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Your data matches 9 different statistics following compositions of up to 3 maps.
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Matching statistic: St001958
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(load all 4 compositions to match this statistic)
St001958: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 1
[2,1] => 1
[1,2,3] => 1
[1,3,2] => 2
[2,1,3] => 2
[2,3,1] => 2
[3,1,2] => 2
[3,2,1] => 1
[1,2,3,4] => 1
[1,2,4,3] => 3
[1,3,2,4] => 3
[1,3,4,2] => 3
[1,4,2,3] => 3
[1,4,3,2] => 3
[2,1,3,4] => 3
[2,1,4,3] => 3
[2,3,1,4] => 3
[2,3,4,1] => 3
[2,4,1,3] => 3
[2,4,3,1] => 3
[3,1,2,4] => 3
[3,1,4,2] => 3
[3,2,1,4] => 3
[3,2,4,1] => 3
[3,4,1,2] => 3
[3,4,2,1] => 3
[4,1,2,3] => 3
[4,1,3,2] => 3
[4,2,1,3] => 3
[4,2,3,1] => 3
[4,3,1,2] => 3
[4,3,2,1] => 1
[1,2,3,4,5] => 1
[1,2,3,5,4] => 4
[1,2,4,3,5] => 4
[1,2,4,5,3] => 3
[1,2,5,3,4] => 4
[1,2,5,4,3] => 4
[1,3,2,4,5] => 4
[1,3,2,5,4] => 4
[1,3,4,2,5] => 4
[1,3,4,5,2] => 4
[1,3,5,2,4] => 4
[1,3,5,4,2] => 4
[1,4,2,3,5] => 4
[1,4,2,5,3] => 4
[1,4,3,2,5] => 3
[1,4,3,5,2] => 4
[1,4,5,2,3] => 4
Description
The degree of the polynomial interpolating the values of a permutation.
Given a permutation $\pi\in\mathfrak S_n$ there is a polynomial $p$ of minimal degree such that $p(n)=\pi(n)$ for $n\in\{1,\dots,n\}$.
This statistic records the degree of $p$.
Matching statistic: St001603
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 14%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 14%
Values
[1] => [1] => [1]
=> []
=> ? = 0 - 4
[1,2] => [1,2] => [1,1]
=> [1]
=> ? = 1 - 4
[2,1] => [2,1] => [2]
=> []
=> ? = 1 - 4
[1,2,3] => [1,3,2] => [2,1]
=> [1]
=> ? = 1 - 4
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? = 2 - 4
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? = 2 - 4
[2,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? = 2 - 4
[3,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? = 2 - 4
[3,2,1] => [3,2,1] => [3]
=> []
=> ? = 1 - 4
[1,2,3,4] => [1,4,3,2] => [3,1]
=> [1]
=> ? = 1 - 4
[1,2,4,3] => [1,4,3,2] => [3,1]
=> [1]
=> ? = 3 - 4
[1,3,2,4] => [1,4,3,2] => [3,1]
=> [1]
=> ? = 3 - 4
[1,3,4,2] => [1,4,3,2] => [3,1]
=> [1]
=> ? = 3 - 4
[1,4,2,3] => [1,4,3,2] => [3,1]
=> [1]
=> ? = 3 - 4
[1,4,3,2] => [1,4,3,2] => [3,1]
=> [1]
=> ? = 3 - 4
[2,1,3,4] => [2,1,4,3] => [2,2]
=> [2]
=> ? = 3 - 4
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> ? = 3 - 4
[2,3,1,4] => [2,4,1,3] => [2,1,1]
=> [1,1]
=> ? = 3 - 4
[2,3,4,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? = 3 - 4
[2,4,1,3] => [2,4,1,3] => [2,1,1]
=> [1,1]
=> ? = 3 - 4
[2,4,3,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? = 3 - 4
[3,1,2,4] => [3,1,4,2] => [2,2]
=> [2]
=> ? = 3 - 4
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> ? = 3 - 4
[3,2,1,4] => [3,2,1,4] => [3,1]
=> [1]
=> ? = 3 - 4
[3,2,4,1] => [3,2,4,1] => [3,1]
=> [1]
=> ? = 3 - 4
[3,4,1,2] => [3,4,1,2] => [2,1,1]
=> [1,1]
=> ? = 3 - 4
[3,4,2,1] => [3,4,2,1] => [3,1]
=> [1]
=> ? = 3 - 4
[4,1,2,3] => [4,1,3,2] => [3,1]
=> [1]
=> ? = 3 - 4
[4,1,3,2] => [4,1,3,2] => [3,1]
=> [1]
=> ? = 3 - 4
[4,2,1,3] => [4,2,1,3] => [3,1]
=> [1]
=> ? = 3 - 4
[4,2,3,1] => [4,2,3,1] => [3,1]
=> [1]
=> ? = 3 - 4
[4,3,1,2] => [4,3,1,2] => [3,1]
=> [1]
=> ? = 3 - 4
[4,3,2,1] => [4,3,2,1] => [4]
=> []
=> ? = 1 - 4
[1,2,3,4,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 1 - 4
[1,2,3,5,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 4 - 4
[1,2,4,3,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 4 - 4
[1,2,4,5,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 3 - 4
[1,2,5,3,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 4 - 4
[1,2,5,4,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 4 - 4
[1,3,2,4,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 4 - 4
[1,3,2,5,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 4 - 4
[1,3,4,2,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 4 - 4
[1,3,4,5,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 4 - 4
[1,3,5,2,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 4 - 4
[1,3,5,4,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 4 - 4
[1,4,2,3,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 4 - 4
[1,4,2,5,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 4 - 4
[1,4,3,2,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 3 - 4
[1,4,3,5,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 4 - 4
[1,4,5,2,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? = 4 - 4
[3,2,1,4,5,6] => [3,2,1,6,5,4] => [3,3]
=> [3]
=> 1 = 5 - 4
[3,2,1,4,6,5] => [3,2,1,6,5,4] => [3,3]
=> [3]
=> 1 = 5 - 4
[3,2,1,5,4,6] => [3,2,1,6,5,4] => [3,3]
=> [3]
=> 1 = 5 - 4
[3,2,1,5,6,4] => [3,2,1,6,5,4] => [3,3]
=> [3]
=> 1 = 5 - 4
[3,2,1,6,4,5] => [3,2,1,6,5,4] => [3,3]
=> [3]
=> 1 = 5 - 4
[3,2,1,6,5,4] => [3,2,1,6,5,4] => [3,3]
=> [3]
=> 1 = 5 - 4
[3,2,4,1,5,6] => [3,2,6,1,5,4] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[3,2,4,1,6,5] => [3,2,6,1,5,4] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[3,2,4,5,1,6] => [3,2,6,5,1,4] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[3,2,4,6,1,5] => [3,2,6,5,1,4] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[3,2,5,1,4,6] => [3,2,6,1,5,4] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[3,2,5,1,6,4] => [3,2,6,1,5,4] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[3,2,5,4,1,6] => [3,2,6,5,1,4] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[3,2,5,6,1,4] => [3,2,6,5,1,4] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[3,2,6,1,4,5] => [3,2,6,1,5,4] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[3,2,6,1,5,4] => [3,2,6,1,5,4] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[3,2,6,4,1,5] => [3,2,6,5,1,4] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[3,2,6,5,1,4] => [3,2,6,5,1,4] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[3,4,2,1,5,6] => [3,6,2,1,5,4] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[3,4,2,1,6,5] => [3,6,2,1,5,4] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[3,4,2,5,1,6] => [3,6,2,5,1,4] => [3,1,1,1]
=> [1,1,1]
=> 1 = 5 - 4
[3,4,2,6,1,5] => [3,6,2,5,1,4] => [3,1,1,1]
=> [1,1,1]
=> 1 = 5 - 4
[3,5,2,1,4,6] => [3,6,2,1,5,4] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[3,5,2,1,6,4] => [3,6,2,1,5,4] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[3,5,2,4,1,6] => [3,6,2,5,1,4] => [3,1,1,1]
=> [1,1,1]
=> 1 = 5 - 4
[3,5,2,6,1,4] => [3,6,2,5,1,4] => [3,1,1,1]
=> [1,1,1]
=> 1 = 5 - 4
[3,6,2,1,4,5] => [3,6,2,1,5,4] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[3,6,2,1,5,4] => [3,6,2,1,5,4] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[3,6,2,4,1,5] => [3,6,2,5,1,4] => [3,1,1,1]
=> [1,1,1]
=> 1 = 5 - 4
[3,6,2,5,1,4] => [3,6,2,5,1,4] => [3,1,1,1]
=> [1,1,1]
=> 1 = 5 - 4
[4,2,1,3,5,6] => [4,2,1,6,5,3] => [3,3]
=> [3]
=> 1 = 5 - 4
[4,2,1,3,6,5] => [4,2,1,6,5,3] => [3,3]
=> [3]
=> 1 = 5 - 4
[4,2,1,5,3,6] => [4,2,1,6,5,3] => [3,3]
=> [3]
=> 1 = 5 - 4
[4,2,1,5,6,3] => [4,2,1,6,5,3] => [3,3]
=> [3]
=> 1 = 5 - 4
[4,2,1,6,3,5] => [4,2,1,6,5,3] => [3,3]
=> [3]
=> 1 = 5 - 4
[4,2,1,6,5,3] => [4,2,1,6,5,3] => [3,3]
=> [3]
=> 1 = 5 - 4
[4,2,3,1,5,6] => [4,2,6,1,5,3] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[4,2,3,1,6,5] => [4,2,6,1,5,3] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[4,2,3,5,1,6] => [4,2,6,5,1,3] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[4,2,3,6,1,5] => [4,2,6,5,1,3] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[4,2,5,1,3,6] => [4,2,6,1,5,3] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[4,2,5,1,6,3] => [4,2,6,1,5,3] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[4,2,5,3,1,6] => [4,2,6,5,1,3] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[4,2,5,6,1,3] => [4,2,6,5,1,3] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[4,2,6,1,3,5] => [4,2,6,1,5,3] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[4,2,6,1,5,3] => [4,2,6,1,5,3] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[4,2,6,3,1,5] => [4,2,6,5,1,3] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[4,2,6,5,1,3] => [4,2,6,5,1,3] => [3,2,1]
=> [2,1]
=> 1 = 5 - 4
[4,3,1,2,5,6] => [4,3,1,6,5,2] => [3,3]
=> [3]
=> 1 = 5 - 4
[4,3,1,2,6,5] => [4,3,1,6,5,2] => [3,3]
=> [3]
=> 1 = 5 - 4
Description
The number of colourings of a polygon such that the multiplicities of a colour are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001879
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 57%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [.,.]
=> ([],1)
=> ? = 0
[1,2] => [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ? = 1
[2,1] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ? = 1
[1,2,3] => [2,3,1] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 1
[1,3,2] => [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 2
[2,1,3] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 2
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 2
[3,1,2] => [3,1,2] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 2
[3,2,1] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 1
[1,2,3,4] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[1,2,4,3] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[1,3,2,4] => [3,2,4,1] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[1,3,4,2] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3
[1,4,2,3] => [3,4,2,1] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[1,4,3,2] => [4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,1,3,4] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3
[2,1,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,3,1,4] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,4,1,3] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[2,4,3,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3
[3,1,2,4] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[3,1,4,2] => [4,1,3,2] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[3,2,1,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[3,2,4,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[3,4,1,2] => [4,1,2,3] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[3,4,2,1] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[4,1,2,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[4,1,3,2] => [4,3,1,2] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 3
[4,2,1,3] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[4,2,3,1] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[4,3,1,2] => [4,2,1,3] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3
[4,3,2,1] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1
[1,2,3,4,5] => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 1
[1,2,3,5,4] => [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4
[1,2,4,3,5] => [2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 4
[1,2,4,5,3] => [2,5,3,4,1] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3
[1,2,5,3,4] => [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 4
[1,2,5,4,3] => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4
[1,3,2,4,5] => [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4
[1,3,2,5,4] => [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4
[1,3,4,2,5] => [4,2,3,5,1] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 4
[1,3,4,5,2] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[1,3,5,2,4] => [4,2,5,3,1] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 4
[1,3,5,4,2] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[1,4,2,3,5] => [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4
[1,4,2,5,3] => [3,5,2,4,1] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4
[1,4,3,2,5] => [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3
[1,4,3,5,2] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[1,4,5,2,3] => [4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 4
[1,4,5,3,2] => [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4
[1,5,2,3,4] => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4
[1,5,2,4,3] => [3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4
[1,5,3,2,4] => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4
[1,5,3,4,2] => [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[1,5,4,2,3] => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3
[1,5,4,3,2] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[2,1,3,4,5] => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[2,1,3,5,4] => [1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3
[2,1,4,3,5] => [1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[2,1,4,5,3] => [1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4
[2,1,5,3,4] => [1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4
[2,1,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[2,3,1,4,5] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4
[2,3,1,5,4] => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[2,3,4,1,5] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[2,3,4,5,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[2,3,5,1,4] => [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[2,4,1,5,3] => [1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[2,4,5,1,3] => [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[2,5,1,4,3] => [1,5,4,2,3] => [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[3,1,5,4,2] => [5,1,4,3,2] => [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[3,4,1,5,2] => [5,1,2,4,3] => [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[3,4,5,1,2] => [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[3,5,1,4,2] => [5,1,4,2,3] => [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[4,1,5,3,2] => [5,4,1,3,2] => [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[4,5,1,3,2] => [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[5,1,4,3,2] => [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4
[1,6,5,4,3,2] => [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[2,1,6,5,4,3] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[2,3,1,6,5,4] => [1,2,6,5,4,3] => [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[2,3,4,1,6,5] => [1,2,3,6,5,4] => [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[2,3,4,5,1,6] => [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[2,3,4,6,1,5] => [1,2,3,6,4,5] => [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[2,3,5,1,6,4] => [1,2,6,3,5,4] => [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[2,3,5,6,1,4] => [1,2,6,3,4,5] => [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[2,3,6,1,5,4] => [1,2,6,5,3,4] => [.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[2,4,1,6,5,3] => [1,6,2,5,4,3] => [.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[2,4,5,1,6,3] => [1,6,2,3,5,4] => [.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[2,4,5,6,1,3] => [1,6,2,3,4,5] => [.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[2,4,6,1,5,3] => [1,6,2,5,3,4] => [.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[2,5,1,6,4,3] => [1,6,5,2,4,3] => [.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[2,5,6,1,4,3] => [1,6,5,2,3,4] => [.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[2,6,1,5,4,3] => [1,6,5,4,2,3] => [.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[3,1,6,5,4,2] => [6,1,5,4,3,2] => [[.,[[[[.,.],.],.],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[3,4,1,6,5,2] => [6,1,2,5,4,3] => [[.,[.,[[[.,.],.],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[3,4,5,1,6,2] => [6,1,2,3,5,4] => [[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[3,4,5,6,1,2] => [6,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 5
[3,4,6,1,5,2] => [6,1,2,5,3,4] => [[.,[.,[[.,[.,.]],.]]],.]
=> ([(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.
Matching statistic: St001880
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 57%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 57%
Values
[1] => [1] => [.,.]
=> ([],1)
=> ? = 0 + 1
[1,2] => [2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ? = 1 + 1
[2,1] => [1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ? = 1 + 1
[1,2,3] => [2,3,1] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 1 + 1
[1,3,2] => [3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,1,3] => [1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[3,1,2] => [3,1,2] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
[3,2,1] => [2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ? = 1 + 1
[1,2,3,4] => [2,3,4,1] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 1
[1,2,4,3] => [2,4,3,1] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 + 1
[1,3,2,4] => [3,2,4,1] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 + 1
[1,3,4,2] => [4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3 + 1
[1,4,2,3] => [3,4,2,1] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 + 1
[1,4,3,2] => [4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,1,3,4] => [1,3,4,2] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3 + 1
[2,1,4,3] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,3,1,4] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,4,1,3] => [1,4,2,3] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[2,4,3,1] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3 + 1
[3,1,2,4] => [3,1,4,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 + 1
[3,1,4,2] => [4,1,3,2] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[3,2,1,4] => [2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 + 1
[3,2,4,1] => [2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 + 1
[3,4,1,2] => [4,1,2,3] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[3,4,2,1] => [3,1,2,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 + 1
[4,1,2,3] => [3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 + 1
[4,1,3,2] => [4,3,1,2] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
[4,2,1,3] => [2,4,1,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 + 1
[4,2,3,1] => [2,3,1,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3 + 1
[4,3,1,2] => [4,2,1,3] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ? = 3 + 1
[4,3,2,1] => [3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 1 + 1
[1,2,3,4,5] => [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 1 + 1
[1,2,3,5,4] => [2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 + 1
[1,2,4,3,5] => [2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 4 + 1
[1,2,4,5,3] => [2,5,3,4,1] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 1
[1,2,5,3,4] => [2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 4 + 1
[1,2,5,4,3] => [2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 + 1
[1,3,2,4,5] => [3,2,4,5,1] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4 + 1
[1,3,2,5,4] => [3,2,5,4,1] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4 + 1
[1,3,4,2,5] => [4,2,3,5,1] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 4 + 1
[1,3,4,5,2] => [5,2,3,4,1] => [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4 + 1
[1,3,5,2,4] => [4,2,5,3,1] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 4 + 1
[1,3,5,4,2] => [5,2,4,3,1] => [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4 + 1
[1,4,2,3,5] => [3,4,2,5,1] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4 + 1
[1,4,2,5,3] => [3,5,2,4,1] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4 + 1
[1,4,3,2,5] => [4,3,2,5,1] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 1
[1,4,3,5,2] => [5,3,2,4,1] => [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4 + 1
[1,4,5,2,3] => [4,5,2,3,1] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ? = 4 + 1
[1,4,5,3,2] => [5,4,2,3,1] => [[[[.,.],[.,.]],.],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4 + 1
[1,5,2,3,4] => [3,4,5,2,1] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4 + 1
[1,5,2,4,3] => [3,5,4,2,1] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 4 + 1
[1,5,3,2,4] => [4,3,5,2,1] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 4 + 1
[1,5,3,4,2] => [5,3,4,2,1] => [[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4 + 1
[1,5,4,2,3] => [4,5,3,2,1] => [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 3 + 1
[1,5,4,3,2] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[2,1,3,4,5] => [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4 + 1
[2,1,3,5,4] => [1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 3 + 1
[2,1,4,3,5] => [1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4 + 1
[2,1,4,5,3] => [1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4 + 1
[2,1,5,3,4] => [1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ? = 4 + 1
[2,1,5,4,3] => [1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[2,3,1,4,5] => [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ? = 4 + 1
[2,3,1,5,4] => [1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[2,3,4,1,5] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[2,3,4,5,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[2,3,5,1,4] => [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[2,4,1,5,3] => [1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[2,4,5,1,3] => [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[2,5,1,4,3] => [1,5,4,2,3] => [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[3,1,5,4,2] => [5,1,4,3,2] => [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[3,4,1,5,2] => [5,1,2,4,3] => [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[3,4,5,1,2] => [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[3,5,1,4,2] => [5,1,4,2,3] => [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[4,1,5,3,2] => [5,4,1,3,2] => [[[.,[[.,.],.]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[4,5,1,3,2] => [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[5,1,4,3,2] => [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
[1,6,5,4,3,2] => [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[2,1,6,5,4,3] => [1,6,5,4,3,2] => [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[2,3,1,6,5,4] => [1,2,6,5,4,3] => [.,[.,[[[[.,.],.],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[2,3,4,1,6,5] => [1,2,3,6,5,4] => [.,[.,[.,[[[.,.],.],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[2,3,4,5,1,6] => [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[2,3,4,6,1,5] => [1,2,3,6,4,5] => [.,[.,[.,[[.,[.,.]],.]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[2,3,5,1,6,4] => [1,2,6,3,5,4] => [.,[.,[[.,[[.,.],.]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[2,3,5,6,1,4] => [1,2,6,3,4,5] => [.,[.,[[.,[.,[.,.]]],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[2,3,6,1,5,4] => [1,2,6,5,3,4] => [.,[.,[[[.,[.,.]],.],.]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[2,4,1,6,5,3] => [1,6,2,5,4,3] => [.,[[.,[[[.,.],.],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[2,4,5,1,6,3] => [1,6,2,3,5,4] => [.,[[.,[.,[[.,.],.]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[2,4,5,6,1,3] => [1,6,2,3,4,5] => [.,[[.,[.,[.,[.,.]]]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[2,4,6,1,5,3] => [1,6,2,5,3,4] => [.,[[.,[[.,[.,.]],.]],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[2,5,1,6,4,3] => [1,6,5,2,4,3] => [.,[[[.,[[.,.],.]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[2,5,6,1,4,3] => [1,6,5,2,3,4] => [.,[[[.,[.,[.,.]]],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[2,6,1,5,4,3] => [1,6,5,4,2,3] => [.,[[[[.,[.,.]],.],.],.]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[3,1,6,5,4,2] => [6,1,5,4,3,2] => [[.,[[[[.,.],.],.],.]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[3,4,1,6,5,2] => [6,1,2,5,4,3] => [[.,[.,[[[.,.],.],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[3,4,5,1,6,2] => [6,1,2,3,5,4] => [[.,[.,[.,[[.,.],.]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[3,4,5,6,1,2] => [6,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
[3,4,6,1,5,2] => [6,1,2,5,3,4] => [[.,[.,[[.,[.,.]],.]]],.]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 5 + 1
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000264
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 14%
Mp00223: Permutations —runsort⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 14%
Values
[1] => [1] => [1] => ([],1)
=> ? = 0 - 1
[1,2] => [1,2] => [1,2] => ([],2)
=> ? = 1 - 1
[2,1] => [2,1] => [1,2] => ([],2)
=> ? = 1 - 1
[1,2,3] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? = 1 - 1
[1,3,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? = 2 - 1
[2,1,3] => [2,1,3] => [1,3,2] => ([(1,2)],3)
=> ? = 2 - 1
[2,3,1] => [2,3,1] => [1,2,3] => ([],3)
=> ? = 2 - 1
[3,1,2] => [3,1,2] => [1,2,3] => ([],3)
=> ? = 2 - 1
[3,2,1] => [3,2,1] => [1,2,3] => ([],3)
=> ? = 1 - 1
[1,2,3,4] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 1 - 1
[1,2,4,3] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 3 - 1
[1,3,2,4] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 3 - 1
[1,3,4,2] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 3 - 1
[1,4,2,3] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 3 - 1
[1,4,3,2] => [1,4,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 3 - 1
[2,1,3,4] => [2,1,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 3 - 1
[2,1,4,3] => [2,1,4,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 3 - 1
[2,3,1,4] => [2,4,1,3] => [1,3,2,4] => ([(2,3)],4)
=> ? = 3 - 1
[2,3,4,1] => [2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ? = 3 - 1
[2,4,1,3] => [2,4,1,3] => [1,3,2,4] => ([(2,3)],4)
=> ? = 3 - 1
[2,4,3,1] => [2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> ? = 3 - 1
[3,1,2,4] => [3,1,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 3 - 1
[3,1,4,2] => [3,1,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 3 - 1
[3,2,1,4] => [3,2,1,4] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 3 - 1
[3,2,4,1] => [3,2,4,1] => [1,2,4,3] => ([(2,3)],4)
=> ? = 3 - 1
[3,4,1,2] => [3,4,1,2] => [1,2,3,4] => ([],4)
=> ? = 3 - 1
[3,4,2,1] => [3,4,2,1] => [1,2,3,4] => ([],4)
=> ? = 3 - 1
[4,1,2,3] => [4,1,3,2] => [1,3,2,4] => ([(2,3)],4)
=> ? = 3 - 1
[4,1,3,2] => [4,1,3,2] => [1,3,2,4] => ([(2,3)],4)
=> ? = 3 - 1
[4,2,1,3] => [4,2,1,3] => [1,3,2,4] => ([(2,3)],4)
=> ? = 3 - 1
[4,2,3,1] => [4,2,3,1] => [1,2,3,4] => ([],4)
=> ? = 3 - 1
[4,3,1,2] => [4,3,1,2] => [1,2,3,4] => ([],4)
=> ? = 3 - 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => ([],4)
=> ? = 1 - 1
[1,2,3,4,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 - 1
[1,2,3,5,4] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,2,4,3,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,2,4,5,3] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3 - 1
[1,2,5,3,4] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,2,5,4,3] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,3,2,4,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,3,2,5,4] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,3,4,2,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,3,4,5,2] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,3,5,2,4] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,3,5,4,2] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,4,2,3,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,4,2,5,3] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,4,3,2,5] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3 - 1
[1,4,3,5,2] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? = 4 - 1
[2,3,1,4,5,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,3,1,4,6,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,3,1,5,4,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,3,1,5,6,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,3,1,6,4,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,3,1,6,5,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,4,1,3,5,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,4,1,3,6,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,4,1,5,3,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,4,1,5,6,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,4,1,6,3,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,4,1,6,5,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,5,1,3,4,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,5,1,3,6,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,5,1,4,3,6] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,5,1,4,6,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,5,1,6,3,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,5,1,6,4,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,6,1,3,4,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,6,1,3,5,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,6,1,4,3,5] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,6,1,4,5,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,6,1,5,3,4] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[2,6,1,5,4,3] => [2,6,1,5,4,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[3,2,4,1,5,6] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[3,2,4,1,6,5] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[3,2,5,1,4,6] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[3,2,5,1,6,4] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[3,2,6,1,4,5] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[3,2,6,1,5,4] => [3,2,6,1,5,4] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[4,2,3,1,5,6] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[4,2,3,1,6,5] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[4,2,5,1,3,6] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[4,2,5,1,6,3] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[4,2,6,1,3,5] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[4,2,6,1,5,3] => [4,2,6,1,5,3] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[4,3,2,5,1,6] => [4,3,2,6,1,5] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
[4,3,2,6,1,5] => [4,3,2,6,1,5] => [1,5,2,6,3,4] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 5 - 1
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001633
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St001633: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 43%
Mp00262: Binary words —poset of factors⟶ Posets
St001633: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 43%
Values
[1] => => ?
=> ? = 0 - 1
[1,2] => 0 => ([(0,1)],2)
=> 0 = 1 - 1
[2,1] => 1 => ([(0,1)],2)
=> 0 = 1 - 1
[1,2,3] => 00 => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,3,2] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,1,3] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,3,1] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,1,2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,2,1] => 11 => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,2,3,4] => 000 => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,4,3] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,4] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[1,3,4,2] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[1,4,2,3] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[1,4,3,2] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,3,4] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,4,3] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[2,3,1,4] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,3,4,1] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,4,1,3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,4,3,1] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[3,1,2,4] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,1,4,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,2,1,4] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,2,4,1] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,4,1,2] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,4,2,1] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,1,2,3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,1,3,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,2,1,3] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,2,3,1] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,3,1,2] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[4,3,2,1] => 111 => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,3,4,5] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,3,5,4] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[1,2,4,3,5] => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,2,4,5,3] => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 3 - 1
[1,2,5,3,4] => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,2,5,4,3] => 0011 => ([(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)
=> ? = 4 - 1
[1,3,2,4,5] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,3,2,5,4] => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[1,3,4,2,5] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,3,4,5,2] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,3,5,2,4] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,3,5,4,2] => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[1,4,2,3,5] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,4,2,5,3] => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[1,4,3,2,5] => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 3 - 1
[1,4,3,5,2] => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[1,4,5,2,3] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,4,5,3,2] => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[1,5,2,3,4] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,5,2,4,3] => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[1,5,3,2,4] => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[1,5,3,4,2] => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[1,5,4,2,3] => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 3 - 1
[1,5,4,3,2] => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,1,3,4,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,1,3,5,4] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 3 - 1
[2,1,4,3,5] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,1,4,5,3] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,1,5,3,4] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,1,5,4,3] => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[2,3,1,4,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,1,5,4] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[2,3,4,1,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,4,5,1] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,5,1,4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,5,4,1] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[2,4,1,3,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,4,1,5,3] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,4,3,1,5] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,4,3,5,1] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,4,5,1,3] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,4,5,3,1] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,5,1,3,4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,5,1,4,3] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,5,3,1,4] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 3 - 1
[2,5,3,4,1] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,5,4,1,3] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[2,5,4,3,1] => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[3,1,2,4,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[3,1,2,5,4] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[5,4,3,2,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,3,4,5,6] => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[6,5,4,3,2,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,3,4,5,6,7] => 000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.
Matching statistic: St000848
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000848: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 43%
Mp00262: Binary words —poset of factors⟶ Posets
St000848: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 43%
Values
[1] => => ?
=> ? = 0 - 1
[1,2] => 0 => ([(0,1)],2)
=> 0 = 1 - 1
[2,1] => 1 => ([(0,1)],2)
=> 0 = 1 - 1
[1,2,3] => 00 => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,3,2] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,1,3] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,3,1] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,1,2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,2,1] => 11 => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,2,3,4] => 000 => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,4,3] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,4] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[1,3,4,2] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[1,4,2,3] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[1,4,3,2] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,3,4] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,4,3] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[2,3,1,4] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,3,4,1] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,4,1,3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,4,3,1] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[3,1,2,4] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,1,4,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,2,1,4] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,2,4,1] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,4,1,2] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,4,2,1] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,1,2,3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,1,3,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,2,1,3] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,2,3,1] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,3,1,2] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[4,3,2,1] => 111 => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,3,4,5] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,3,5,4] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[1,2,4,3,5] => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,2,4,5,3] => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 3 - 1
[1,2,5,3,4] => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,2,5,4,3] => 0011 => ([(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)
=> ? = 4 - 1
[1,3,2,4,5] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,3,2,5,4] => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[1,3,4,2,5] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,3,4,5,2] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,3,5,2,4] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,3,5,4,2] => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[1,4,2,3,5] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,4,2,5,3] => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[1,4,3,2,5] => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 3 - 1
[1,4,3,5,2] => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[1,4,5,2,3] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,4,5,3,2] => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[1,5,2,3,4] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,5,2,4,3] => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[1,5,3,2,4] => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[1,5,3,4,2] => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[1,5,4,2,3] => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 3 - 1
[1,5,4,3,2] => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,1,3,4,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,1,3,5,4] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 3 - 1
[2,1,4,3,5] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,1,4,5,3] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,1,5,3,4] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,1,5,4,3] => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[2,3,1,4,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,1,5,4] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[2,3,4,1,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,4,5,1] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,5,1,4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,5,4,1] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[2,4,1,3,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,4,1,5,3] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,4,3,1,5] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,4,3,5,1] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,4,5,1,3] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,4,5,3,1] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,5,1,3,4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,5,1,4,3] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,5,3,1,4] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 3 - 1
[2,5,3,4,1] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,5,4,1,3] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[2,5,4,3,1] => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[3,1,2,4,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[3,1,2,5,4] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[5,4,3,2,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,3,4,5,6] => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[6,5,4,3,2,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
Description
The balance constant multiplied with the number of linear extensions of a poset.
A pair of elements $x,y$ of a poset is $\alpha$-balanced if the proportion $P(x,y)$ of linear extensions where $x$ comes before $y$ is between $\alpha$ and $1-\alpha$. The balance constant of a poset is $\max\min(P(x,y), P(y,x)).$
Kislitsyn [1] conjectured that every poset which is not a chain is $1/3$-balanced. Brightwell, Felsner and Trotter [2] show that it is at least $(1-\sqrt 5)/10$-balanced.
Olson and Sagan [3] exhibit various posets that are $1/2$-balanced.
Matching statistic: St000849
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00131: Permutations —descent bottoms⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000849: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 43%
Mp00262: Binary words —poset of factors⟶ Posets
St000849: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 43%
Values
[1] => => ?
=> ? = 0 - 1
[1,2] => 0 => ([(0,1)],2)
=> 0 = 1 - 1
[2,1] => 1 => ([(0,1)],2)
=> 0 = 1 - 1
[1,2,3] => 00 => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,3,2] => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,1,3] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,3,1] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,1,2] => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,2,1] => 11 => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,2,3,4] => 000 => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,4,3] => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[1,3,2,4] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[1,3,4,2] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[1,4,2,3] => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[1,4,3,2] => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,3,4] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,1,4,3] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[2,3,1,4] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,3,4,1] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,4,1,3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[2,4,3,1] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[3,1,2,4] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,1,4,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,2,1,4] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,2,4,1] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,4,1,2] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[3,4,2,1] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,1,2,3] => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,1,3,2] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,2,1,3] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,2,3,1] => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 3 - 1
[4,3,1,2] => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 3 - 1
[4,3,2,1] => 111 => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,3,4,5] => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,3,5,4] => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[1,2,4,3,5] => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,2,4,5,3] => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 3 - 1
[1,2,5,3,4] => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,2,5,4,3] => 0011 => ([(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)
=> ? = 4 - 1
[1,3,2,4,5] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,3,2,5,4] => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[1,3,4,2,5] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,3,4,5,2] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,3,5,2,4] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,3,5,4,2] => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[1,4,2,3,5] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,4,2,5,3] => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[1,4,3,2,5] => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 3 - 1
[1,4,3,5,2] => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[1,4,5,2,3] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,4,5,3,2] => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[1,5,2,3,4] => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[1,5,2,4,3] => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[1,5,3,2,4] => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[1,5,3,4,2] => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[1,5,4,2,3] => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 3 - 1
[1,5,4,3,2] => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,1,3,4,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,1,3,5,4] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 3 - 1
[2,1,4,3,5] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,1,4,5,3] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,1,5,3,4] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,1,5,4,3] => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[2,3,1,4,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,1,5,4] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[2,3,4,1,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,4,5,1] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,5,1,4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,3,5,4,1] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[2,4,1,3,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,4,1,5,3] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,4,3,1,5] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,4,3,5,1] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,4,5,1,3] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,4,5,3,1] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,5,1,3,4] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 - 1
[2,5,1,4,3] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,5,3,1,4] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 3 - 1
[2,5,3,4,1] => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 4 - 1
[2,5,4,1,3] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[2,5,4,3,1] => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 4 - 1
[3,1,2,4,5] => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 - 1
[3,1,2,5,4] => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 4 - 1
[5,4,3,2,1] => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,3,4,5,6] => 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[6,5,4,3,2,1] => 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
Description
The number of 1/3-balanced pairs in a poset.
A pair of elements $x,y$ of a poset is $\alpha$-balanced if the proportion of linear extensions where $x$ comes before $y$ is between $\alpha$ and $1-\alpha$.
Kislitsyn [1] conjectured that every poset which is not a chain has a $1/3$-balanced pair. Brightwell, Felsner and Trotter [2] show that at least a $(1-\sqrt 5)/10$-balanced pair exists in posets which are not chains.
Olson and Sagan [3] show that posets corresponding to skew Ferrers diagrams have a $1/3$-balanced pair.
Matching statistic: St001704
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> ? = 0 + 1
[1,2] => ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[2,1] => ([(0,1)],2)
=> ([],2)
=> 2 = 1 + 1
[1,2,3] => ([(0,2),(2,1)],3)
=> ([],3)
=> 2 = 1 + 1
[1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3 = 2 + 1
[2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3 = 2 + 1
[2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3 = 2 + 1
[3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 3 = 2 + 1
[3,2,1] => ([(0,2),(2,1)],3)
=> ([],3)
=> 2 = 1 + 1
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 2 = 1 + 1
[1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 4 = 3 + 1
[2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ([(2,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 4 = 3 + 1
[3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[4,1,3,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ([(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> 4 = 3 + 1
[4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 4 = 3 + 1
[4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 2 = 1 + 1
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 2 = 1 + 1
[1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4 + 1
[1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 + 1
[1,2,4,5,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 3 + 1
[1,2,5,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 + 1
[1,2,5,4,3] => ([(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)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ? = 4 + 1
[1,3,2,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 + 1
[1,3,2,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,5),(3,6),(3,9),(4,7),(4,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 4 + 1
[1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(2,3),(3,7),(3,11),(4,5),(4,6),(4,10),(4,11),(5,6),(5,9),(5,11),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4 + 1
[1,3,4,5,2] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 + 1
[1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 4 + 1
[1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4 + 1
[1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> ([(2,3),(3,7),(3,11),(4,5),(4,6),(4,10),(4,11),(5,6),(5,9),(5,11),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4 + 1
[1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 4 + 1
[1,4,3,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 3 + 1
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 + 1
[1,4,5,2,3] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> ([(2,3),(3,10),(4,5),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 4 + 1
[1,4,5,3,2] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 4 + 1
[1,5,2,3,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 + 1
[1,5,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4 + 1
[1,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 + 1
[1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 + 1
[1,5,4,2,3] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 3 + 1
[1,5,4,3,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4 + 1
[2,1,3,4,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4 + 1
[2,1,3,5,4] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,3),(3,9),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,9),(7,9),(8,9)],10)
=> ? = 3 + 1
[2,1,4,3,5] => ([(0,1),(0,2),(0,3),(1,7),(1,8),(2,5),(2,8),(3,5),(3,7),(3,8),(5,9),(6,4),(7,6),(7,9),(8,6),(8,9),(9,4)],10)
=> ([(2,5),(3,6),(3,9),(4,7),(4,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 4 + 1
[2,1,4,5,3] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 4 + 1
[2,1,5,3,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 4 + 1
[2,1,5,4,3] => ([(0,3),(0,4),(1,8),(2,7),(2,8),(3,1),(3,5),(4,2),(4,5),(5,7),(5,8),(7,6),(8,6)],9)
=> ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
=> ? = 4 + 1
[2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 + 1
[2,3,1,5,4] => ([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(3,10),(4,6),(4,7),(4,8),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 4 + 1
[2,3,4,1,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 + 1
[2,3,4,5,1] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4 + 1
[2,3,5,1,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(2,3),(3,11),(4,6),(4,9),(4,10),(4,11),(5,7),(5,8),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,12),(11,12)],13)
=> ? = 4 + 1
[2,3,5,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ([(2,3),(2,10),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,10),(8,10),(9,10)],11)
=> ? = 4 + 1
[2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ([(2,3),(3,9),(4,9),(4,11),(4,12),(4,13),(5,6),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,10),(6,12),(6,13),(7,8),(7,10),(7,11),(7,13),(8,10),(8,11),(8,12),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,13),(12,13)],14)
=> ? = 4 + 1
[2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ([(2,3),(3,13),(4,5),(4,6),(4,11),(4,12),(4,13),(5,6),(5,10),(5,11),(5,13),(6,9),(6,11),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,9),(8,10),(8,12),(8,13),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12),(11,13),(12,13)],14)
=> ? = 4 + 1
[2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 + 1
[2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 + 1
[2,4,5,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(2,3),(3,12),(4,7),(4,9),(4,11),(4,12),(5,6),(5,8),(5,10),(5,12),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 + 1
[2,4,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> ([(2,3),(2,11),(3,10),(4,9),(4,10),(4,11),(5,6),(5,7),(5,8),(5,10),(6,7),(6,8),(6,11),(7,8),(7,9),(7,10),(8,9),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 4 + 1
[2,5,1,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(2,3),(3,11),(4,6),(4,9),(4,10),(4,11),(5,7),(5,8),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,12),(11,12)],13)
=> ? = 4 + 1
[2,5,1,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ([(2,3),(3,12),(4,7),(4,9),(4,11),(4,12),(5,6),(5,8),(5,10),(5,12),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 + 1
[2,5,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> ([(2,3),(2,14),(3,13),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,7),(5,8),(5,10),(5,12),(5,13),(5,14),(6,7),(6,8),(6,10),(6,11),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,9),(8,12),(8,13),(8,14),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13),(11,12),(11,14),(12,14),(13,14)],15)
=> ? = 3 + 1
[2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ([(2,3),(2,12),(3,11),(4,5),(4,6),(4,7),(4,8),(4,12),(5,6),(5,8),(5,10),(5,11),(6,8),(6,9),(6,11),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 + 1
[2,5,4,1,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> ([(2,3),(3,11),(4,6),(4,9),(4,10),(4,11),(5,7),(5,8),(5,10),(5,11),(5,12),(6,8),(6,9),(6,10),(6,12),(7,8),(7,9),(7,10),(7,11),(7,12),(8,9),(8,11),(8,12),(9,10),(9,12),(10,12),(11,12)],13)
=> ? = 4 + 1
[2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ([(2,7),(3,6),(3,9),(4,5),(4,7),(4,8),(5,8),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4 + 1
[5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> 2 = 1 + 1
[1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 2 = 1 + 1
[6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([],6)
=> 2 = 1 + 1
[1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([],7)
=> 2 = 1 + 1
Description
The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph.
The deck of a graph is the multiset of induced subgraphs obtained by deleting a single vertex.
The graph reconstruction conjecture states that the deck of a graph with at least three vertices determines the graph.
This statistic is only defined for graphs with at least two vertices, because there is only a single graph of the given size otherwise.
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