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Matching statistic: St000054
Mp00252: Permutations —restriction⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2,3] => [1,2] => [1] => 1
[1,3,2] => [1,2] => [1] => 1
[2,1,3] => [2,1] => [1] => 1
[2,3,1] => [2,1] => [1] => 1
[3,1,2] => [1,2] => [1] => 1
[3,2,1] => [2,1] => [1] => 1
[1,2,3,4] => [1,2,3] => [1,2] => 1
[1,2,4,3] => [1,2,3] => [1,2] => 1
[1,3,2,4] => [1,3,2] => [1,2] => 1
[1,3,4,2] => [1,3,2] => [1,2] => 1
[1,4,2,3] => [1,2,3] => [1,2] => 1
[1,4,3,2] => [1,3,2] => [1,2] => 1
[2,1,3,4] => [2,1,3] => [2,1] => 2
[2,1,4,3] => [2,1,3] => [2,1] => 2
[2,3,1,4] => [2,3,1] => [2,1] => 2
[2,3,4,1] => [2,3,1] => [2,1] => 2
[2,4,1,3] => [2,1,3] => [2,1] => 2
[2,4,3,1] => [2,3,1] => [2,1] => 2
[3,1,2,4] => [3,1,2] => [1,2] => 1
[3,1,4,2] => [3,1,2] => [1,2] => 1
[3,2,1,4] => [3,2,1] => [2,1] => 2
[3,2,4,1] => [3,2,1] => [2,1] => 2
[3,4,1,2] => [3,1,2] => [1,2] => 1
[3,4,2,1] => [3,2,1] => [2,1] => 2
[4,1,2,3] => [1,2,3] => [1,2] => 1
[4,1,3,2] => [1,3,2] => [1,2] => 1
[4,2,1,3] => [2,1,3] => [2,1] => 2
[4,2,3,1] => [2,3,1] => [2,1] => 2
[4,3,1,2] => [3,1,2] => [1,2] => 1
[4,3,2,1] => [3,2,1] => [2,1] => 2
[1,2,3,4,5] => [1,2,3,4] => [1,2,3] => 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3] => 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,3] => 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,3] => 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3] => 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,3] => 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2] => 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2] => 1
[1,3,4,2,5] => [1,3,4,2] => [1,3,2] => 1
[1,3,4,5,2] => [1,3,4,2] => [1,3,2] => 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2] => 1
[1,3,5,4,2] => [1,3,4,2] => [1,3,2] => 1
[1,4,2,3,5] => [1,4,2,3] => [1,2,3] => 1
[1,4,2,5,3] => [1,4,2,3] => [1,2,3] => 1
[1,4,3,2,5] => [1,4,3,2] => [1,3,2] => 1
[1,4,3,5,2] => [1,4,3,2] => [1,3,2] => 1
[1,4,5,2,3] => [1,4,2,3] => [1,2,3] => 1
[1,4,5,3,2] => [1,4,3,2] => [1,3,2] => 1
[1,5,2,3,4] => [1,2,3,4] => [1,2,3] => 1
[1,5,2,4,3] => [1,2,4,3] => [1,2,3] => 1
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
Matching statistic: St000740
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 70% ●values known / values provided: 75%●distinct values known / distinct values provided: 70%
Mp00252: Permutations —restriction⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 70% ●values known / values provided: 75%●distinct values known / distinct values provided: 70%
Values
[1,2,3] => [3,2,1] => [2,1] => [1] => 1
[1,3,2] => [2,3,1] => [2,1] => [1] => 1
[2,1,3] => [3,1,2] => [1,2] => [1] => 1
[2,3,1] => [1,3,2] => [1,2] => [1] => 1
[3,1,2] => [2,1,3] => [2,1] => [1] => 1
[3,2,1] => [1,2,3] => [1,2] => [1] => 1
[1,2,3,4] => [4,3,2,1] => [3,2,1] => [2,1] => 1
[1,2,4,3] => [3,4,2,1] => [3,2,1] => [2,1] => 1
[1,3,2,4] => [4,2,3,1] => [2,3,1] => [2,1] => 1
[1,3,4,2] => [2,4,3,1] => [2,3,1] => [2,1] => 1
[1,4,2,3] => [3,2,4,1] => [3,2,1] => [2,1] => 1
[1,4,3,2] => [2,3,4,1] => [2,3,1] => [2,1] => 1
[2,1,3,4] => [4,3,1,2] => [3,1,2] => [1,2] => 2
[2,1,4,3] => [3,4,1,2] => [3,1,2] => [1,2] => 2
[2,3,1,4] => [4,1,3,2] => [1,3,2] => [1,2] => 2
[2,3,4,1] => [1,4,3,2] => [1,3,2] => [1,2] => 2
[2,4,1,3] => [3,1,4,2] => [3,1,2] => [1,2] => 2
[2,4,3,1] => [1,3,4,2] => [1,3,2] => [1,2] => 2
[3,1,2,4] => [4,2,1,3] => [2,1,3] => [2,1] => 1
[3,1,4,2] => [2,4,1,3] => [2,1,3] => [2,1] => 1
[3,2,1,4] => [4,1,2,3] => [1,2,3] => [1,2] => 2
[3,2,4,1] => [1,4,2,3] => [1,2,3] => [1,2] => 2
[3,4,1,2] => [2,1,4,3] => [2,1,3] => [2,1] => 1
[3,4,2,1] => [1,2,4,3] => [1,2,3] => [1,2] => 2
[4,1,2,3] => [3,2,1,4] => [3,2,1] => [2,1] => 1
[4,1,3,2] => [2,3,1,4] => [2,3,1] => [2,1] => 1
[4,2,1,3] => [3,1,2,4] => [3,1,2] => [1,2] => 2
[4,2,3,1] => [1,3,2,4] => [1,3,2] => [1,2] => 2
[4,3,1,2] => [2,1,3,4] => [2,1,3] => [2,1] => 1
[4,3,2,1] => [1,2,3,4] => [1,2,3] => [1,2] => 2
[1,2,3,4,5] => [5,4,3,2,1] => [4,3,2,1] => [3,2,1] => 1
[1,2,3,5,4] => [4,5,3,2,1] => [4,3,2,1] => [3,2,1] => 1
[1,2,4,3,5] => [5,3,4,2,1] => [3,4,2,1] => [3,2,1] => 1
[1,2,4,5,3] => [3,5,4,2,1] => [3,4,2,1] => [3,2,1] => 1
[1,2,5,3,4] => [4,3,5,2,1] => [4,3,2,1] => [3,2,1] => 1
[1,2,5,4,3] => [3,4,5,2,1] => [3,4,2,1] => [3,2,1] => 1
[1,3,2,4,5] => [5,4,2,3,1] => [4,2,3,1] => [2,3,1] => 1
[1,3,2,5,4] => [4,5,2,3,1] => [4,2,3,1] => [2,3,1] => 1
[1,3,4,2,5] => [5,2,4,3,1] => [2,4,3,1] => [2,3,1] => 1
[1,3,4,5,2] => [2,5,4,3,1] => [2,4,3,1] => [2,3,1] => 1
[1,3,5,2,4] => [4,2,5,3,1] => [4,2,3,1] => [2,3,1] => 1
[1,3,5,4,2] => [2,4,5,3,1] => [2,4,3,1] => [2,3,1] => 1
[1,4,2,3,5] => [5,3,2,4,1] => [3,2,4,1] => [3,2,1] => 1
[1,4,2,5,3] => [3,5,2,4,1] => [3,2,4,1] => [3,2,1] => 1
[1,4,3,2,5] => [5,2,3,4,1] => [2,3,4,1] => [2,3,1] => 1
[1,4,3,5,2] => [2,5,3,4,1] => [2,3,4,1] => [2,3,1] => 1
[1,4,5,2,3] => [3,2,5,4,1] => [3,2,4,1] => [3,2,1] => 1
[1,4,5,3,2] => [2,3,5,4,1] => [2,3,4,1] => [2,3,1] => 1
[1,5,2,3,4] => [4,3,2,5,1] => [4,3,2,1] => [3,2,1] => 1
[1,5,2,4,3] => [3,4,2,5,1] => [3,4,2,1] => [3,2,1] => 1
[7,8,5,6,4,3,2,1] => [1,2,3,4,6,5,8,7] => ? => ? => ? = 5
[7,5,6,8,4,3,2,1] => [1,2,3,4,8,6,5,7] => ? => ? => ? = 5
[6,5,7,8,4,3,2,1] => [1,2,3,4,8,7,5,6] => ? => ? => ? = 6
[7,8,6,4,5,3,2,1] => [1,2,3,5,4,6,8,7] => ? => ? => ? = 6
[7,6,4,5,8,3,2,1] => [1,2,3,8,5,4,6,7] => ? => ? => ? = 6
[7,5,4,6,8,3,2,1] => [1,2,3,8,6,4,5,7] => ? => ? => ? = 5
[7,4,5,6,8,3,2,1] => [1,2,3,8,6,5,4,7] => ? => ? => ? = 4
[8,7,6,5,3,4,2,1] => [1,2,4,3,5,6,7,8] => ? => ? => ? = 6
[7,8,6,5,3,4,2,1] => [1,2,4,3,5,6,8,7] => ? => ? => ? = 6
[8,6,7,5,3,4,2,1] => [1,2,4,3,5,7,6,8] => ? => ? => ? = 6
[6,7,8,3,4,5,2,1] => [1,2,5,4,3,8,7,6] => ? => ? => ? = 6
[8,7,5,4,3,6,2,1] => [1,2,6,3,4,5,7,8] => ? => ? => ? = 5
[7,6,5,4,3,8,2,1] => [1,2,8,3,4,5,6,7] => ? => ? => ? = 6
[6,4,5,7,3,8,2,1] => [1,2,8,3,7,5,4,6] => ? => ? => ? = 6
[7,6,4,3,5,8,2,1] => [1,2,8,5,3,4,6,7] => ? => ? => ? = 6
[6,3,4,5,7,8,2,1] => [1,2,8,7,5,4,3,6] => ? => ? => ? = 6
[4,3,5,6,7,8,2,1] => [1,2,8,7,6,5,3,4] => ? => ? => ? = 4
[7,8,6,5,4,2,3,1] => [1,3,2,4,5,6,8,7] => ? => ? => ? = 6
[8,7,5,6,4,2,3,1] => [1,3,2,4,6,5,7,8] => ? => ? => ? = 5
[8,7,6,5,2,3,4,1] => [1,4,3,2,5,6,7,8] => ? => ? => ? = 6
[6,7,8,5,2,3,4,1] => [1,4,3,2,5,8,7,6] => ? => ? => ? = 6
[6,5,3,4,7,2,8,1] => [1,8,2,7,4,3,5,6] => ? => ? => ? = 6
[7,6,5,3,2,4,8,1] => [1,8,4,2,3,5,6,7] => ? => ? => ? = 6
[7,6,5,2,3,4,8,1] => [1,8,4,3,2,5,6,7] => ? => ? => ? = 6
[3,4,5,2,6,7,8,1] => [1,8,7,6,2,5,4,3] => ? => ? => ? = 3
[4,3,2,5,6,7,8,1] => [1,8,7,6,5,2,3,4] => ? => ? => ? = 4
[3,4,2,5,6,7,8,1] => [1,8,7,6,5,2,4,3] => ? => ? => ? = 3
[4,2,3,5,6,7,8,1] => [1,8,7,6,5,3,2,4] => ? => ? => ? = 4
[8,6,7,5,4,3,1,2] => [2,1,3,4,5,7,6,8] => ? => ? => ? = 6
[8,7,5,6,4,3,1,2] => [2,1,3,4,6,5,7,8] => ? => ? => ? = 5
[6,5,7,8,4,3,1,2] => [2,1,3,4,8,7,5,6] => ? => ? => ? = 6
[8,7,6,4,5,3,1,2] => [2,1,3,5,4,6,7,8] => ? => ? => ? = 6
[7,8,6,4,5,3,1,2] => [2,1,3,5,4,6,8,7] => ? => ? => ? = 6
[8,7,6,5,3,4,1,2] => [2,1,4,3,5,6,7,8] => ? => ? => ? = 6
[8,7,5,6,3,4,1,2] => [2,1,4,3,6,5,7,8] => ? => ? => ? = 5
[8,3,4,5,6,7,1,2] => [2,1,7,6,5,4,3,8] => ? => ? => ? = 3
[6,5,7,4,3,8,1,2] => [2,1,8,3,4,7,5,6] => ? => ? => ? = 6
[5,4,6,3,7,8,1,2] => [2,1,8,7,3,6,4,5] => ? => ? => ? = 5
[6,5,7,8,4,1,2,3] => [3,2,1,4,8,7,5,6] => ? => ? => ? = 6
[8,7,6,4,5,1,2,3] => [3,2,1,5,4,6,7,8] => ? => ? => ? = 6
[8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8] => ? => ? => ? = 4
[7,8,4,5,6,1,2,3] => [3,2,1,6,5,4,8,7] => ? => ? => ? = 4
[8,7,6,5,3,1,2,4] => [4,2,1,3,5,6,7,8] => ? => ? => ? = 6
[8,7,6,5,2,1,3,4] => [4,3,1,2,5,6,7,8] => ? => ? => ? = 6
[6,7,8,4,3,2,1,5] => [5,1,2,3,4,8,7,6] => ? => ? => ? = 6
[6,7,8,3,4,2,1,5] => [5,1,2,4,3,8,7,6] => ? => ? => ? = 6
[8,7,6,4,2,3,1,5] => [5,1,3,2,4,6,7,8] => ? => ? => ? = 6
[6,7,8,4,2,3,1,5] => [5,1,3,2,4,8,7,6] => ? => ? => ? = 6
[8,7,6,3,2,4,1,5] => [5,1,4,2,3,6,7,8] => ? => ? => ? = 6
[6,7,8,3,2,4,1,5] => [5,1,4,2,3,8,7,6] => ? => ? => ? = 6
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St000260
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 10%
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 10%
Values
[1,2,3] => [.,[.,[.,.]]]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,3,2] => [.,[[.,.],.]]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[2,1,3] => [[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> ? = 1
[2,3,1] => [[.,.],[.,.]]
=> [1,3,2] => ([(1,2)],3)
=> ? = 1
[3,1,2] => [[.,[.,.]],.]
=> [2,1,3] => ([(1,2)],3)
=> ? = 1
[3,2,1] => [[[.,.],.],.]
=> [1,2,3] => ([],3)
=> ? = 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
=> [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 2
[2,3,1,4] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[2,3,4,1] => [[.,.],[.,[.,.]]]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 2
[2,4,1,3] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 2
[2,4,3,1] => [[.,.],[[.,.],.]]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 2
[3,1,2,4] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 2
[3,2,4,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 1
[3,4,2,1] => [[[.,.],.],[.,.]]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 2
[4,1,2,3] => [[.,[.,[.,.]]],.]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? = 1
[4,1,3,2] => [[.,[[.,.],.]],.]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 1
[4,2,1,3] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 2
[4,3,1,2] => [[[.,[.,.]],.],.]
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,2,3,4] => ([],4)
=> ? = 2
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [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)
=> 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[1,4,5,3,2] => [.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,2,4,3] => [.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,3,2,4] => [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,4,2,3] => [.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,1,5,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,3,1,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,3,5,1,4] => [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,3,5,4,1] => [[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,4,3,1,5] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,4,3,5,1] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,4,5,3,1] => [[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,5,1,4,3] => [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[2,5,3,1,4] => [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,5,3,4,1] => [[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,5,4,1,3] => [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[2,5,4,3,1] => [[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 3
[3,1,2,5,4] => [[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ? = 3
[3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 3
[3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> [6,5,4,3,2,1] => ([(0,1),(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)
=> 1
[1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
=> [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)
=> 1
[1,2,3,5,4,6] => [.,[.,[.,[[.,.],[.,.]]]]]
=> [4,6,5,3,2,1] => ([(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)
=> 1
[1,2,3,5,6,4] => [.,[.,[.,[[.,.],[.,.]]]]]
=> [4,6,5,3,2,1] => ([(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)
=> 1
[1,2,3,6,4,5] => [.,[.,[.,[[.,[.,.]],.]]]]
=> [5,4,6,3,2,1] => ([(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)
=> 1
[1,2,3,6,5,4] => [.,[.,[.,[[[.,.],.],.]]]]
=> [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)
=> 1
[1,2,4,3,5,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> [3,6,5,4,2,1] => ([(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)
=> 1
[1,2,4,3,6,5] => [.,[.,[[.,.],[[.,.],.]]]]
=> [3,5,6,4,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,2,4,5,3,6] => [.,[.,[[.,.],[.,[.,.]]]]]
=> [3,6,5,4,2,1] => ([(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)
=> 1
[1,2,4,5,6,3] => [.,[.,[[.,.],[.,[.,.]]]]]
=> [3,6,5,4,2,1] => ([(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)
=> 1
[1,2,4,6,3,5] => [.,[.,[[.,.],[[.,.],.]]]]
=> [3,5,6,4,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,2,4,6,5,3] => [.,[.,[[.,.],[[.,.],.]]]]
=> [3,5,6,4,2,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,2,5,3,4,6] => [.,[.,[[.,[.,.]],[.,.]]]]
=> [4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,2,5,3,6,4] => [.,[.,[[.,[.,.]],[.,.]]]]
=> [4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,2,5,4,3,6] => [.,[.,[[[.,.],.],[.,.]]]]
=> [3,4,6,5,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,2,5,4,6,3] => [.,[.,[[[.,.],.],[.,.]]]]
=> [3,4,6,5,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,2,5,6,3,4] => [.,[.,[[.,[.,.]],[.,.]]]]
=> [4,3,6,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[1,2,5,6,4,3] => [.,[.,[[[.,.],.],[.,.]]]]
=> [3,4,6,5,2,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St001964
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 10%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 10%
Values
[1,2,3] => [1,3,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[1,3,2] => [1,3,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[2,1,3] => [2,1,3] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[2,3,1] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[3,1,2] => [3,1,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[3,2,1] => [3,2,1] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[1,2,3,4] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,2,4,3] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,3,2,4] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,3,4,2] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,4,2,3] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[1,4,3,2] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[2,1,3,4] => [2,1,4,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 - 1
[2,1,4,3] => [2,1,4,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 - 1
[2,3,1,4] => [2,4,1,3] => [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 - 1
[2,3,4,1] => [2,4,3,1] => [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 - 1
[2,4,1,3] => [2,4,1,3] => [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 - 1
[2,4,3,1] => [2,4,3,1] => [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 - 1
[3,1,2,4] => [3,1,4,2] => [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)
=> ? = 1 - 1
[3,1,4,2] => [3,1,4,2] => [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)
=> ? = 1 - 1
[3,2,1,4] => [3,2,1,4] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2 - 1
[3,2,4,1] => [3,2,4,1] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 - 1
[3,4,1,2] => [3,4,1,2] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 - 1
[3,4,2,1] => [3,4,2,1] => [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 - 1
[4,1,2,3] => [4,1,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 - 1
[4,1,3,2] => [4,1,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 - 1
[4,2,1,3] => [4,2,1,3] => [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 - 1
[4,2,3,1] => [4,2,3,1] => [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 - 1
[4,3,1,2] => [4,3,1,2] => [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)
=> ? = 1 - 1
[4,3,2,1] => [4,3,2,1] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 - 1
[1,2,3,4,5] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,3,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,4,3,5] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,4,5,3] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,5,3,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,2,5,4,3] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,3,2,4,5] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,3,2,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,3,4,2,5] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,3,4,5,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,3,5,2,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,3,5,4,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,4,2,3,5] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,4,2,5,3] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,4,3,2,5] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,4,3,5,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,4,5,2,3] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,4,5,3,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,5,2,3,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,5,2,4,3] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,5,3,2,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,5,3,4,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,5,4,2,3] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[2,1,3,4,5] => [2,1,5,4,3] => [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 - 1
[2,1,3,5,4] => [2,1,5,4,3] => [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 - 1
[2,1,4,3,5] => [2,1,5,4,3] => [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 - 1
[2,1,4,5,3] => [2,1,5,4,3] => [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 - 1
[2,1,5,3,4] => [2,1,5,4,3] => [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 - 1
[2,1,5,4,3] => [2,1,5,4,3] => [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 - 1
[2,3,1,4,5] => [2,5,1,4,3] => [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 - 1
[2,3,1,5,4] => [2,5,1,4,3] => [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 - 1
[2,3,4,1,5] => [2,5,4,1,3] => [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 - 1
[2,3,4,5,1] => [2,5,4,3,1] => [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 - 1
[2,3,5,1,4] => [2,5,4,1,3] => [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 - 1
[2,3,5,4,1] => [2,5,4,3,1] => [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 - 1
[2,4,1,3,5] => [2,5,1,4,3] => [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 - 1
[2,4,1,5,3] => [2,5,1,4,3] => [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 - 1
[2,4,3,1,5] => [2,5,4,1,3] => [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 - 1
[2,4,3,5,1] => [2,5,4,3,1] => [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 - 1
[2,4,5,1,3] => [2,5,4,1,3] => [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 - 1
[2,4,5,3,1] => [2,5,4,3,1] => [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 - 1
[2,5,1,3,4] => [2,5,1,4,3] => [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 - 1
[2,5,1,4,3] => [2,5,1,4,3] => [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 - 1
[2,5,3,1,4] => [2,5,4,1,3] => [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 - 1
[2,5,3,4,1] => [2,5,4,3,1] => [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 - 1
[2,5,4,1,3] => [2,5,4,1,3] => [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 - 1
[2,5,4,3,1] => [2,5,4,3,1] => [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 - 1
[3,1,2,4,5] => [3,1,5,4,2] => [5,1,4,3,2] => ([(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
[3,1,2,5,4] => [3,1,5,4,2] => [5,1,4,3,2] => ([(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
[3,1,4,2,5] => [3,1,5,4,2] => [5,1,4,3,2] => ([(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
[3,1,4,5,2] => [3,1,5,4,2] => [5,1,4,3,2] => ([(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
[3,1,5,2,4] => [3,1,5,4,2] => [5,1,4,3,2] => ([(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
[3,1,5,4,2] => [3,1,5,4,2] => [5,1,4,3,2] => ([(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
[3,2,1,4,5] => [3,2,1,5,4] => [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)
=> ? = 3 - 1
[3,2,1,5,4] => [3,2,1,5,4] => [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)
=> ? = 3 - 1
[1,2,3,4,5,6] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,3,4,6,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,3,5,4,6] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,3,5,6,4] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,3,6,4,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,3,6,5,4] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,4,3,5,6] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,4,3,6,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,4,5,3,6] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,4,5,6,3] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,4,6,3,5] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,4,6,5,3] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,5,3,4,6] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[1,2,5,3,6,4] => [1,6,5,4,3,2] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
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: St000181
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 10%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 10%
Values
[1,2,3] => [1,3,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[1,3,2] => [1,3,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[2,1,3] => [2,1,3] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,3,1] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[3,1,2] => [3,1,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,2,1] => [3,2,1] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,2,4] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,4,2] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,2,3] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,3,2] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,3,4] => [2,1,4,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[2,1,4,3] => [2,1,4,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[2,3,1,4] => [2,4,1,3] => [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
[2,3,4,1] => [2,4,3,1] => [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
[2,4,1,3] => [2,4,1,3] => [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
[2,4,3,1] => [2,4,3,1] => [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
[3,1,2,4] => [3,1,4,2] => [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)
=> ? = 1
[3,1,4,2] => [3,1,4,2] => [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)
=> ? = 1
[3,2,1,4] => [3,2,1,4] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2
[3,2,4,1] => [3,2,4,1] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[3,4,1,2] => [3,4,1,2] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[3,4,2,1] => [3,4,2,1] => [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
[4,1,2,3] => [4,1,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[4,1,3,2] => [4,1,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[4,2,1,3] => [4,2,1,3] => [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
[4,2,3,1] => [4,2,3,1] => [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
[4,3,1,2] => [4,3,1,2] => [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)
=> ? = 1
[4,3,2,1] => [4,3,2,1] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[1,2,3,4,5] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,4,3,5] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,4,5,3] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,5,3,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,5,4,3] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,2,4,5] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,2,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,4,2,5] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,4,5,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,5,2,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,4,2,3,5] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,4,2,5,3] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,4,3,2,5] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,4,3,5,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,4,5,3,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,5,2,3,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,5,2,4,3] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,5,3,2,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,5,3,4,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,5,4,2,3] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,1,3,4,5] => [2,1,5,4,3] => [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
[2,1,3,5,4] => [2,1,5,4,3] => [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
[2,1,4,3,5] => [2,1,5,4,3] => [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
[2,1,4,5,3] => [2,1,5,4,3] => [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
[2,1,5,3,4] => [2,1,5,4,3] => [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
[2,1,5,4,3] => [2,1,5,4,3] => [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
[2,3,1,4,5] => [2,5,1,4,3] => [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
[2,3,1,5,4] => [2,5,1,4,3] => [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
[2,3,4,1,5] => [2,5,4,1,3] => [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
[2,3,4,5,1] => [2,5,4,3,1] => [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
[2,3,5,1,4] => [2,5,4,1,3] => [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
[2,3,5,4,1] => [2,5,4,3,1] => [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
[2,4,1,3,5] => [2,5,1,4,3] => [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
[2,4,1,5,3] => [2,5,1,4,3] => [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
[2,4,3,1,5] => [2,5,4,1,3] => [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
[2,4,3,5,1] => [2,5,4,3,1] => [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
[2,4,5,1,3] => [2,5,4,1,3] => [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
[2,4,5,3,1] => [2,5,4,3,1] => [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
[2,5,1,3,4] => [2,5,1,4,3] => [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
[2,5,1,4,3] => [2,5,1,4,3] => [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
[2,5,3,1,4] => [2,5,4,1,3] => [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
[2,5,3,4,1] => [2,5,4,3,1] => [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
[2,5,4,1,3] => [2,5,4,1,3] => [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
[2,5,4,3,1] => [2,5,4,3,1] => [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
[3,1,2,4,5] => [3,1,5,4,2] => [5,1,4,3,2] => ([(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
[3,1,2,5,4] => [3,1,5,4,2] => [5,1,4,3,2] => ([(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
[3,1,4,2,5] => [3,1,5,4,2] => [5,1,4,3,2] => ([(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
[3,1,4,5,2] => [3,1,5,4,2] => [5,1,4,3,2] => ([(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
[3,1,5,2,4] => [3,1,5,4,2] => [5,1,4,3,2] => ([(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
[3,1,5,4,2] => [3,1,5,4,2] => [5,1,4,3,2] => ([(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
[3,2,1,4,5] => [3,2,1,5,4] => [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)
=> ? = 3
[3,2,1,5,4] => [3,2,1,5,4] => [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)
=> ? = 3
Description
The number of connected components of the Hasse diagram for the poset.
Matching statistic: St001890
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001890: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 10%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001890: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 10%
Values
[1,2,3] => [1,3,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[1,3,2] => [1,3,2] => [3,2,1] => ([(0,2),(2,1)],3)
=> 1
[2,1,3] => [2,1,3] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[2,3,1] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[3,1,2] => [3,1,2] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[3,2,1] => [3,2,1] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,2,4] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,4,2] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,2,3] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,3,2] => [1,4,3,2] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,3,4] => [2,1,4,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[2,1,4,3] => [2,1,4,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[2,3,1,4] => [2,4,1,3] => [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
[2,3,4,1] => [2,4,3,1] => [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
[2,4,1,3] => [2,4,1,3] => [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
[2,4,3,1] => [2,4,3,1] => [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
[3,1,2,4] => [3,1,4,2] => [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)
=> ? = 1
[3,1,4,2] => [3,1,4,2] => [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)
=> ? = 1
[3,2,1,4] => [3,2,1,4] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2
[3,2,4,1] => [3,2,4,1] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[3,4,1,2] => [3,4,1,2] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[3,4,2,1] => [3,4,2,1] => [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
[4,1,2,3] => [4,1,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[4,1,3,2] => [4,1,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[4,2,1,3] => [4,2,1,3] => [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
[4,2,3,1] => [4,2,3,1] => [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
[4,3,1,2] => [4,3,1,2] => [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)
=> ? = 1
[4,3,2,1] => [4,3,2,1] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
[1,2,3,4,5] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,4,3,5] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,4,5,3] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,5,3,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,5,4,3] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,2,4,5] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,2,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,4,2,5] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,4,5,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,5,2,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,5,4,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,4,2,3,5] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,4,2,5,3] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,4,3,2,5] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,4,3,5,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,4,5,2,3] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,4,5,3,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,5,2,3,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,5,2,4,3] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,5,3,2,4] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,5,3,4,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,5,4,2,3] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,1,3,4,5] => [2,1,5,4,3] => [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
[2,1,3,5,4] => [2,1,5,4,3] => [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
[2,1,4,3,5] => [2,1,5,4,3] => [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
[2,1,4,5,3] => [2,1,5,4,3] => [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
[2,1,5,3,4] => [2,1,5,4,3] => [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
[2,1,5,4,3] => [2,1,5,4,3] => [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
[2,3,1,4,5] => [2,5,1,4,3] => [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
[2,3,1,5,4] => [2,5,1,4,3] => [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
[2,3,4,1,5] => [2,5,4,1,3] => [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
[2,3,4,5,1] => [2,5,4,3,1] => [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
[2,3,5,1,4] => [2,5,4,1,3] => [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
[2,3,5,4,1] => [2,5,4,3,1] => [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
[2,4,1,3,5] => [2,5,1,4,3] => [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
[2,4,1,5,3] => [2,5,1,4,3] => [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
[2,4,3,1,5] => [2,5,4,1,3] => [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
[2,4,3,5,1] => [2,5,4,3,1] => [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
[2,4,5,1,3] => [2,5,4,1,3] => [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
[2,4,5,3,1] => [2,5,4,3,1] => [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
[2,5,1,3,4] => [2,5,1,4,3] => [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
[2,5,1,4,3] => [2,5,1,4,3] => [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
[2,5,3,1,4] => [2,5,4,1,3] => [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
[2,5,3,4,1] => [2,5,4,3,1] => [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
[2,5,4,1,3] => [2,5,4,1,3] => [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
[2,5,4,3,1] => [2,5,4,3,1] => [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
[3,1,2,4,5] => [3,1,5,4,2] => [5,1,4,3,2] => ([(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
[3,1,2,5,4] => [3,1,5,4,2] => [5,1,4,3,2] => ([(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
[3,1,4,2,5] => [3,1,5,4,2] => [5,1,4,3,2] => ([(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
[3,1,4,5,2] => [3,1,5,4,2] => [5,1,4,3,2] => ([(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
[3,1,5,2,4] => [3,1,5,4,2] => [5,1,4,3,2] => ([(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
[3,1,5,4,2] => [3,1,5,4,2] => [5,1,4,3,2] => ([(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
[3,2,1,4,5] => [3,2,1,5,4] => [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)
=> ? = 3
[3,2,1,5,4] => [3,2,1,5,4] => [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)
=> ? = 3
Description
The maximum magnitude of the Möbius function of a poset.
The '''Möbius function''' of a poset is the multiplicative inverse of the zeta function in the incidence algebra. The Möbius value $\mu(x, y)$ is equal to the signed sum of chains from $x$ to $y$, where odd-length chains are counted with a minus sign, so this statistic is bounded above by the total number of chains in the poset.
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