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Your data matches 3 different statistics following compositions of up to 3 maps.
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Matching statistic: St001641
St001641: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> 0
{{1,2}}
=> 1
{{1},{2}}
=> 1
{{1,2,3}}
=> 2
{{1,2},{3}}
=> 2
{{1,3},{2}}
=> 0
{{1},{2,3}}
=> 2
{{1},{2},{3}}
=> 2
{{1,2,3,4}}
=> 3
{{1,2,3},{4}}
=> 3
{{1,2,4},{3}}
=> 1
{{1,2},{3,4}}
=> 3
{{1,2},{3},{4}}
=> 3
{{1,3,4},{2}}
=> 0
{{1,3},{2,4}}
=> 1
{{1,3},{2},{4}}
=> 1
{{1,4},{2,3}}
=> 1
{{1},{2,3,4}}
=> 3
{{1},{2,3},{4}}
=> 3
{{1,4},{2},{3}}
=> 1
{{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> 3
{{1},{2},{3},{4}}
=> 3
{{1,2,3,4,5}}
=> 4
{{1,2,3,4},{5}}
=> 4
{{1,2,3,5},{4}}
=> 2
{{1,2,3},{4,5}}
=> 4
{{1,2,3},{4},{5}}
=> 4
{{1,2,4,5},{3}}
=> 1
{{1,2,4},{3,5}}
=> 2
{{1,2,4},{3},{5}}
=> 2
{{1,2,5},{3,4}}
=> 2
{{1,2},{3,4,5}}
=> 4
{{1,2},{3,4},{5}}
=> 4
{{1,2,5},{3},{4}}
=> 2
{{1,2},{3,5},{4}}
=> 2
{{1,2},{3},{4,5}}
=> 4
{{1,2},{3},{4},{5}}
=> 4
{{1,3,4,5},{2}}
=> 0
{{1,3,4},{2,5}}
=> 1
{{1,3,4},{2},{5}}
=> 1
{{1,3,5},{2,4}}
=> 1
{{1,3},{2,4,5}}
=> 2
{{1,3},{2,4},{5}}
=> 2
{{1,3,5},{2},{4}}
=> 1
{{1,3},{2,5},{4}}
=> 0
{{1,3},{2},{4,5}}
=> 2
{{1,3},{2},{4},{5}}
=> 2
{{1,4,5},{2,3}}
=> 1
{{1,4},{2,3,5}}
=> 2
Description
The number of ascent tops in the flattened set partition such that all smaller elements appear before.
Let $P$ be a set partition. The flattened set partition is the permutation obtained by sorting the set of blocks of $P$ according to their minimal element and the elements in each block in increasing order.
Given a set partition $P$, this statistic is the binary logarithm of the number of set partitions that flatten to the same permutation as $P$.
Matching statistic: St001640
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 0
{{1,2}}
=> [2,1] => [1,2] => 1
{{1},{2}}
=> [1,2] => [1,2] => 1
{{1,2,3}}
=> [2,3,1] => [1,2,3] => 2
{{1,2},{3}}
=> [2,1,3] => [1,2,3] => 2
{{1,3},{2}}
=> [3,2,1] => [1,3,2] => 0
{{1},{2,3}}
=> [1,3,2] => [1,2,3] => 2
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 2
{{1,2,3,4}}
=> [2,3,4,1] => [1,2,3,4] => 3
{{1,2,3},{4}}
=> [2,3,1,4] => [1,2,3,4] => 3
{{1,2,4},{3}}
=> [2,4,3,1] => [1,2,4,3] => 1
{{1,2},{3,4}}
=> [2,1,4,3] => [1,2,3,4] => 3
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,2,3,4] => 3
{{1,3,4},{2}}
=> [3,2,4,1] => [1,3,4,2] => 0
{{1,3},{2,4}}
=> [3,4,1,2] => [1,3,2,4] => 1
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,3,2,4] => 1
{{1,4},{2,3}}
=> [4,3,2,1] => [1,4,2,3] => 1
{{1},{2,3,4}}
=> [1,3,4,2] => [1,2,3,4] => 3
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,2,3,4] => 3
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,4,2,3] => 1
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,2,4,3] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,3,4] => 3
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 3
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,2,3,4,5] => 4
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,2,3,4,5] => 4
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,2,3,5,4] => 2
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,2,3,4,5] => 4
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,2,3,4,5] => 4
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,2,4,5,3] => 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,2,4,3,5] => 2
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,2,4,3,5] => 2
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,2,5,3,4] => 2
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,2,3,4,5] => 4
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,2,3,4,5] => 4
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,2,5,3,4] => 2
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,2,3,5,4] => 2
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,2,3,4,5] => 4
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,2,3,4,5] => 4
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [1,3,4,5,2] => 0
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [1,3,4,2,5] => 1
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [1,3,4,2,5] => 1
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [1,3,5,2,4] => 1
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [1,3,2,4,5] => 2
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [1,3,2,4,5] => 2
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [1,3,5,2,4] => 1
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [1,3,2,5,4] => 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [1,3,2,4,5] => 2
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [1,3,2,4,5] => 2
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [1,4,5,2,3] => 1
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [1,4,2,3,5] => 2
Description
The number of ascent tops in the permutation such that all smaller elements appear before.
Matching statistic: St001879
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 67%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 67%
Values
{{1}}
=> [1] => [.,.]
=> ([],1)
=> ? = 0
{{1,2}}
=> [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ? ∊ {1,1}
{{1},{2}}
=> [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ? ∊ {1,1}
{{1,2,3}}
=> [2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 2
{{1,2},{3}}
=> [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 0
{{1,3},{2}}
=> [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 2
{{1},{2,3}}
=> [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 2
{{1},{2},{3}}
=> [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
{{1,2,3,4}}
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
{{1,2,3},{4}}
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,3}
{{1,2,4},{3}}
=> [2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
{{1,2},{3,4}}
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,3}
{{1,2},{3},{4}}
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,3}
{{1,3,4},{2}}
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {0,1,1,1,1,1,1,3}
{{1,3},{2,4}}
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,3}
{{1,3},{2},{4}}
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,1,1,1,1,1,1,3}
{{1,4},{2,3}}
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
{{1},{2,3,4}}
=> [1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
{{1},{2,3},{4}}
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {0,1,1,1,1,1,1,3}
{{1,4},{2},{3}}
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? ∊ {0,1,1,1,1,1,1,3}
{{1},{2,4},{3}}
=> [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
{{1},{2},{3,4}}
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
{{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] => [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,3,4},{2,5}}
=> [3,5,4,1,2] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,3,5},{2,4}}
=> [3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
{{1,3},{2,4,5}}
=> [3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,4,5},{2,3}}
=> [4,3,2,5,1] => [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,4},{2,3,5}}
=> [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,5},{2,3,4}}
=> [5,3,4,2,1] => [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,4,4,4}
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
{{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}}
=> [2,3,4,5,6,1] => [[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [[.,[.,[[[.,.],.],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
{{1,2,4,6},{3,5}}
=> [2,4,5,6,3,1] => [[.,[[.,[.,[.,.]]],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
{{1,2,6},{3,5},{4}}
=> [2,6,5,4,3,1] => [[.,[[[[.,.],.],.],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
{{1,3,6},{2,4,5}}
=> [3,4,6,5,2,1] => [[[.,[.,[[.,.],.]]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
{{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => [[[.,[[.,[.,.]],.]],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
{{1},{2,3,4,5,6}}
=> [1,3,4,5,6,2] => [.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
{{1},{2,3,4,6},{5}}
=> [1,3,4,6,5,2] => [.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
{{1},{2,3,6},{4,5}}
=> [1,3,6,5,4,2] => [.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
{{1},{2,4,6},{3,5}}
=> [1,4,5,6,3,2] => [.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
{{1,6},{2,5},{3,4}}
=> [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
{{1},{2},{3,4,5,6}}
=> [1,2,4,5,6,3] => [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
{{1},{2},{3,4,6},{5}}
=> [1,2,4,6,5,3] => [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
{{1},{2,6},{3,5},{4}}
=> [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
{{1},{2},{3,6},{4,5}}
=> [1,2,6,5,4,3] => [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
{{1},{2},{3},{4,5,6}}
=> [1,2,3,5,6,4] => [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
{{1},{2},{3},{4,6},{5}}
=> [1,2,3,6,5,4] => [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
{{1},{2},{3},{4},{5,6}}
=> [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
{{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
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.
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