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Your data matches 11 different statistics following compositions of up to 3 maps.
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Matching statistic: St000645
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000645: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [.,.]
=> [1,0]
=> 0
[1,2] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[2,1] => [1,2] => [.,[.,.]]
=> [1,1,0,0]
=> 0
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[2,1,3] => [1,3,2] => [.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[3,1,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,3,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,3,4] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,4,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,1,2,4] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,2,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,4,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[3,4,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[4,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[4,1,3,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[4,2,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 2
[4,2,3,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[4,3,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[1,2,4,5,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[1,3,2,5,4] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,3,4,2,5] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[1,3,4,5,2] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[2,1,3,4,5] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[2,3,4,5,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[2,3,5,4,1] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[2,4,3,5,1] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[2,4,5,1,3] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[2,4,5,3,1] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[2,5,1,3,4] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[2,5,4,1,3] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[3,1,2,4,5] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,2,4,5,1] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,4,5,1,2] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[3,4,5,2,1] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[3,5,1,2,4] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[3,5,2,4,1] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[3,5,4,1,2] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
Description
The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between.
For a Dyck path $D = D_1 \cdots D_{2n}$ with peaks in positions $i_1 < \ldots < i_k$ and valleys in positions $j_1 < \ldots < j_{k-1}$, this statistic is given by
$$
\sum_{a=1}^{k-1} (j_a-i_a)(i_{a+1}-j_a)
$$
Matching statistic: St001232
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [1,0]
=> 0
[1,2] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[2,1] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,3] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,3,1] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[3,1,2] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,3,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,3,4] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,3,4,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,4,1,3] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,4,3,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,1,2,4] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,2,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,4,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[3,4,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[4,1,2,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[4,1,3,2] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,2,1,3] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,2,3,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[4,3,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,2,4,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[1,3,4,2,5] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[1,3,4,5,2] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[2,1,3,4,5] => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[2,3,4,5,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[2,3,5,4,1] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[2,4,3,5,1] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[2,4,5,1,3] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[2,4,5,3,1] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[2,5,1,3,4] => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 4
[2,5,4,1,3] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[3,1,2,4,5] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[3,2,4,5,1] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[3,4,5,1,2] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[3,4,5,2,1] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[3,5,1,2,4] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[3,5,2,4,1] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[3,5,4,1,2] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000632
Mp00223: Permutations —runsort⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 30%
Mp00209: Permutations —pattern poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 30%
Values
[1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => ([(0,1)],2)
=> 0
[2,1] => [1,2] => ([(0,1)],2)
=> 0
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,1,3] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[1,3,2,4] => [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
[1,3,4,2] => [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
[2,1,3,4] => [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
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[2,4,1,3] => [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
[2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[3,1,2,4] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[3,2,4,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[4,1,3,2] => [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
[4,2,1,3] => [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
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,3,5,4] => [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)
=> ? = 1
[1,2,4,3,5] => [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
[1,2,4,5,3] => [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
[1,3,2,4,5] => [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)
=> ? = 3
[1,3,2,5,4] => [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)
=> ? = 3
[1,3,4,2,5] => [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)
=> ? = 4
[1,3,4,5,2] => [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)
=> ? = 3
[2,1,3,4,5] => [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)
=> ? = 3
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,3,5,4,1] => [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)
=> ? = 1
[2,4,3,5,1] => [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
[2,4,5,1,3] => [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)
=> ? = 3
[2,4,5,3,1] => [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
[2,5,1,3,4] => [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)
=> ? = 4
[2,5,4,1,3] => [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)
=> ? = 3
[3,1,2,4,5] => [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
[3,2,4,5,1] => [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
[3,4,5,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[3,5,1,2,4] => [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
[3,5,2,4,1] => [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
[3,5,4,1,2] => [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)
=> ? = 1
[3,5,4,2,1] => [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)
=> ? = 1
[4,1,2,3,5] => [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)
=> ? = 1
[4,1,3,2,5] => [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)
=> ? = 3
[4,2,3,5,1] => [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)
=> ? = 1
[4,2,5,1,3] => [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)
=> ? = 3
[4,3,5,1,2] => [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)
=> ? = 1
[4,3,5,2,1] => [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)
=> ? = 1
[4,5,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,5,1,3,2] => [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)
=> ? = 3
[4,5,2,1,3] => [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)
=> ? = 3
[4,5,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,5,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,1,2,3,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,1,2,4,3] => [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
[5,1,3,2,4] => [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)
=> ? = 3
[5,1,3,4,2] => [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)
=> ? = 4
[5,2,1,3,4] => [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)
=> ? = 4
[5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,2,4,1,3] => [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)
=> ? = 3
[5,2,4,3,1] => [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
[5,3,1,2,4] => [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
[5,3,2,4,1] => [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
[5,3,4,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,3,4,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,4,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,4,1,3,2] => [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)
=> ? = 3
[5,4,2,1,3] => [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)
=> ? = 3
[5,4,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,4,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[1,2,3,4,6,5] => [1,2,3,4,6,5] => ([(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)
=> ? = 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 2
[1,2,3,5,6,4] => [1,2,3,5,6,4] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 2
[1,2,4,3,5,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 3
[1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(0,2),(0,3),(0,4),(1,7),(1,13),(2,6),(2,12),(3,1),(3,9),(3,12),(4,6),(4,9),(4,12),(6,10),(7,8),(7,11),(8,5),(9,7),(9,10),(9,13),(10,11),(11,5),(12,10),(12,13),(13,8),(13,11)],14)
=> ? = 3
[1,2,4,5,3,6] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 4
[1,2,4,5,6,3] => [1,2,4,5,6,3] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 3
[1,3,2,4,5,6] => [1,3,2,4,5,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 4
[1,3,2,4,6,5] => [1,3,2,4,6,5] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,13),(2,8),(2,9),(2,13),(3,7),(3,9),(3,13),(4,6),(4,7),(4,8),(6,15),(7,12),(7,15),(8,11),(8,12),(8,15),(9,11),(9,12),(10,5),(11,10),(11,14),(12,10),(12,14),(13,11),(13,15),(14,5),(15,14)],16)
=> ? = 4
[1,3,2,5,4,6] => [1,3,2,5,4,6] => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,15),(2,7),(2,9),(2,15),(3,6),(3,7),(3,15),(4,6),(4,8),(4,15),(6,14),(7,11),(7,14),(8,12),(8,14),(9,11),(9,12),(10,5),(11,10),(11,13),(12,10),(12,13),(13,5),(14,13),(15,11),(15,12),(15,14)],16)
=> ? = 4
[1,3,2,5,6,4] => [1,3,2,5,6,4] => ([(0,2),(0,3),(0,4),(0,5),(1,14),(1,16),(2,7),(2,8),(2,9),(3,7),(3,10),(3,12),(4,8),(4,11),(4,12),(5,9),(5,10),(5,11),(7,17),(8,15),(8,17),(9,1),(9,15),(9,17),(10,13),(10,17),(11,13),(11,15),(12,13),(12,15),(12,17),(13,14),(13,16),(14,6),(15,14),(15,16),(16,6),(17,16)],18)
=> ? = 5
[1,3,4,2,5,6] => [1,3,4,2,5,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 6
[2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[3,4,5,6,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[3,4,5,6,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[4,5,6,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[4,5,6,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
[4,5,6,3,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0
Description
The jump number of the poset.
A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.
Matching statistic: St000307
Mp00223: Permutations —runsort⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 20%
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 20%
Values
[1] => [1] => ([],1)
=> 1 = 0 + 1
[1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[1,3,2,4] => [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 + 1
[1,3,4,2] => [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 + 1
[2,1,3,4] => [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 + 1
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,4,1,3] => [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 + 1
[2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,1,2,4] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,2,4,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,1,3,2] => [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 + 1
[4,2,1,3] => [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 + 1
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [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)
=> ? = 1 + 1
[1,2,4,3,5] => [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 + 1
[1,2,4,5,3] => [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 + 1
[1,3,2,4,5] => [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)
=> ? = 3 + 1
[1,3,2,5,4] => [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)
=> ? = 3 + 1
[1,3,4,2,5] => [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)
=> ? = 4 + 1
[1,3,4,5,2] => [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)
=> ? = 3 + 1
[2,1,3,4,5] => [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)
=> ? = 3 + 1
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,5,4,1] => [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)
=> ? = 1 + 1
[2,4,3,5,1] => [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 + 1
[2,4,5,1,3] => [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)
=> ? = 3 + 1
[2,4,5,3,1] => [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 + 1
[2,5,1,3,4] => [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)
=> ? = 4 + 1
[2,5,4,1,3] => [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)
=> ? = 3 + 1
[3,1,2,4,5] => [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 + 1
[3,2,4,5,1] => [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 + 1
[3,4,5,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,5,1,2,4] => [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 + 1
[3,5,2,4,1] => [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 + 1
[3,5,4,1,2] => [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)
=> ? = 1 + 1
[3,5,4,2,1] => [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)
=> ? = 1 + 1
[4,1,2,3,5] => [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)
=> ? = 1 + 1
[4,1,3,2,5] => [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)
=> ? = 3 + 1
[4,2,3,5,1] => [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)
=> ? = 1 + 1
[4,2,5,1,3] => [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)
=> ? = 3 + 1
[4,3,5,1,2] => [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)
=> ? = 1 + 1
[4,3,5,2,1] => [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)
=> ? = 1 + 1
[4,5,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,1,3,2] => [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)
=> ? = 3 + 1
[4,5,2,1,3] => [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)
=> ? = 3 + 1
[4,5,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,1,2,3,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,1,2,4,3] => [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 + 1
[5,1,3,2,4] => [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)
=> ? = 3 + 1
[5,1,3,4,2] => [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)
=> ? = 4 + 1
[5,2,1,3,4] => [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)
=> ? = 4 + 1
[5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,2,4,1,3] => [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)
=> ? = 3 + 1
[5,2,4,3,1] => [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 + 1
[5,3,1,2,4] => [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 + 1
[5,3,2,4,1] => [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 + 1
[5,3,4,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,3,4,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,1,3,2] => [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)
=> ? = 3 + 1
[5,4,2,1,3] => [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)
=> ? = 3 + 1
[5,4,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => ([(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)
=> ? = 1 + 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 2 + 1
[1,2,3,5,6,4] => [1,2,3,5,6,4] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 2 + 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 3 + 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(0,2),(0,3),(0,4),(1,7),(1,13),(2,6),(2,12),(3,1),(3,9),(3,12),(4,6),(4,9),(4,12),(6,10),(7,8),(7,11),(8,5),(9,7),(9,10),(9,13),(10,11),(11,5),(12,10),(12,13),(13,8),(13,11)],14)
=> ? = 3 + 1
[1,2,4,5,3,6] => [1,2,4,5,3,6] => ([(0,1),(0,3),(0,4),(0,5),(1,14),(2,7),(2,8),(2,16),(3,9),(3,11),(3,14),(4,9),(4,10),(4,14),(5,2),(5,10),(5,11),(5,14),(7,13),(7,15),(8,13),(8,15),(9,12),(9,16),(10,7),(10,12),(10,16),(11,8),(11,12),(11,16),(12,13),(12,15),(13,6),(14,16),(15,6),(16,15)],17)
=> ? = 4 + 1
[2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[3,4,5,6,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[3,4,5,6,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[4,5,6,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[4,5,6,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[4,5,6,3,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[4,5,6,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,1,2,3,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,2,3,4,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,3,4,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,3,4,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,4,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
Description
The number of rowmotion orbits of a poset.
Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
Matching statistic: St001632
Mp00223: Permutations —runsort⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 20%
Mp00209: Permutations —pattern poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 20%
Values
[1] => [1] => ([],1)
=> ? = 0 + 1
[1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,1,3] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,1,2] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[1,3,2,4] => [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 + 1
[1,3,4,2] => [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 + 1
[2,1,3,4] => [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 + 1
[2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,4,1,3] => [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 + 1
[2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,1,2,4] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,2,4,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[3,4,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,1,2,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,1,3,2] => [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 + 1
[4,2,1,3] => [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 + 1
[4,2,3,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,3,1,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [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)
=> ? = 1 + 1
[1,2,4,3,5] => [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 + 1
[1,2,4,5,3] => [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 + 1
[1,3,2,4,5] => [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)
=> ? = 3 + 1
[1,3,2,5,4] => [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)
=> ? = 3 + 1
[1,3,4,2,5] => [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)
=> ? = 4 + 1
[1,3,4,5,2] => [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)
=> ? = 3 + 1
[2,1,3,4,5] => [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)
=> ? = 3 + 1
[2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[2,3,5,4,1] => [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)
=> ? = 1 + 1
[2,4,3,5,1] => [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 + 1
[2,4,5,1,3] => [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)
=> ? = 3 + 1
[2,4,5,3,1] => [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 + 1
[2,5,1,3,4] => [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)
=> ? = 4 + 1
[2,5,4,1,3] => [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)
=> ? = 3 + 1
[3,1,2,4,5] => [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 + 1
[3,2,4,5,1] => [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 + 1
[3,4,5,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[3,5,1,2,4] => [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 + 1
[3,5,2,4,1] => [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 + 1
[3,5,4,1,2] => [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)
=> ? = 1 + 1
[3,5,4,2,1] => [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)
=> ? = 1 + 1
[4,1,2,3,5] => [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)
=> ? = 1 + 1
[4,1,3,2,5] => [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)
=> ? = 3 + 1
[4,2,3,5,1] => [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)
=> ? = 1 + 1
[4,2,5,1,3] => [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)
=> ? = 3 + 1
[4,3,5,1,2] => [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)
=> ? = 1 + 1
[4,3,5,2,1] => [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)
=> ? = 1 + 1
[4,5,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,1,3,2] => [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)
=> ? = 3 + 1
[4,5,2,1,3] => [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)
=> ? = 3 + 1
[4,5,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,1,2,3,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,1,2,4,3] => [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 + 1
[5,1,3,2,4] => [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)
=> ? = 3 + 1
[5,1,3,4,2] => [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)
=> ? = 4 + 1
[5,2,1,3,4] => [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)
=> ? = 4 + 1
[5,2,3,4,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,2,4,1,3] => [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)
=> ? = 3 + 1
[5,2,4,3,1] => [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 + 1
[5,3,1,2,4] => [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 + 1
[5,3,2,4,1] => [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 + 1
[5,3,4,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,3,4,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,1,2,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,1,3,2] => [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)
=> ? = 3 + 1
[5,4,2,1,3] => [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)
=> ? = 3 + 1
[5,4,2,3,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,3,1,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,2,3,4,6,5] => [1,2,3,4,6,5] => ([(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)
=> ? = 1 + 1
[1,2,3,5,4,6] => [1,2,3,5,4,6] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 2 + 1
[1,2,3,5,6,4] => [1,2,3,5,6,4] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 2 + 1
[1,2,4,3,5,6] => [1,2,4,3,5,6] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 3 + 1
[1,2,4,3,6,5] => [1,2,4,3,6,5] => ([(0,2),(0,3),(0,4),(1,7),(1,13),(2,6),(2,12),(3,1),(3,9),(3,12),(4,6),(4,9),(4,12),(6,10),(7,8),(7,11),(8,5),(9,7),(9,10),(9,13),(10,11),(11,5),(12,10),(12,13),(13,8),(13,11)],14)
=> ? = 3 + 1
[2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[3,4,5,6,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[3,4,5,6,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[4,5,6,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[4,5,6,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[4,5,6,3,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[4,5,6,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,1,2,3,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,2,3,4,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,3,4,1,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,3,4,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,4,1,2,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[5,6,4,2,3,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Matching statistic: St000422
Values
[1] => ([],1)
=> ([],1)
=> ([],1)
=> 0
[1,2] => ([],2)
=> ([],1)
=> ([],1)
=> 0
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
[1,2,3] => ([],3)
=> ([],1)
=> ([],1)
=> 0
[1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1
[2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[1,2,3,4] => ([],4)
=> ([],1)
=> ([],1)
=> 0
[1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1
[1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 2
[2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0
[1,2,3,4,5] => ([],5)
=> ([],1)
=> ([],1)
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 2
[1,3,2,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 3
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 3
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 4
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 3
[2,1,3,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 3
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 2
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 2
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 1
[4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1
[4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1
[4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1
[4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 1
[4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
[4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 3
[4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 3
[4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0
[5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
[5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2
[5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 3
[5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 4
[5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 4
[5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 3
[5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2
[5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2
[5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2
[5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0
[5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 3
[5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 3
[5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0
[5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0
[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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 0
[1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ([],1)
=> 0
[2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
[3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
[3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
[4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[4,5,6,3,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0
[5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
[5,6,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0
[5,6,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0
[5,6,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0
[5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 0
[6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
[6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
Description
The energy of a graph, if it is integral.
The energy of a graph is the sum of the absolute values of its eigenvalues. This statistic is only defined for graphs with integral energy. It is known, that the energy is never an odd integer [2]. In fact, it is never the square root of an odd integer [3].
The energy of a graph is the sum of the energies of the connected components of a graph. The energy of the complete graph $K_n$ equals $2n-2$. For this reason, we do not define the energy of the empty graph.
Matching statistic: St000454
Values
[1] => ([],1)
=> ([],1)
=> ([],1)
=> 0
[1,2] => ([],2)
=> ([],1)
=> ([],1)
=> 0
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0
[1,2,3] => ([],3)
=> ([],1)
=> ([],1)
=> 0
[1,3,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1
[2,1,3] => ([(1,2)],3)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 0
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[1,2,3,4] => ([],4)
=> ([],1)
=> ([],1)
=> 0
[1,2,4,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1
[1,3,2,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 2
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 2
[2,1,3,4] => ([(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 2
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 0
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0
[1,2,3,4,5] => ([],5)
=> ([],1)
=> ([],1)
=> 0
[1,2,3,5,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1
[1,2,4,3,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 2
[1,3,2,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 3
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ? = 3
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 4
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 3
[2,1,3,4,5] => ([(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 3
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 3
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? = 3
[3,1,2,4,5] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 2
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2
[3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 2
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 1
[4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ? = 1
[4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1
[4,2,5,1,3] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 3
[4,3,5,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 1
[4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 1
[4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
[4,5,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 3
[4,5,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 3
[4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[4,5,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0
[5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 0
[5,1,2,4,3] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2
[5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 3
[5,1,3,4,2] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 4
[5,2,1,3,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 4
[5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[5,2,4,1,3] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ? = 3
[5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2
[5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ? = 2
[5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 2
[5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0
[5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[5,4,1,3,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 3
[5,4,2,1,3] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ? = 3
[5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0
[5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0
[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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 0
[1,2,3,4,5,6] => ([],6)
=> ([],1)
=> ([],1)
=> 0
[2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
[3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
[3,4,5,6,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
[4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[4,5,6,3,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[4,5,6,3,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0
[5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
[5,6,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0
[5,6,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0
[5,6,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 0
[5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> 0
[6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> 0
[6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
[6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 0
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001964
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 10%
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 10%
Values
[1] => [1] => [1] => ([],1)
=> 0
[1,2] => [1,2] => [2,1] => ([],2)
=> 0
[2,1] => [1,2] => [2,1] => ([],2)
=> 0
[1,2,3] => [1,2,3] => [3,2,1] => ([],3)
=> 0
[1,3,2] => [1,3,2] => [2,3,1] => ([(1,2)],3)
=> ? = 1
[2,1,3] => [1,3,2] => [2,3,1] => ([(1,2)],3)
=> ? = 1
[2,3,1] => [1,2,3] => [3,2,1] => ([],3)
=> 0
[3,1,2] => [1,2,3] => [3,2,1] => ([],3)
=> 0
[3,2,1] => [1,2,3] => [3,2,1] => ([],3)
=> 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> ? = 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> ? = 2
[1,3,4,2] => [1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ? = 2
[2,1,3,4] => [1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ? = 2
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> ? = 2
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> ? = 1
[3,1,2,4] => [1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> ? = 1
[3,2,4,1] => [1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
=> ? = 1
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> ? = 2
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
=> ? = 2
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => ([],4)
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> ? = 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> ? = 2
[1,2,4,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> ? = 2
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> ? = 3
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ? = 3
[1,3,4,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ? = 4
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ? = 3
[2,1,3,4,5] => [1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ? = 3
[2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[2,3,5,4,1] => [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> ? = 1
[2,4,3,5,1] => [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> ? = 2
[2,4,5,1,3] => [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> ? = 3
[2,4,5,3,1] => [1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> ? = 2
[2,5,1,3,4] => [1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ? = 4
[2,5,4,1,3] => [1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ? = 3
[3,1,2,4,5] => [1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> ? = 2
[3,2,4,5,1] => [1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
=> ? = 2
[3,4,5,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[3,4,5,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[3,5,1,2,4] => [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> ? = 2
[3,5,2,4,1] => [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> ? = 2
[3,5,4,1,2] => [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> ? = 1
[3,5,4,2,1] => [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> ? = 1
[4,1,2,3,5] => [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> ? = 1
[4,1,3,2,5] => [1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ? = 3
[4,2,3,5,1] => [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> ? = 1
[4,2,5,1,3] => [1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> ? = 3
[4,3,5,1,2] => [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> ? = 1
[4,3,5,2,1] => [1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
=> ? = 1
[4,5,1,2,3] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[4,5,1,3,2] => [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> ? = 3
[4,5,2,1,3] => [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> ? = 3
[4,5,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[4,5,3,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[4,5,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[5,1,2,3,4] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[5,1,2,4,3] => [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> ? = 2
[5,1,3,2,4] => [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> ? = 3
[5,1,3,4,2] => [1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ? = 4
[5,2,1,3,4] => [1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
=> ? = 4
[5,2,3,4,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[5,2,4,1,3] => [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> ? = 3
[5,2,4,3,1] => [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> ? = 2
[5,3,1,2,4] => [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> ? = 2
[5,3,2,4,1] => [1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
=> ? = 2
[5,3,4,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[5,3,4,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[5,4,1,2,3] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[5,4,1,3,2] => [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> ? = 3
[5,4,2,1,3] => [1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
=> ? = 3
[5,4,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[5,4,3,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[5,4,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => ([],5)
=> 0
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6)
=> 0
[2,3,4,5,6,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6)
=> 0
[3,4,5,6,1,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6)
=> 0
[3,4,5,6,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6)
=> 0
[4,5,6,1,2,3] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6)
=> 0
[4,5,6,2,3,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6)
=> 0
[4,5,6,3,1,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6)
=> 0
[4,5,6,3,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6)
=> 0
[5,6,1,2,3,4] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6)
=> 0
[5,6,2,3,4,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6)
=> 0
[5,6,3,4,1,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6)
=> 0
[5,6,3,4,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6)
=> 0
[5,6,4,1,2,3] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6)
=> 0
[5,6,4,2,3,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6)
=> 0
[5,6,4,3,1,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6)
=> 0
[5,6,4,3,2,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6)
=> 0
[6,1,2,3,4,5] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6)
=> 0
[6,2,3,4,5,1] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6)
=> 0
[6,3,4,5,1,2] => [1,2,3,4,5,6] => [6,5,4,3,2,1] => ([],6)
=> 0
Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Matching statistic: St001209
Mp00223: Permutations —runsort⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001209: Parking functions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 30%
Mp00064: Permutations —reverse⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001209: Parking functions ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 30%
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 0
[2,1] => [1,2] => [2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[1,3,2] => [1,3,2] => [2,3,1] => [2,3,1] => 1
[2,1,3] => [1,3,2] => [2,3,1] => [2,3,1] => 1
[2,3,1] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[3,1,2] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[3,2,1] => [1,2,3] => [3,2,1] => [3,2,1] => 0
[1,2,3,4] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,4,3] => [1,2,4,3] => [3,4,2,1] => [3,4,2,1] => 1
[1,3,2,4] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 2
[1,3,4,2] => [1,3,4,2] => [2,4,3,1] => [2,4,3,1] => 2
[2,1,3,4] => [1,3,4,2] => [2,4,3,1] => [2,4,3,1] => 2
[2,3,4,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[2,4,1,3] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 2
[2,4,3,1] => [1,2,4,3] => [3,4,2,1] => [3,4,2,1] => 1
[3,1,2,4] => [1,2,4,3] => [3,4,2,1] => [3,4,2,1] => 1
[3,2,4,1] => [1,2,4,3] => [3,4,2,1] => [3,4,2,1] => 1
[3,4,1,2] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[3,4,2,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[4,1,2,3] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[4,1,3,2] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 2
[4,2,1,3] => [1,3,2,4] => [4,2,3,1] => [4,2,3,1] => 2
[4,2,3,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[4,3,1,2] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[4,3,2,1] => [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 1
[1,2,4,3,5] => [1,2,4,3,5] => [5,3,4,2,1] => [5,3,4,2,1] => ? = 2
[1,2,4,5,3] => [1,2,4,5,3] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 2
[1,3,2,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 3
[1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 3
[1,3,4,2,5] => [1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 4
[1,3,4,5,2] => [1,3,4,5,2] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 3
[2,1,3,4,5] => [1,3,4,5,2] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 3
[2,3,4,5,1] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[2,3,5,4,1] => [1,2,3,5,4] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 1
[2,4,3,5,1] => [1,2,4,3,5] => [5,3,4,2,1] => [5,3,4,2,1] => ? = 2
[2,4,5,1,3] => [1,3,2,4,5] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 3
[2,4,5,3,1] => [1,2,4,5,3] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 2
[2,5,1,3,4] => [1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 4
[2,5,4,1,3] => [1,3,2,5,4] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 3
[3,1,2,4,5] => [1,2,4,5,3] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 2
[3,2,4,5,1] => [1,2,4,5,3] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 2
[3,4,5,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[3,4,5,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[3,5,1,2,4] => [1,2,4,3,5] => [5,3,4,2,1] => [5,3,4,2,1] => ? = 2
[3,5,2,4,1] => [1,2,4,3,5] => [5,3,4,2,1] => [5,3,4,2,1] => ? = 2
[3,5,4,1,2] => [1,2,3,5,4] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 1
[3,5,4,2,1] => [1,2,3,5,4] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 1
[4,1,2,3,5] => [1,2,3,5,4] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 1
[4,1,3,2,5] => [1,3,2,5,4] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 3
[4,2,3,5,1] => [1,2,3,5,4] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 1
[4,2,5,1,3] => [1,3,2,5,4] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 3
[4,3,5,1,2] => [1,2,3,5,4] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 1
[4,3,5,2,1] => [1,2,3,5,4] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 1
[4,5,1,2,3] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[4,5,1,3,2] => [1,3,2,4,5] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 3
[4,5,2,1,3] => [1,3,2,4,5] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 3
[4,5,2,3,1] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[4,5,3,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[4,5,3,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[5,1,2,3,4] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[5,1,2,4,3] => [1,2,4,3,5] => [5,3,4,2,1] => [5,3,4,2,1] => ? = 2
[5,1,3,2,4] => [1,3,2,4,5] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 3
[5,1,3,4,2] => [1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 4
[5,2,1,3,4] => [1,3,4,2,5] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 4
[5,2,3,4,1] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[5,2,4,1,3] => [1,3,2,4,5] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 3
[5,2,4,3,1] => [1,2,4,3,5] => [5,3,4,2,1] => [5,3,4,2,1] => ? = 2
[5,3,1,2,4] => [1,2,4,3,5] => [5,3,4,2,1] => [5,3,4,2,1] => ? = 2
[5,3,2,4,1] => [1,2,4,3,5] => [5,3,4,2,1] => [5,3,4,2,1] => ? = 2
[5,3,4,1,2] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[5,3,4,2,1] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[5,4,1,2,3] => [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[5,4,1,3,2] => [1,3,2,4,5] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 3
Description
The pmaj statistic of a parking function.
This is the parking function analogue of the bounce statistic, see [1, Section 1.3].
The definition given there is equivalent to the following: One again and again scans the parking function from right to left, keeping track of how often one has started over again. At step $i$, one marks the next position whose value is at most $i$ with the number of restarts from the end of the parking function. The pmaj statistic is then the sum of the markings.
For example, consider the parking function $[6,2,4,1,4,1,7,3]$.
In the first round, we mark positions $6,4,2$ with $0$'s.
In the second round, we mark positions $8,5,3,1$ with $1$'s.
In the third round, we mark position $7$ with a $2$.
In total, we obtain that
$$\operatorname{pmaj}([6,2,4,1,4,1,7,3]) = 6.$$
This statistic is equidistributed with the area statistic [[St000188]] and the dinv statistic [[St000136]].
Matching statistic: St001207
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 30%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 30%
Values
[1] => [1] => [1] => [1] => ? = 0
[1,2] => [1,2] => [2,1] => [1,2] => 0
[2,1] => [1,2] => [2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => [2,3,1] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,2,1] => [1,3,2] => 1
[2,1,3] => [1,3,2] => [3,2,1] => [1,3,2] => 1
[2,3,1] => [1,2,3] => [2,3,1] => [1,2,3] => 0
[3,1,2] => [1,2,3] => [2,3,1] => [1,2,3] => 0
[3,2,1] => [1,2,3] => [2,3,1] => [1,2,3] => 0
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [2,4,3,1] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => [3,2,4,1] => [1,3,4,2] => 2
[1,3,4,2] => [1,3,4,2] => [4,2,3,1] => [1,4,2,3] => 2
[2,1,3,4] => [1,3,4,2] => [4,2,3,1] => [1,4,2,3] => 2
[2,3,4,1] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 0
[2,4,1,3] => [1,3,2,4] => [3,2,4,1] => [1,3,4,2] => 2
[2,4,3,1] => [1,2,4,3] => [2,4,3,1] => [1,2,4,3] => 1
[3,1,2,4] => [1,2,4,3] => [2,4,3,1] => [1,2,4,3] => 1
[3,2,4,1] => [1,2,4,3] => [2,4,3,1] => [1,2,4,3] => 1
[3,4,1,2] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 0
[3,4,2,1] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 0
[4,1,2,3] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 0
[4,1,3,2] => [1,3,2,4] => [3,2,4,1] => [1,3,4,2] => 2
[4,2,1,3] => [1,3,2,4] => [3,2,4,1] => [1,3,4,2] => 2
[4,2,3,1] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 0
[4,3,1,2] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 0
[4,3,2,1] => [1,2,3,4] => [2,3,4,1] => [1,2,3,4] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => ? = 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => [1,2,3,5,4] => ? = 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => [1,2,4,5,3] => ? = 2
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,1] => [1,2,5,3,4] => ? = 2
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,1] => [1,3,4,5,2] => ? = 3
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,1] => [1,3,5,2,4] => ? = 3
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,1] => [1,4,5,2,3] => ? = 4
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,1] => [1,5,2,3,4] => ? = 3
[2,1,3,4,5] => [1,3,4,5,2] => [5,2,3,4,1] => [1,5,2,3,4] => ? = 3
[2,3,4,5,1] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => ? = 0
[2,3,5,4,1] => [1,2,3,5,4] => [2,3,5,4,1] => [1,2,3,5,4] => ? = 1
[2,4,3,5,1] => [1,2,4,3,5] => [2,4,3,5,1] => [1,2,4,5,3] => ? = 2
[2,4,5,1,3] => [1,3,2,4,5] => [3,2,4,5,1] => [1,3,4,5,2] => ? = 3
[2,4,5,3,1] => [1,2,4,5,3] => [2,5,3,4,1] => [1,2,5,3,4] => ? = 2
[2,5,1,3,4] => [1,3,4,2,5] => [4,2,3,5,1] => [1,4,5,2,3] => ? = 4
[2,5,4,1,3] => [1,3,2,5,4] => [3,2,5,4,1] => [1,3,5,2,4] => ? = 3
[3,1,2,4,5] => [1,2,4,5,3] => [2,5,3,4,1] => [1,2,5,3,4] => ? = 2
[3,2,4,5,1] => [1,2,4,5,3] => [2,5,3,4,1] => [1,2,5,3,4] => ? = 2
[3,4,5,1,2] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => ? = 0
[3,4,5,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => ? = 0
[3,5,1,2,4] => [1,2,4,3,5] => [2,4,3,5,1] => [1,2,4,5,3] => ? = 2
[3,5,2,4,1] => [1,2,4,3,5] => [2,4,3,5,1] => [1,2,4,5,3] => ? = 2
[3,5,4,1,2] => [1,2,3,5,4] => [2,3,5,4,1] => [1,2,3,5,4] => ? = 1
[3,5,4,2,1] => [1,2,3,5,4] => [2,3,5,4,1] => [1,2,3,5,4] => ? = 1
[4,1,2,3,5] => [1,2,3,5,4] => [2,3,5,4,1] => [1,2,3,5,4] => ? = 1
[4,1,3,2,5] => [1,3,2,5,4] => [3,2,5,4,1] => [1,3,5,2,4] => ? = 3
[4,2,3,5,1] => [1,2,3,5,4] => [2,3,5,4,1] => [1,2,3,5,4] => ? = 1
[4,2,5,1,3] => [1,3,2,5,4] => [3,2,5,4,1] => [1,3,5,2,4] => ? = 3
[4,3,5,1,2] => [1,2,3,5,4] => [2,3,5,4,1] => [1,2,3,5,4] => ? = 1
[4,3,5,2,1] => [1,2,3,5,4] => [2,3,5,4,1] => [1,2,3,5,4] => ? = 1
[4,5,1,2,3] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => ? = 0
[4,5,1,3,2] => [1,3,2,4,5] => [3,2,4,5,1] => [1,3,4,5,2] => ? = 3
[4,5,2,1,3] => [1,3,2,4,5] => [3,2,4,5,1] => [1,3,4,5,2] => ? = 3
[4,5,2,3,1] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => ? = 0
[4,5,3,1,2] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => ? = 0
[4,5,3,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => ? = 0
[5,1,2,3,4] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => ? = 0
[5,1,2,4,3] => [1,2,4,3,5] => [2,4,3,5,1] => [1,2,4,5,3] => ? = 2
[5,1,3,2,4] => [1,3,2,4,5] => [3,2,4,5,1] => [1,3,4,5,2] => ? = 3
[5,1,3,4,2] => [1,3,4,2,5] => [4,2,3,5,1] => [1,4,5,2,3] => ? = 4
[5,2,1,3,4] => [1,3,4,2,5] => [4,2,3,5,1] => [1,4,5,2,3] => ? = 4
[5,2,3,4,1] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => ? = 0
[5,2,4,1,3] => [1,3,2,4,5] => [3,2,4,5,1] => [1,3,4,5,2] => ? = 3
[5,2,4,3,1] => [1,2,4,3,5] => [2,4,3,5,1] => [1,2,4,5,3] => ? = 2
[5,3,1,2,4] => [1,2,4,3,5] => [2,4,3,5,1] => [1,2,4,5,3] => ? = 2
[5,3,2,4,1] => [1,2,4,3,5] => [2,4,3,5,1] => [1,2,4,5,3] => ? = 2
[5,3,4,1,2] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => ? = 0
[5,3,4,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => ? = 0
[5,4,1,2,3] => [1,2,3,4,5] => [2,3,4,5,1] => [1,2,3,4,5] => ? = 0
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
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