3,5},{4}}
=> 4
{{1,2,3},{4,5}}
=> 3
{{1,2,3},{4},{5}}
=> 7
{{1,2,4,5},{3}}
=> 3
{{1,2,4},{3,5}}
=> 4
{{1,2,4},{3},{5}}
=> 7
{{1,2,5},{3,4}}
=> 4
{{1,2},{3,4,5}}
=> 2
{{1,2},{3,4},{5}}
=> 6
{{1,2,5},{3},{4}}
=> 7
{{1,2},{3,5},{4}}
=> 6
{{1,2},{3},{4,5}}
=> 5
{{1,2},{3},{4},{5}}
=> 9
{{1,3,4,5},{2}}
=> 2
{{1,3,4},{2,5}}
=> 4
{{1,3,4},{2},{5}}
=> 6
{{1,3,5},{2,4}}
=> 4
{{1,3},{2,4,5}}
=> 3
{{1,3},{2,4},{5}}
=> 7
{{1,3,5},{2},{4}}
=> 6
{{1,3},{2,5},{4}}
=> 7
{{1,3},{2},{4,5}}
=> 5
{{1,3},{2},{4},{5}}
=> 9
{{1,4,5},{2,3}}
=> 3
{{1,4},{2,3,5}}
=> 4
{{1,4},{2,3},{5}}
=> 7
Description
The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element.
This is the number of pairs $i\lt j$ in different blocks such that $j$ is the maximal element of a block.
Matching statistic: St001161
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001161: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001161: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2,1] => [2,1] => [1,1,0,0]
=> 0
{{1},{2}}
=> [1,2] => [1,2] => [1,0,1,0]
=> 1
{{1,2,3}}
=> [2,3,1] => [3,1,2] => [1,1,1,0,0,0]
=> 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 5
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 3
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 5
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 4
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 5
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 4
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 6
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 4
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 4
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 3
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 7
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 3
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 4
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 7
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 4
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 2
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 6
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 7
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> 6
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 5
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 9
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 2
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 4
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 6
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 4
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> 3
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> 7
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> 6
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 7
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> 5
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 9
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> 3
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> 4
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> 7
Description
The major index north count of a Dyck path.
The descent set $\operatorname{des}(D)$ of a Dyck path $D = D_1 \cdots D_{2n}$ with $D_i \in \{N,E\}$ is given by all indices $i$ such that $D_i = E$ and $D_{i+1} = N$. This is, the positions of the valleys of $D$.
The '''major index''' of a Dyck path is then the sum of the positions of the valleys, $\sum_{i \in \operatorname{des}(D)} i$, see [[St000027]].
The '''major index north count''' is given by $\sum_{i \in \operatorname{des}(D)} \#\{ j \leq i \mid D_j = N\}$.
Matching statistic: St001232
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 59%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 59%
Values
{{1,2}}
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 1
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 3
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {3,3,4,5,5,5,6}
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 3
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {3,3,4,5,5,5,6}
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,4,5,5,5,6}
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,4,5,5,5,6}
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {3,3,4,5,5,5,6}
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 4
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {3,3,4,5,5,5,6}
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> ? ∊ {3,3,4,5,5,5,6}
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 5
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 6
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 6
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,15}
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 7
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 4
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 6
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,15}
{{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,15}
{{1,2,3},{4,5},{6}}
=> [2,3,1,5,4,6] => [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,15}
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,15}
{{1,2,3},{4,6},{5}}
=> [2,3,1,6,5,4] => [1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> ? ∊ {3,3,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,13,13,13,13,14,14,14,14,14,15}
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 7
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 8
{{1,2,4,6},{3,5}}
=> [2,4,5,6,3,1] => [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> 3
{{1,2,4},{3,5,6}}
=> [2,4,5,1,6,3] => [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 5
{{1,2,4},{3,5},{6}}
=> [2,4,5,1,3,6] => [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> 5
{{1,2,4},{3,6},{5}}
=> [2,4,6,1,5,3] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 7
{{1,2},{3},{4},{5},{6}}
=> [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
{{1},{2,3,4,5,6}}
=> [1,3,4,5,6,2] => [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
{{1},{2,3,4,5},{6}}
=> [1,3,4,5,2,6] => [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 4
{{1},{2,3,4,6},{5}}
=> [1,3,4,6,5,2] => [1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001879
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 55%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 55%
Values
{{1,2}}
=> [2,1] => [2,1] => ([],2)
=> ? ∊ {0,1}
{{1},{2}}
=> [1,2] => [1,2] => ([(0,1)],2)
=> ? ∊ {0,1}
{{1,2,3}}
=> [2,3,1] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {0,1,2,3}
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ? ∊ {0,1,2,3}
{{1,3},{2}}
=> [3,2,1] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {0,1,2,3}
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ? ∊ {0,1,2,3}
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? ∊ {0,1,2,2,3,3,3,3,4,5,5,5,6}
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,1,2,2,3,3,3,3,4,5,5,5,6}
{{1,2,4},{3}}
=> [2,4,3,1] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {0,1,2,2,3,3,3,3,4,5,5,5,6}
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ? ∊ {0,1,2,2,3,3,3,3,4,5,5,5,6}
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {0,1,2,2,3,3,3,3,4,5,5,5,6}
{{1,3,4},{2}}
=> [3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {0,1,2,2,3,3,3,3,4,5,5,5,6}
{{1,3},{2,4}}
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {0,1,2,2,3,3,3,3,4,5,5,5,6}
{{1,3},{2},{4}}
=> [3,2,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,1,2,2,3,3,3,3,4,5,5,5,6}
{{1,4},{2,3}}
=> [4,3,2,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> ? ∊ {0,1,2,2,3,3,3,3,4,5,5,5,6}
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {0,1,2,2,3,3,3,3,4,5,5,5,6}
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
{{1,4},{2},{3}}
=> [4,2,3,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ? ∊ {0,1,2,2,3,3,3,3,4,5,5,5,6}
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {0,1,2,2,3,3,3,3,4,5,5,5,6}
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ? ∊ {0,1,2,2,3,3,3,3,4,5,5,5,6}
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,5,3,4] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [4,1,3,5,2] => ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [3,1,5,2,4] => ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [4,1,5,2,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 6
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
{{1},{2,3,4,5},{6}}
=> [1,3,4,5,2,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 8
{{1},{2,3,4},{5},{6}}
=> [1,3,4,2,5,6] => [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 7
{{1},{2,3,5},{4},{6}}
=> [1,3,5,4,2,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 8
{{1},{2,3},{4},{5},{6}}
=> [1,3,2,4,5,6] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 6
{{1},{2,4,5},{3},{6}}
=> [1,4,3,5,2,6] => [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 7
{{1},{2,4},{3,5},{6}}
=> [1,4,5,2,3,6] => [1,4,2,5,3,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 7
{{1},{2,4},{3},{5},{6}}
=> [1,4,3,2,5,6] => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 7
{{1},{2,5},{3,4},{6}}
=> [1,5,4,3,2,6] => [1,4,3,5,2,6] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 10
{{1},{2},{3,4,5},{6}}
=> [1,2,4,5,3,6] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 7
{{1},{2},{3,4},{5},{6}}
=> [1,2,4,3,5,6] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 6
{{1},{2,5},{3},{4},{6}}
=> [1,5,3,4,2,6] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 8
{{1},{2},{3,5},{4},{6}}
=> [1,2,5,4,3,6] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 7
{{1},{2},{3},{4,5},{6}}
=> [1,2,3,5,4,6] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 6
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
{{1},{2,3,4,5,6},{7}}
=> [1,3,4,5,6,2,7] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7)
=> 10
{{1},{2,3,4,5},{6},{7}}
=> [1,3,4,5,2,6,7] => [1,5,2,3,4,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> 9
{{1},{2,3,4,6},{5},{7}}
=> [1,3,4,6,5,2,7] => [1,5,6,2,3,4,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
=> 10
{{1},{2,3,4},{5},{6},{7}}
=> [1,3,4,2,5,6,7] => [1,4,2,3,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> 8
{{1},{2,3,5,6},{4},{7}}
=> [1,3,5,4,6,2,7] => [1,4,6,2,3,5,7] => ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7)
=> 9
{{1},{2,3,5},{4,6},{7}}
=> [1,3,5,6,2,4,7] => [1,5,2,3,6,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=> 9
{{1},{2,3,5},{4},{6},{7}}
=> [1,3,5,4,2,6,7] => [1,4,5,2,3,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7)
=> 9
{{1},{2,3,6},{4,5},{7}}
=> [1,3,6,5,4,2,7] => [1,5,4,6,2,3,7] => ([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> 13
{{1},{2,3,6},{4},{5},{7}}
=> [1,3,6,4,5,2,7] => [1,4,5,6,2,3,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7)
=> 10
{{1},{2,3},{4},{5,6},{7}}
=> [1,3,2,4,6,5,7] => [1,3,2,4,6,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 8
{{1},{2,3},{4},{5},{6},{7}}
=> [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 7
{{1},{2,4,5,6},{3},{7}}
=> [1,4,3,5,6,2,7] => [1,3,6,2,4,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> 9
{{1},{2,4,5},{3,6},{7}}
=> [1,4,6,5,2,3,7] => [1,5,2,4,6,3,7] => ([(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(4,2),(5,6)],7)
=> 11
{{1},{2,4,5},{3},{6},{7}}
=> [1,4,3,5,2,6,7] => [1,3,5,2,4,6,7] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 8
{{1},{2,4,6},{3,5},{7}}
=> [1,4,5,6,3,2,7] => [1,5,3,6,2,4,7] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 11
{{1},{2,4},{3,5},{6},{7}}
=> [1,4,5,2,3,6,7] => [1,4,2,5,3,6,7] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 8
{{1},{2,4,6},{3},{5},{7}}
=> [1,4,3,6,5,2,7] => [1,3,5,6,2,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> 9
{{1},{2,4},{3,6},{5},{7}}
=> [1,4,6,2,5,3,7] => [1,4,2,5,6,3,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7)
=> 9
{{1},{2,4},{3},{5},{6},{7}}
=> [1,4,3,2,5,6,7] => [1,3,4,2,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7)
=> 8
{{1},{2,5},{3,4,6},{7}}
=> [1,5,4,6,2,3,7] => [1,5,2,6,3,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7)
=> 9
{{1},{2,5},{3,4},{6},{7}}
=> [1,5,4,3,2,6,7] => [1,4,3,5,2,6,7] => ([(0,2),(0,3),(0,4),(2,6),(3,5),(4,5),(5,6),(6,1)],7)
=> 11
{{1},{2,6},{3,4,5},{7}}
=> [1,6,4,5,3,2,7] => [1,5,3,4,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,5),(4,1),(5,6)],7)
=> 13
{{1},{2},{3,4,5,6},{7}}
=> [1,2,4,5,6,3,7] => [1,2,6,3,4,5,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7)
=> 9
{{1},{2},{3,4,5},{6},{7}}
=> [1,2,4,5,3,6,7] => [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7)
=> 8
{{1},{2,6},{3,4},{5},{7}}
=> [1,6,4,3,5,2,7] => [1,4,3,5,6,2,7] => ([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> 12
{{1},{2},{3,4,6},{5},{7}}
=> [1,2,4,6,5,3,7] => [1,2,5,6,3,4,7] => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7)
=> 9
{{1},{2},{3,4},{5},{6},{7}}
=> [1,2,4,3,5,6,7] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 7
{{1},{2,5,6},{3},{4},{7}}
=> [1,5,3,4,6,2,7] => [1,3,4,6,2,5,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7)
=> 9
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St001583
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 32%
Mp00064: Permutations —reverse⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 32%
Values
{{1,2}}
=> [2,1] => [1,2] => [1,2] => 1
{{1},{2}}
=> [1,2] => [2,1] => [2,1] => 0
{{1,2,3}}
=> [2,3,1] => [1,3,2] => [1,3,2] => 2
{{1,2},{3}}
=> [2,1,3] => [3,1,2] => [3,1,2] => 1
{{1,3},{2}}
=> [3,2,1] => [1,2,3] => [1,2,3] => 3
{{1},{2,3}}
=> [1,3,2] => [2,3,1] => [1,3,2] => 2
{{1},{2},{3}}
=> [1,2,3] => [3,2,1] => [3,2,1] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [1,4,3,2] => [1,4,3,2] => 3
{{1,2,3},{4}}
=> [2,3,1,4] => [4,1,3,2] => [4,1,3,2] => 2
{{1,2,4},{3}}
=> [2,4,3,1] => [1,3,4,2] => [1,2,4,3] => 5
{{1,2},{3,4}}
=> [2,1,4,3] => [3,4,1,2] => [2,4,1,3] => 3
{{1,2},{3},{4}}
=> [2,1,3,4] => [4,3,1,2] => [4,3,1,2] => 1
{{1,3,4},{2}}
=> [3,2,4,1] => [1,4,2,3] => [1,4,2,3] => 4
{{1,3},{2,4}}
=> [3,4,1,2] => [2,1,4,3] => [2,1,4,3] => 4
{{1,3},{2},{4}}
=> [3,2,1,4] => [4,1,2,3] => [4,1,2,3] => 3
{{1,4},{2,3}}
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 6
{{1},{2,3,4}}
=> [1,3,4,2] => [2,4,3,1] => [1,4,3,2] => 3
{{1},{2,3},{4}}
=> [1,3,2,4] => [4,2,3,1] => [4,1,3,2] => 2
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,3,2,4] => [1,3,2,4] => 5
{{1},{2,4},{3}}
=> [1,4,3,2] => [2,3,4,1] => [1,2,4,3] => 5
{{1},{2},{3,4}}
=> [1,2,4,3] => [3,4,2,1] => [1,4,3,2] => 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,5,4,3,2] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [5,1,4,3,2] => [5,1,4,3,2] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,4,5,3,2] => [1,2,5,4,3] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [4,5,1,3,2] => [2,5,1,4,3] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [5,4,1,3,2] => [5,4,1,3,2] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,5,3,4,2] => [1,5,2,4,3] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [3,1,5,4,2] => [2,1,5,4,3] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [5,1,3,4,2] => [5,1,2,4,3] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,3,4,5,2] => [1,2,3,5,4] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [3,5,4,1,2] => [2,5,4,1,3] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [5,3,4,1,2] => [5,2,4,1,3] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,4,3,5,2] => [1,3,2,5,4] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [3,4,5,1,2] => [1,3,5,2,4] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [4,5,3,1,2] => [2,5,4,1,3] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [5,4,3,1,2] => [5,4,3,1,2] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,5,4,2,3] => [1,5,4,2,3] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [2,1,4,5,3] => [2,1,3,5,4] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [5,1,4,2,3] => [5,1,4,2,3] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,2,5,4,3] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [2,5,1,4,3] => [2,5,1,4,3] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [5,2,1,4,3] => [5,2,1,4,3] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,4,5,2,3] => [1,3,5,2,4] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [2,4,1,5,3] => [1,3,2,5,4] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [4,5,1,2,3] => [3,5,1,2,4] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [5,4,1,2,3] => [5,4,1,2,3] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,5,2,3,4] => [1,5,2,3,4] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [2,1,5,3,4] => [2,1,5,3,4] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [5,1,2,3,4] => [5,1,2,3,4] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [1,2,4,3,5] => [1,2,4,3,5] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [2,5,4,3,1] => [1,5,4,3,2] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [5,2,4,3,1] => [5,1,4,3,2] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [1,4,2,3,5] => [1,4,2,3,5] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [2,4,5,3,1] => [1,2,5,4,3] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [4,5,2,3,1] => [2,5,1,4,3] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [5,4,2,3,1] => [5,4,1,3,2] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [1,5,3,2,4] => [1,5,3,2,4] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [2,1,3,5,4] => [2,1,3,5,4] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [3,1,5,2,4] => [3,1,5,2,4] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [5,1,3,2,4] => [5,1,3,2,4] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [2,5,3,4,1] => [1,5,2,4,3] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [3,2,5,4,1] => [2,1,5,4,3] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [5,2,3,4,1] => [5,1,2,4,3] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [1,3,4,2,5] => [1,2,4,3,5] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [2,3,4,5,1] => [1,2,3,5,4] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [3,5,4,2,1] => [1,5,4,3,2] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [5,3,4,2,1] => [5,1,4,3,2] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [1,4,3,2,5] => [1,4,3,2,5] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [2,4,3,5,1] => [1,3,2,5,4] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [3,4,5,2,1] => [1,2,5,4,3] => ? ∊ {0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,9,9,9,9,10}
Description
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
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