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Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St001074
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Mp00201: Dyck paths —Ringel⟶ Permutations
St001074: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001074: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => 2
[1,0,1,0]
=> [3,1,2] => 3
[1,1,0,0]
=> [2,3,1] => 5
[1,0,1,0,1,0]
=> [4,1,2,3] => 4
[1,0,1,1,0,0]
=> [3,1,4,2] => 6
[1,1,0,0,1,0]
=> [2,4,1,3] => 6
[1,1,0,1,0,0]
=> [4,3,1,2] => 4
[1,1,1,0,0,0]
=> [2,3,4,1] => 8
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 5
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 7
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 9
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 9
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 9
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 9
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 11
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 5
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 5
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 9
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 9
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 9
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 7
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 11
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => 6
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 8
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => 10
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 10
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => 10
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => 12
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => 12
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => 10
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 12
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => 12
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => 10
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => 12
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => 12
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => 12
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => 12
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => 12
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => 14
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => 16
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => 14
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => 6
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => 8
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => 6
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 8
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => 10
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => 12
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => 10
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => 10
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => 12
Description
The number of inversions of the cyclic embedding of a permutation.
The cyclic embedding of a permutation $\pi$ of length $n$ is given by the permutation of length $n+1$ represented in cycle notation by $(\pi_1,\ldots,\pi_n,n+1)$.
This reflects in particular the fact that the number of long cycles of length $n+1$ equals $n!$.
This statistic counts the number of inversions of this embedding, see [1]. As shown in [2], the sum of this statistic on all permutations of length $n$ equals $n!\cdot(3n-1)/12$.
Matching statistic: St001880
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 31%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 31%
Values
[1,0]
=> [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ? = 2
[1,0,1,0]
=> [3,1,2] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 3
[1,1,0,0]
=> [2,3,1] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 5
[1,0,1,0,1,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,0,1,1,0,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {6,6,8}
[1,1,0,0,1,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {6,6,8}
[1,1,0,1,0,0]
=> [4,3,1,2] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,1,1,0,0,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {6,6,8}
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {7,7,9,9,9,9,9,9,9,11,11}
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {7,7,9,9,9,9,9,9,9,11,11}
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {7,7,9,9,9,9,9,9,9,11,11}
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {7,7,9,9,9,9,9,9,9,11,11}
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? ∊ {7,7,9,9,9,9,9,9,9,11,11}
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {7,7,9,9,9,9,9,9,9,11,11}
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {7,7,9,9,9,9,9,9,9,11,11}
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {7,7,9,9,9,9,9,9,9,11,11}
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {7,7,9,9,9,9,9,9,9,11,11}
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? ∊ {7,7,9,9,9,9,9,9,9,11,11}
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {7,7,9,9,9,9,9,9,9,11,11}
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [[.,[.,[.,[.,.]]]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [[.,[.,[.,.]]],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [[.,[.,[.,.]]],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [[.,[.,.]],[[.,[.,.]],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [[.,[.,.]],[[.,.],[.,.]]]
=> ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [[.,[[.,[.,.]],[.,.]]],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [[.,[[.,[.,[.,.]]],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [[.,[[.,[.,.]],.]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [[.,[.,.]],[.,[[.,.],.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [[.,[.,.]],[[[.,.],.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [[.,[[.,[.,.]],[.,.]]],.]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [[.,[.,.]],[.,[.,[.,.]]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [[.,.],[[.,[.,[.,.]]],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [[.,.],[[.,[.,.]],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [[.,.],[[.,.],[[.,.],.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [[.,.],[[[.,[.,.]],.],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [[.,.],[[.,.],[.,[.,.]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [[[.,[.,.]],[.,[.,.]]],.]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [[[.,[.,.]],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [[[.,[.,[.,.]]],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [[.,[.,[.,[.,.]]]],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [[[.,[.,[.,.]]],.],[.,.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [[[.,[.,.]],.],[[.,.],.]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [[[.,[.,.]],[[.,.],.]],.]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [[[.,[.,[.,.]]],[.,.]],.]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [[[.,[.,.]],.],[.,[.,.]]]
=> ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => [[.,.],[.,[[.,[.,.]],.]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [[.,.],[.,[[.,.],[.,.]]]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => [[.,.],[[[.,.],[.,.]],.]]
=> ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => [[.,.],[[[.,[.,.]],.],.]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [[.,.],[[[.,.],.],[.,.]]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => [[[.,[.,.]],[.,[.,.]]],.]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,1,0,1,0,0,1,0,0]
=> [6,3,5,1,2,4] => [[[.,[.,.]],[[.,.],.]],.]
=> ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => [[[[.,[.,[.,.]]],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [[[.,[.,.]],[.,.]],[.,.]]
=> ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => [[.,.],[.,[.,[[.,.],.]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001645
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 38%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 38%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1 = 2 - 1
[1,0,1,0]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 2 = 3 - 1
[1,1,0,0]
=> [1,2] => [1,2] => ([],2)
=> ? = 5 - 1
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {6,6,8} - 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,1,0,0,1,0]
=> [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 4 - 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {6,6,8} - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {6,6,8} - 1
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {5,7,7,9,9,9,9,9,9,9,11,11} - 1
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 5 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {5,7,7,9,9,9,9,9,9,9,11,11} - 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {5,7,7,9,9,9,9,9,9,9,11,11} - 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {5,7,7,9,9,9,9,9,9,9,11,11} - 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {5,7,7,9,9,9,9,9,9,9,11,11} - 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {5,7,7,9,9,9,9,9,9,9,11,11} - 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {5,7,7,9,9,9,9,9,9,9,11,11} - 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {5,7,7,9,9,9,9,9,9,9,11,11} - 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {5,7,7,9,9,9,9,9,9,9,11,11} - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {5,7,7,9,9,9,9,9,9,9,11,11} - 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {5,7,7,9,9,9,9,9,9,9,11,11} - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {5,7,7,9,9,9,9,9,9,9,11,11} - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 6 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,5,2,3] => [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [4,2,3,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,4,2,3,5] => [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
Description
The pebbling number of a connected graph.
Matching statistic: St001232
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 38%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 38%
Values
[1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 5
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {4,6,6,8}
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {4,6,6,8}
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {4,6,6,8}
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {4,6,6,8}
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 5
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {5,7,7,9,9,9,9,9,9,9,11,11}
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {5,7,7,9,9,9,9,9,9,9,11,11}
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {5,7,7,9,9,9,9,9,9,9,11,11}
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> ? ∊ {5,7,7,9,9,9,9,9,9,9,11,11}
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {5,7,7,9,9,9,9,9,9,9,11,11}
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {5,7,7,9,9,9,9,9,9,9,11,11}
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {5,7,7,9,9,9,9,9,9,9,11,11}
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {5,7,7,9,9,9,9,9,9,9,11,11}
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {5,7,7,9,9,9,9,9,9,9,11,11}
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {5,7,7,9,9,9,9,9,9,9,11,11}
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {5,7,7,9,9,9,9,9,9,9,11,11}
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {5,7,7,9,9,9,9,9,9,9,11,11}
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> 6
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> 6
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> 8
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {6,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16}
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000957
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000957: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 46%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000957: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 46%
Values
[1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 6 = 5 + 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4 = 3 + 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [6,5,4,1,2,3] => 5 = 4 + 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => 7 = 6 + 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2,6,4,5,1,3] => 7 = 6 + 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => 9 = 8 + 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 5 = 4 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,7,5,1,2,3,4] => ? ∊ {5,5,5,7,7,9,9,9,9,9,9,9,11,11} + 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,5,4,1,2,7,3] => ? ∊ {5,5,5,7,7,9,9,9,9,9,9,9,11,11} + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [7,3,5,1,6,2,4] => ? ∊ {5,5,5,7,7,9,9,9,9,9,9,9,11,11} + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [7,5,4,1,6,2,3] => ? ∊ {5,5,5,7,7,9,9,9,9,9,9,9,11,11} + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => ? ∊ {5,5,5,7,7,9,9,9,9,9,9,9,11,11} + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [2,7,6,5,1,3,4] => ? ∊ {5,5,5,7,7,9,9,9,9,9,9,9,11,11} + 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [2,6,4,5,1,7,3] => ? ∊ {5,5,5,7,7,9,9,9,9,9,9,9,11,11} + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => ? ∊ {5,5,5,7,7,9,9,9,9,9,9,9,11,11} + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [7,6,4,5,1,2,3] => ? ∊ {5,5,5,7,7,9,9,9,9,9,9,9,11,11} + 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,4,5,1,7,2] => ? ∊ {5,5,5,7,7,9,9,9,9,9,9,9,11,11} + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [2,3,7,5,6,1,4] => ? ∊ {5,5,5,7,7,9,9,9,9,9,9,9,11,11} + 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,4,5,6,1,3] => ? ∊ {5,5,5,7,7,9,9,9,9,9,9,9,11,11} + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [7,3,4,5,6,1,2] => ? ∊ {5,5,5,7,7,9,9,9,9,9,9,9,11,11} + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? ∊ {5,5,5,7,7,9,9,9,9,9,9,9,11,11} + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [6,7,8,1,2,3,4,5] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [6,7,5,1,2,3,8,4] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> [6,8,4,1,2,7,3,5] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [6,8,5,1,2,7,3,4] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [6,5,4,1,2,7,8,3] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,1,0,0,0]
=> [7,3,8,1,6,2,4,5] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [7,3,5,1,6,2,8,4] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,1,0,0,1,0,0,0]
=> [7,8,4,1,6,2,3,5] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [7,8,5,1,6,2,3,4] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [7,5,4,1,6,2,8,3] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [8,3,4,1,6,7,2,5] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [8,3,5,1,6,7,2,4] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [8,5,4,1,6,7,2,3] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [5,3,4,1,6,7,8,2] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [2,7,8,6,1,3,4,5] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [2,7,6,5,1,3,8,4] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [2,8,4,6,1,7,3,5] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [2,8,6,5,1,7,3,4] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [2,6,4,5,1,7,8,3] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [7,3,8,6,1,2,4,5] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [7,3,6,5,1,2,8,4] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,0]
=> [7,8,4,6,1,2,3,5] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [7,8,6,5,1,2,3,4] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [7,6,4,5,1,2,8,3] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [8,3,4,6,1,7,2,5] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0]
=> [8,3,6,5,1,7,2,4] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [8,6,4,5,1,7,2,3] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [6,3,4,5,1,7,8,2] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [2,3,8,7,6,1,4,5] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> [2,3,7,5,6,1,8,4] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0]
=> [2,8,4,7,6,1,3,5] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [2,8,7,5,6,1,3,4] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [2,7,4,5,6,1,8,3] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [8,3,4,7,6,1,2,5] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> [8,3,7,5,6,1,2,4] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [8,7,4,5,6,1,2,3] => ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} + 1
Description
The number of Bruhat lower covers of a permutation.
This is, for a permutation $\pi$, the number of permutations $\tau$ with $\operatorname{inv}(\tau) = \operatorname{inv}(\pi) - 1$ such that $\tau*t = \pi$ for a transposition $t$.
This is also the number of occurrences of the boxed pattern $21$: occurrences of the pattern $21$ such that any entry between the two matched entries is either larger or smaller than both of the matched entries.
Matching statistic: St000327
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000327: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 38%
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000327: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 38%
Values
[1,0]
=> [1,1,0,0]
=> ([(0,1)],2)
=> 1 = 2 - 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4 = 5 - 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 5 = 6 - 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 5 = 6 - 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? ∊ {4,8} - 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? ∊ {4,8} - 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ? ∊ {5,5,7,7,9,9,9,9,9,9,9,11,11} - 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ? ∊ {5,5,7,7,9,9,9,9,9,9,9,11,11} - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? ∊ {5,5,7,7,9,9,9,9,9,9,9,11,11} - 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? ∊ {5,5,7,7,9,9,9,9,9,9,9,11,11} - 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ? ∊ {5,5,7,7,9,9,9,9,9,9,9,11,11} - 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ? ∊ {5,5,7,7,9,9,9,9,9,9,9,11,11} - 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? ∊ {5,5,7,7,9,9,9,9,9,9,9,11,11} - 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {5,5,7,7,9,9,9,9,9,9,9,11,11} - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? ∊ {5,5,7,7,9,9,9,9,9,9,9,11,11} - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? ∊ {5,5,7,7,9,9,9,9,9,9,9,11,11} - 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? ∊ {5,5,7,7,9,9,9,9,9,9,9,11,11} - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? ∊ {5,5,7,7,9,9,9,9,9,9,9,11,11} - 1
[1,1,1,1,0,0,0,0]
=> [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)
=> ? ∊ {5,5,7,7,9,9,9,9,9,9,9,11,11} - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ([(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)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,4),(5,6),(6,9),(7,8),(9,1),(9,7)],10)
=> ? ∊ {6,6,6,6,8,8,8,8,8,10,10,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,12,14,14,14,14,14,14,16,16} - 1
Description
The number of cover relations in a poset.
Equivalently, this is also the number of edges in the Hasse diagram [1].
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