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Your data matches 19 different statistics following compositions of up to 3 maps.
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Matching statistic: St000069
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Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1] => ([],1)
=> 1
[[.,.],.]
=> [1,2] => [1] => ([],1)
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => ([],2)
=> 2
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => ([],2)
=> 2
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => ([(0,1)],2)
=> 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => ([],2)
=> 2
[[[.,.],.],.]
=> [1,2,3] => [1,2] => ([(0,1)],2)
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => ([],3)
=> 3
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => ([],3)
=> 3
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => ([(1,2)],3)
=> 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => ([],3)
=> 3
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => ([(1,2)],3)
=> 2
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => ([(1,2)],3)
=> 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => ([(1,2)],3)
=> 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => ([],3)
=> 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => ([(1,2)],3)
=> 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => ([(1,2)],3)
=> 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => ([],4)
=> 4
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => ([],4)
=> 4
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
=> 3
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => ([],4)
=> 4
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => ([(2,3)],4)
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> 3
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => ([],4)
=> 4
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => ([(2,3)],4)
=> 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => ([(2,3)],4)
=> 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 3
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 3
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => ([],4)
=> 4
Description
The number of maximal elements of a poset.
Matching statistic: St000908
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1] => ([],1)
=> 1
[[.,.],.]
=> [1,2] => [1] => ([],1)
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => ([],2)
=> 2
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => ([],2)
=> 2
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => ([(0,1)],2)
=> 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => ([],2)
=> 2
[[[.,.],.],.]
=> [1,2,3] => [1,2] => ([(0,1)],2)
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => ([],3)
=> 3
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => ([],3)
=> 3
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => ([(1,2)],3)
=> 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => ([],3)
=> 3
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => ([(1,2)],3)
=> 2
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => ([(1,2)],3)
=> 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => ([(1,2)],3)
=> 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => ([],3)
=> 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => ([(1,2)],3)
=> 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => ([(1,2)],3)
=> 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => ([],4)
=> 4
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => ([],4)
=> 4
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
=> 3
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => ([],4)
=> 4
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => ([(2,3)],4)
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> 3
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => ([],4)
=> 4
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => ([(2,3)],4)
=> 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => ([(2,3)],4)
=> 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 3
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 3
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => ([],4)
=> 4
Description
The length of the shortest maximal antichain in a poset.
Matching statistic: St000914
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000914: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000914: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1] => ([],1)
=> ? = 1
[[.,.],.]
=> [1,2] => [1] => ([],1)
=> ? = 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => ([],2)
=> 2
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => ([],2)
=> 2
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => ([(0,1)],2)
=> 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => ([],2)
=> 2
[[[.,.],.],.]
=> [1,2,3] => [1,2] => ([(0,1)],2)
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => ([],3)
=> 3
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => ([],3)
=> 3
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => ([(1,2)],3)
=> 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => ([],3)
=> 3
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => ([(1,2)],3)
=> 2
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => ([(1,2)],3)
=> 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => ([(1,2)],3)
=> 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => ([],3)
=> 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => ([(1,2)],3)
=> 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => ([(1,2)],3)
=> 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => ([],4)
=> 4
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => ([],4)
=> 4
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
=> 3
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => ([],4)
=> 4
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => ([(2,3)],4)
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> 3
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => ([],4)
=> 4
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => ([(2,3)],4)
=> 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => ([(2,3)],4)
=> 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 3
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 3
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => ([],4)
=> 4
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [3,4,2,1] => ([(2,3)],4)
=> 3
[[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [4,2,3,1] => ([(2,3)],4)
=> 3
Description
The sum of the values of the Möbius function of a poset.
The Möbius function $\mu$ of a finite poset is defined as
$$\mu (x,y)=\begin{cases} 1& \text{if }x = y\\
-\sum _{z: x\leq z < y}\mu (x,z)& \text{for }x < y\\
0&\text{otherwise}.
\end{cases}
$$
Since $\mu(x,y)=0$ whenever $x\not\leq y$, this statistic is
$$
\sum_{x\leq y} \mu(x,y).
$$
If the poset has a minimal or a maximal element, then the definition implies immediately that the statistic equals $1$. Moreover, the statistic equals the sum of the statistics of the connected components.
This statistic is also called the magnitude of a poset.
Matching statistic: St000234
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1] => [1] => 0 = 1 - 1
[[.,.],.]
=> [1,2] => [1] => [1] => 0 = 1 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [1,2] => 1 = 2 - 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [1,2] => 1 = 2 - 1
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => [2,1] => 0 = 1 - 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [1,2] => 1 = 2 - 1
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [2,1] => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [1,2,3] => 2 = 3 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [1,2,3] => 2 = 3 - 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [1,2,3] => 2 = 3 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => [2,3,1] => 0 = 1 - 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [1,2,3] => 2 = 3 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [2,1,3] => 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => [1,3,2] => 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [2,3,1] => 0 = 1 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [3,2,1] => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 3 = 4 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [1,2,3,4] => 3 = 4 - 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => [2,1,3,4] => 2 = 3 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [1,2,3,4] => 3 = 4 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [2,1,3,4] => 2 = 3 - 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 2 = 3 - 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => [1,3,2,4] => 2 = 3 - 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => [2,3,1,4] => 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => [3,2,1,4] => 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [1,2,3,4] => 3 = 4 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [2,1,3,4] => 2 = 3 - 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => [1,3,2,4] => 2 = 3 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [2,3,1,4] => 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [3,2,1,4] => 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 2 = 3 - 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => [1,2,4,3] => 2 = 3 - 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => [1,2,4,3] => 2 = 3 - 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => [2,1,4,3] => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => [1,3,4,2] => 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => [1,3,4,2] => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => [1,4,3,2] => 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 0 = 1 - 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => [3,2,4,1] => 0 = 1 - 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => [2,4,3,1] => 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0 = 1 - 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => [1,2,3,4] => 3 = 4 - 1
[.,[[.,[.,.]],[[[[.,.],.],.],.]]]
=> [5,6,7,8,3,2,4,1] => [5,6,7,3,2,4,1] => [3,2,1,5,6,4,7] => ? = 3 - 1
[.,[[[[.,[.,.]],.],.],[[.,.],.]]]
=> [7,8,3,2,4,5,6,1] => [7,3,2,4,5,6,1] => [1,5,6,4,3,2,7] => ? = 3 - 1
[.,[[.,[[[.,.],[.,[.,.]]],.]],.]]
=> [6,5,3,4,7,2,8,1] => [6,5,3,4,7,2,1] => [2,3,5,4,1,6,7] => ? = 3 - 1
[[.,.],[[[.,[.,.]],.],[[.,.],.]]]
=> [7,8,4,3,5,6,1,2] => [7,4,3,5,6,1,2] => [1,4,5,3,2,7,6] => ? = 3 - 1
[[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> [8,4,3,5,6,7,1,2] => [4,3,5,6,7,1,2] => [4,5,3,2,1,7,6] => ? = 2 - 1
[[.,.],[[.,[[[.,[.,.]],.],.]],.]]
=> [5,4,6,7,3,8,1,2] => [5,4,6,7,3,1,2] => [3,4,2,1,5,7,6] => ? = 3 - 1
[[.,.],[[[.,[[.,[.,.]],.]],.],.]]
=> [5,4,6,3,7,8,1,2] => [5,4,6,3,7,1,2] => [3,4,2,5,1,7,6] => ? = 2 - 1
[[.,.],[[[[.,[.,.]],[.,.]],.],.]]
=> [6,4,3,5,7,8,1,2] => [6,4,3,5,7,1,2] => [2,4,5,3,1,7,6] => ? = 2 - 1
[[.,.],[[[[.,[.,[.,.]]],.],.],.]]
=> [5,4,3,6,7,8,1,2] => [5,4,3,6,7,1,2] => [3,4,5,2,1,7,6] => ? = 2 - 1
[[.,.],[[[[[.,[.,.]],.],.],.],.]]
=> [4,3,5,6,7,8,1,2] => [4,3,5,6,7,1,2] => [4,5,3,2,1,7,6] => ? = 2 - 1
[[.,[.,.]],[.,[[[[.,.],.],.],.]]]
=> [5,6,7,8,4,2,1,3] => [5,6,7,4,2,1,3] => [3,2,1,4,6,7,5] => ? = 3 - 1
[[.,[.,.]],[[[[[.,.],.],.],.],.]]
=> [4,5,6,7,8,2,1,3] => [4,5,6,7,2,1,3] => [4,3,2,1,6,7,5] => ? = 2 - 1
[[.,[.,[.,.]]],[[[[.,.],.],.],.]]
=> [5,6,7,8,3,2,1,4] => [5,6,7,3,2,1,4] => [3,2,1,5,6,7,4] => ? = 2 - 1
[[[.,[.,.]],.],[.,[[[.,.],.],.]]]
=> [6,7,8,5,2,1,3,4] => [6,7,5,2,1,3,4] => [2,1,3,6,7,5,4] => ? = 3 - 1
[[[.,[.,.]],.],[[.,.],[[.,.],.]]]
=> [7,8,5,6,2,1,3,4] => [7,5,6,2,1,3,4] => [1,3,2,6,7,5,4] => ? = 3 - 1
[[[.,[.,.]],.],[[[.,[.,.]],.],.]]
=> [6,5,7,8,2,1,3,4] => [6,5,7,2,1,3,4] => [2,3,1,6,7,5,4] => ? = 2 - 1
[[[.,[.,.]],.],[[[[.,.],.],.],.]]
=> [5,6,7,8,2,1,3,4] => [5,6,7,2,1,3,4] => [3,2,1,6,7,5,4] => ? = 2 - 1
[[[[.,.],.],.],[[.,[.,.]],[.,.]]]
=> [8,6,5,7,1,2,3,4] => [6,5,7,1,2,3,4] => [2,3,1,7,6,5,4] => ? = 2 - 1
[[[[.,.],.],.],[[[.,[.,.]],.],.]]
=> [6,5,7,8,1,2,3,4] => [6,5,7,1,2,3,4] => [2,3,1,7,6,5,4] => ? = 2 - 1
[[.,[.,[[.,.],.]]],[[[.,.],.],.]]
=> [6,7,8,3,4,2,1,5] => [6,7,3,4,2,1,5] => [2,1,5,4,6,7,3] => ? = 2 - 1
[[.,[[.,.],[.,.]]],[.,[.,[.,.]]]]
=> [8,7,6,4,2,3,1,5] => [7,6,4,2,3,1,5] => [1,2,4,6,5,7,3] => ? = 3 - 1
[[.,[[.,.],[.,.]]],[[[.,.],.],.]]
=> [6,7,8,4,2,3,1,5] => [6,7,4,2,3,1,5] => [2,1,4,6,5,7,3] => ? = 2 - 1
[[.,[[.,[.,.]],.]],[[[.,.],.],.]]
=> [6,7,8,3,2,4,1,5] => [6,7,3,2,4,1,5] => [2,1,5,6,4,7,3] => ? = 2 - 1
[[[.,.],[.,[.,.]]],[[[.,.],.],.]]
=> [6,7,8,4,3,1,2,5] => [6,7,4,3,1,2,5] => [2,1,4,5,7,6,3] => ? = 2 - 1
[[[.,[.,.]],[.,.]],[[[.,.],.],.]]
=> [6,7,8,4,2,1,3,5] => [6,7,4,2,1,3,5] => [2,1,4,6,7,5,3] => ? = 2 - 1
[[[.,[.,[.,.]]],.],[[[.,.],.],.]]
=> [6,7,8,3,2,1,4,5] => [6,7,3,2,1,4,5] => [2,1,5,6,7,4,3] => ? = 2 - 1
[[[[.,[.,.]],.],.],[[[.,.],.],.]]
=> [6,7,8,2,1,3,4,5] => [6,7,2,1,3,4,5] => [2,1,6,7,5,4,3] => ? = 2 - 1
[[.,[[.,[.,.]],[[[.,.],.],.]]],.]
=> [5,6,7,3,2,4,1,8] => [5,6,7,3,2,4,1] => [3,2,1,5,6,4,7] => ? = 3 - 1
[[.,[[[.,.],.],[[.,[.,.]],.]]],.]
=> [6,5,7,2,3,4,1,8] => [6,5,7,2,3,4,1] => [2,3,1,6,5,4,7] => ? = 3 - 1
[[[.,[.,.]],[[[.,.],.],[.,.]]],.]
=> [7,4,5,6,2,1,3,8] => [7,4,5,6,2,1,3] => [1,4,3,2,6,7,5] => ? = 3 - 1
[[[[.,.],.],[[[.,[.,.]],.],.]],.]
=> [5,4,6,7,1,2,3,8] => [5,4,6,7,1,2,3] => [3,4,2,1,7,6,5] => ? = 2 - 1
[[[[.,[.,.]],.],[[[.,.],.],.]],.]
=> [5,6,7,2,1,3,4,8] => [5,6,7,2,1,3,4] => [3,2,1,6,7,5,4] => ? = 2 - 1
[[[[[.,.],.],.],[[.,[.,.]],.]],.]
=> [6,5,7,1,2,3,4,8] => [6,5,7,1,2,3,4] => [2,3,1,7,6,5,4] => ? = 2 - 1
Description
The number of global ascents of a permutation.
The global ascents are the integers $i$ such that
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i < k \leq n: \pi(j) < \pi(k)\}.$$
Equivalently, by the pigeonhole principle,
$$C(\pi)=\{i\in [n-1] \mid \forall 1 \leq j \leq i: \pi(j) \leq i \}.$$
For $n > 1$ it can also be described as an occurrence of the mesh pattern
$$([1,2], \{(0,2),(1,0),(1,1),(2,0),(2,1) \})$$
or equivalently
$$([1,2], \{(0,1),(0,2),(1,1),(1,2),(2,0) \}),$$
see [3].
According to [2], this is also the cardinality of the connectivity set of a permutation. The permutation is connected, when the connectivity set is empty. This gives [[oeis:A003319]].
Matching statistic: St000740
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1] => [1] => 1
[[.,.],.]
=> [1,2] => [1] => [1] => 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [1,2] => 2
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [1,2] => 2
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => [2,1] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [1,2] => 2
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [2,1] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [1,2,3] => 3
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [1,2,3] => 3
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => [3,1,2] => 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [1,2,3] => 3
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [3,1,2] => 2
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => [1,3,2] => 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => [1,3,2] => 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => [3,2,1] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => [2,3,1] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [1,2,3] => 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [3,1,2] => 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => [1,3,2] => 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [3,2,1] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [2,3,1] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 4
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [1,2,3,4] => 4
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => [4,1,2,3] => 3
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [1,2,3,4] => 4
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [4,1,2,3] => 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => [1,4,2,3] => 3
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => [1,4,2,3] => 3
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => [4,3,1,2] => 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => [3,4,1,2] => 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [1,2,3,4] => 4
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [4,1,2,3] => 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => [1,4,2,3] => 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [4,3,1,2] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [3,4,1,2] => 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 3
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => [1,2,4,3] => 3
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => [4,1,3,2] => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => [1,2,4,3] => 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => [4,1,3,2] => 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => [1,4,3,2] => 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => [1,4,3,2] => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => [1,3,4,2] => 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => [1,3,4,2] => 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => [4,3,2,1] => 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => [3,4,2,1] => 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => [4,2,3,1] => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => [3,2,4,1] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => [1,2,3,4] => 4
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [6,7,8,5,4,3,2,1] => [6,7,5,4,3,2,1] => [7,1,2,3,4,5,6] => ? = 6
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [7,8,5,6,4,3,2,1] => [7,5,6,4,3,2,1] => [1,7,2,3,4,5,6] => ? = 6
[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> [7,5,6,8,4,3,2,1] => [7,5,6,4,3,2,1] => [1,7,2,3,4,5,6] => ? = 6
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [6,5,7,8,4,3,2,1] => [6,5,7,4,3,2,1] => [7,6,1,2,3,4,5] => ? = 5
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [5,6,7,8,4,3,2,1] => [5,6,7,4,3,2,1] => [6,7,1,2,3,4,5] => ? = 5
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [8,6,7,4,5,3,2,1] => [6,7,4,5,3,2,1] => [7,1,6,2,3,4,5] => ? = 5
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [6,5,7,4,8,3,2,1] => [6,5,7,4,3,2,1] => [7,6,1,2,3,4,5] => ? = 5
[.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> [7,5,4,6,8,3,2,1] => [7,5,4,6,3,2,1] => [1,7,6,2,3,4,5] => ? = 5
[.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> [7,4,5,6,8,3,2,1] => [7,4,5,6,3,2,1] => [1,6,7,2,3,4,5] => ? = 5
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [4,5,6,7,8,3,2,1] => [4,5,6,7,3,2,1] => [5,6,7,1,2,3,4] => ? = 4
[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [8,6,7,5,3,4,2,1] => [6,7,5,3,4,2,1] => [7,1,2,6,3,4,5] => ? = 5
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [8,7,5,6,3,4,2,1] => [7,5,6,3,4,2,1] => [1,7,2,6,3,4,5] => ? = 5
[.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [8,7,5,4,3,6,2,1] => [7,5,4,3,6,2,1] => [1,7,6,5,2,3,4] => ? = 4
[.,[.,[[.,[[[.,[.,.]],.],.]],.]]]
=> [5,4,6,7,3,8,2,1] => [5,4,6,7,3,2,1] => [6,5,7,1,2,3,4] => ? = 4
[.,[.,[[[[[.,[.,.]],.],.],.],.]]]
=> [4,3,5,6,7,8,2,1] => [4,3,5,6,7,2,1] => [5,4,6,7,1,2,3] => ? = 3
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [8,6,7,5,4,2,3,1] => [6,7,5,4,2,3,1] => [7,1,2,3,6,4,5] => ? = 5
[.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> [8,7,5,6,4,2,3,1] => [7,5,6,4,2,3,1] => [1,7,2,3,6,4,5] => ? = 5
[.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [8,6,5,7,3,2,4,1] => [6,5,7,3,2,4,1] => [7,6,1,5,4,2,3] => ? = 3
[.,[[.,[.,.]],[[[[.,.],.],.],.]]]
=> [5,6,7,8,3,2,4,1] => [5,6,7,3,2,4,1] => [6,7,1,5,4,2,3] => ? = 3
[.,[[[[.,[.,.]],.],.],[[.,.],.]]]
=> [7,8,3,2,4,5,6,1] => [7,3,2,4,5,6,1] => [1,5,4,6,7,2,3] => ? = 3
[.,[[.,[.,[.,[.,[[.,.],.]]]]],.]]
=> [6,7,5,4,3,2,8,1] => [6,7,5,4,3,2,1] => [7,1,2,3,4,5,6] => ? = 6
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [6,5,7,4,3,2,8,1] => [6,5,7,4,3,2,1] => [7,6,1,2,3,4,5] => ? = 5
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [6,5,4,3,7,2,8,1] => [6,5,4,3,7,2,1] => [7,6,5,4,1,2,3] => ? = 3
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [5,4,6,3,7,2,8,1] => [5,4,6,3,7,2,1] => [6,5,7,4,1,2,3] => ? = 3
[.,[[.,[[[.,.],[.,[.,.]]],.]],.]]
=> [6,5,3,4,7,2,8,1] => [6,5,3,4,7,2,1] => [7,6,4,5,1,2,3] => ? = 3
[.,[[[[.,[.,[.,[.,.]]]],.],.],.]]
=> [5,4,3,2,6,7,8,1] => [5,4,3,2,6,7,1] => [6,5,4,3,7,1,2] => ? = 2
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [4,3,2,5,6,7,8,1] => [4,3,2,5,6,7,1] => [5,4,3,6,7,1,2] => ? = 2
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,1] => [3,2,4,5,6,7,1] => [4,3,5,6,7,1,2] => ? = 2
[[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> [8,6,7,5,4,3,1,2] => [6,7,5,4,3,1,2] => [7,1,2,3,4,6,5] => ? = 5
[[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> [8,7,5,6,4,3,1,2] => [7,5,6,4,3,1,2] => [1,7,2,3,4,6,5] => ? = 5
[[.,.],[.,[.,[[[.,[.,.]],.],.]]]]
=> [6,5,7,8,4,3,1,2] => [6,5,7,4,3,1,2] => [7,6,1,2,3,5,4] => ? = 4
[[.,.],[[.,.],[[[.,[.,.]],.],.]]]
=> [6,5,7,8,3,4,1,2] => [6,5,7,3,4,1,2] => [7,6,1,5,2,4,3] => ? = 3
[[.,.],[[.,[.,[.,.]]],[[.,.],.]]]
=> [7,8,5,4,3,6,1,2] => [7,5,4,3,6,1,2] => [1,7,6,5,2,4,3] => ? = 3
[[.,.],[[[.,[.,.]],.],[[.,.],.]]]
=> [7,8,4,3,5,6,1,2] => [7,4,3,5,6,1,2] => [1,6,5,7,2,4,3] => ? = 3
[[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> [8,4,3,5,6,7,1,2] => [4,3,5,6,7,1,2] => [5,4,6,7,1,3,2] => ? = 2
[[.,.],[[.,[.,[[.,[.,.]],.]]],.]]
=> [6,5,7,4,3,8,1,2] => [6,5,7,4,3,1,2] => [7,6,1,2,3,5,4] => ? = 4
[[.,.],[[.,[[[.,[.,.]],.],.]],.]]
=> [5,4,6,7,3,8,1,2] => [5,4,6,7,3,1,2] => [6,5,7,1,2,4,3] => ? = 3
[[.,.],[[[.,[[.,[.,.]],.]],.],.]]
=> [5,4,6,3,7,8,1,2] => [5,4,6,3,7,1,2] => [6,5,7,4,1,3,2] => ? = 2
[[.,.],[[[[.,[.,.]],[.,.]],.],.]]
=> [6,4,3,5,7,8,1,2] => [6,4,3,5,7,1,2] => [7,5,4,6,1,3,2] => ? = 2
[[.,.],[[[[.,[.,[.,.]]],.],.],.]]
=> [5,4,3,6,7,8,1,2] => [5,4,3,6,7,1,2] => [6,5,4,7,1,3,2] => ? = 2
[[.,.],[[[[[.,[.,.]],.],.],.],.]]
=> [4,3,5,6,7,8,1,2] => [4,3,5,6,7,1,2] => [5,4,6,7,1,3,2] => ? = 2
[[.,[.,.]],[.,[[[[.,.],.],.],.]]]
=> [5,6,7,8,4,2,1,3] => [5,6,7,4,2,1,3] => [6,7,1,2,5,4,3] => ? = 3
[[.,[.,.]],[[[[[.,.],.],.],.],.]]
=> [4,5,6,7,8,2,1,3] => [4,5,6,7,2,1,3] => [5,6,7,1,4,3,2] => ? = 2
[[[.,.],.],[.,[[[.,[.,.]],.],.]]]
=> [6,5,7,8,4,1,2,3] => [6,5,7,4,1,2,3] => [7,6,1,2,4,5,3] => ? = 3
[[[.,.],.],[[[.,[.,[.,.]]],.],.]]
=> [6,5,4,7,8,1,2,3] => [6,5,4,7,1,2,3] => [7,6,5,1,3,4,2] => ? = 2
[[.,[.,[.,.]]],[[.,.],[[.,.],.]]]
=> [7,8,5,6,3,2,1,4] => [7,5,6,3,2,1,4] => [1,7,2,6,5,4,3] => ? = 3
[[.,[.,[.,.]]],[[[[.,.],.],.],.]]
=> [5,6,7,8,3,2,1,4] => [5,6,7,3,2,1,4] => [6,7,1,5,4,3,2] => ? = 2
[[[.,[.,.]],.],[.,[[[.,.],.],.]]]
=> [6,7,8,5,2,1,3,4] => [6,7,5,2,1,3,4] => [7,1,2,5,4,6,3] => ? = 3
[[[.,[.,.]],.],[[.,.],[[.,.],.]]]
=> [7,8,5,6,2,1,3,4] => [7,5,6,2,1,3,4] => [1,7,2,5,4,6,3] => ? = 3
[[[.,[.,.]],.],[[[.,[.,.]],.],.]]
=> [6,5,7,8,2,1,3,4] => [6,5,7,2,1,3,4] => [7,6,1,4,3,5,2] => ? = 2
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St000007
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1] => 1
[[.,.],.]
=> [1,2] => [1] => 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => 2
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => 2
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => 2
[[[.,.],.],.]
=> [1,2,3] => [1,2] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => 3
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => 3
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => 3
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => 2
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => 4
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => 4
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => 3
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => 4
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => 3
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => 3
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => 4
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => 3
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => 3
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => 4
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [6,7,8,5,4,3,2,1] => [6,7,5,4,3,2,1] => ? = 6
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [7,8,5,6,4,3,2,1] => [7,5,6,4,3,2,1] => ? = 6
[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> [7,5,6,8,4,3,2,1] => [7,5,6,4,3,2,1] => ? = 6
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [6,5,7,8,4,3,2,1] => [6,5,7,4,3,2,1] => ? = 5
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [5,6,7,8,4,3,2,1] => [5,6,7,4,3,2,1] => ? = 5
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [8,7,6,4,5,3,2,1] => [7,6,4,5,3,2,1] => ? = 6
[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [7,8,6,4,5,3,2,1] => [7,6,4,5,3,2,1] => ? = 6
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [8,6,7,4,5,3,2,1] => [6,7,4,5,3,2,1] => ? = 5
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [6,5,7,4,8,3,2,1] => [6,5,7,4,3,2,1] => ? = 5
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [7,6,4,5,8,3,2,1] => [7,6,4,5,3,2,1] => ? = 6
[.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> [7,5,4,6,8,3,2,1] => [7,5,4,6,3,2,1] => ? = 5
[.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> [7,4,5,6,8,3,2,1] => [7,4,5,6,3,2,1] => ? = 5
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [4,5,6,7,8,3,2,1] => [4,5,6,7,3,2,1] => ? = 4
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [8,7,6,5,3,4,2,1] => [7,6,5,3,4,2,1] => ? = 6
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [7,8,6,5,3,4,2,1] => [7,6,5,3,4,2,1] => ? = 6
[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [8,6,7,5,3,4,2,1] => [6,7,5,3,4,2,1] => ? = 5
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [8,7,5,6,3,4,2,1] => [7,5,6,3,4,2,1] => ? = 5
[.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [8,7,5,4,3,6,2,1] => [7,5,4,3,6,2,1] => ? = 4
[.,[.,[[.,[[[.,[.,.]],.],.]],.]]]
=> [5,4,6,7,3,8,2,1] => [5,4,6,7,3,2,1] => ? = 4
[.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]
=> [7,6,5,3,4,8,2,1] => [7,6,5,3,4,2,1] => ? = 6
[.,[.,[[[.,[.,.]],[.,[.,.]]],.]]]
=> [7,6,4,3,5,8,2,1] => [7,6,4,3,5,2,1] => ? = 5
[.,[.,[[[[.,.],.],[.,[.,.]]],.]]]
=> [7,6,3,4,5,8,2,1] => [7,6,3,4,5,2,1] => ? = 5
[.,[.,[[[[[.,[.,.]],.],.],.],.]]]
=> [4,3,5,6,7,8,2,1] => [4,3,5,6,7,2,1] => ? = 3
[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [8,7,6,5,4,2,3,1] => [7,6,5,4,2,3,1] => ? = 6
[.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> [7,8,6,5,4,2,3,1] => [7,6,5,4,2,3,1] => ? = 6
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [8,6,7,5,4,2,3,1] => [6,7,5,4,2,3,1] => ? = 5
[.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> [8,7,5,6,4,2,3,1] => [7,5,6,4,2,3,1] => ? = 5
[.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> [8,7,6,4,5,2,3,1] => [7,6,4,5,2,3,1] => ? = 5
[.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> [8,7,6,5,3,2,4,1] => [7,6,5,3,2,4,1] => ? = 5
[.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [8,6,5,7,3,2,4,1] => [6,5,7,3,2,4,1] => ? = 3
[.,[[.,[.,.]],[[[[.,.],.],.],.]]]
=> [5,6,7,8,3,2,4,1] => [5,6,7,3,2,4,1] => ? = 3
[.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]
=> [8,7,6,5,2,3,4,1] => [7,6,5,2,3,4,1] => ? = 5
[.,[[[[.,[.,.]],.],.],[[.,.],.]]]
=> [7,8,3,2,4,5,6,1] => [7,3,2,4,5,6,1] => ? = 3
[.,[[.,[.,[.,[.,[[.,.],.]]]]],.]]
=> [6,7,5,4,3,2,8,1] => [6,7,5,4,3,2,1] => ? = 6
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [6,5,7,4,3,2,8,1] => [6,5,7,4,3,2,1] => ? = 5
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [6,5,4,3,7,2,8,1] => [6,5,4,3,7,2,1] => ? = 3
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [5,4,6,3,7,2,8,1] => [5,4,6,3,7,2,1] => ? = 3
[.,[[.,[[[.,.],[.,[.,.]]],.]],.]]
=> [6,5,3,4,7,2,8,1] => [6,5,3,4,7,2,1] => ? = 3
[.,[[[.,.],[.,[.,[.,[.,.]]]]],.]]
=> [7,6,5,4,2,3,8,1] => [7,6,5,4,2,3,1] => ? = 6
[.,[[[.,[.,.]],[.,[.,[.,.]]]],.]]
=> [7,6,5,3,2,4,8,1] => [7,6,5,3,2,4,1] => ? = 5
[.,[[[[.,.],.],[.,[.,[.,.]]]],.]]
=> [7,6,5,2,3,4,8,1] => [7,6,5,2,3,4,1] => ? = 5
[.,[[[[.,[.,[.,[.,.]]]],.],.],.]]
=> [5,4,3,2,6,7,8,1] => [5,4,3,2,6,7,1] => ? = 2
[[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> [8,6,7,5,4,3,1,2] => [6,7,5,4,3,1,2] => ? = 5
[[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> [8,7,5,6,4,3,1,2] => [7,5,6,4,3,1,2] => ? = 5
[[.,.],[.,[.,[[[.,[.,.]],.],.]]]]
=> [6,5,7,8,4,3,1,2] => [6,5,7,4,3,1,2] => ? = 4
[[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
=> [8,7,6,4,5,3,1,2] => [7,6,4,5,3,1,2] => ? = 5
[[.,.],[.,[[.,.],[.,[[.,.],.]]]]]
=> [7,8,6,4,5,3,1,2] => [7,6,4,5,3,1,2] => ? = 5
[[.,.],[[.,[.,[.,.]]],[[.,.],.]]]
=> [7,8,5,4,3,6,1,2] => [7,5,4,3,6,1,2] => ? = 3
[[.,.],[[[.,[.,.]],.],[[.,.],.]]]
=> [7,8,4,3,5,6,1,2] => [7,4,3,5,6,1,2] => ? = 3
[[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> [8,4,3,5,6,7,1,2] => [4,3,5,6,7,1,2] => ? = 2
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000546
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000546: Permutations ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
St000546: Permutations ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1] => 0 = 1 - 1
[[.,.],.]
=> [1,2] => [1] => 0 = 1 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => 1 = 2 - 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => 1 = 2 - 1
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => 0 = 1 - 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => 1 = 2 - 1
[[[.,.],.],.]
=> [1,2,3] => [1,2] => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => 2 = 3 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => 2 = 3 - 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => 2 = 3 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => 0 = 1 - 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => 0 = 1 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => 2 = 3 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => 0 = 1 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => 3 = 4 - 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => 2 = 3 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => 3 = 4 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => 2 = 3 - 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => 2 = 3 - 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => 2 = 3 - 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => 3 = 4 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => 2 = 3 - 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => 2 = 3 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => 2 = 3 - 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => 2 = 3 - 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => 2 = 3 - 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => 0 = 1 - 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => 3 = 4 - 1
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [6,7,8,5,4,3,2,1] => [6,7,5,4,3,2,1] => ? = 6 - 1
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [7,8,5,6,4,3,2,1] => [7,5,6,4,3,2,1] => ? = 6 - 1
[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> [7,5,6,8,4,3,2,1] => [7,5,6,4,3,2,1] => ? = 6 - 1
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [6,5,7,8,4,3,2,1] => [6,5,7,4,3,2,1] => ? = 5 - 1
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [5,6,7,8,4,3,2,1] => [5,6,7,4,3,2,1] => ? = 5 - 1
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [8,7,6,4,5,3,2,1] => [7,6,4,5,3,2,1] => ? = 6 - 1
[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [7,8,6,4,5,3,2,1] => [7,6,4,5,3,2,1] => ? = 6 - 1
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [8,6,7,4,5,3,2,1] => [6,7,4,5,3,2,1] => ? = 5 - 1
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [6,5,7,4,8,3,2,1] => [6,5,7,4,3,2,1] => ? = 5 - 1
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [7,6,4,5,8,3,2,1] => [7,6,4,5,3,2,1] => ? = 6 - 1
[.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> [7,5,4,6,8,3,2,1] => [7,5,4,6,3,2,1] => ? = 5 - 1
[.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> [7,4,5,6,8,3,2,1] => [7,4,5,6,3,2,1] => ? = 5 - 1
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [4,5,6,7,8,3,2,1] => [4,5,6,7,3,2,1] => ? = 4 - 1
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [8,7,6,5,3,4,2,1] => [7,6,5,3,4,2,1] => ? = 6 - 1
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [7,8,6,5,3,4,2,1] => [7,6,5,3,4,2,1] => ? = 6 - 1
[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [8,6,7,5,3,4,2,1] => [6,7,5,3,4,2,1] => ? = 5 - 1
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [8,7,5,6,3,4,2,1] => [7,5,6,3,4,2,1] => ? = 5 - 1
[.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [8,7,5,4,3,6,2,1] => [7,5,4,3,6,2,1] => ? = 4 - 1
[.,[.,[[.,[[[.,[.,.]],.],.]],.]]]
=> [5,4,6,7,3,8,2,1] => [5,4,6,7,3,2,1] => ? = 4 - 1
[.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]
=> [7,6,5,3,4,8,2,1] => [7,6,5,3,4,2,1] => ? = 6 - 1
[.,[.,[[[.,[.,.]],[.,[.,.]]],.]]]
=> [7,6,4,3,5,8,2,1] => [7,6,4,3,5,2,1] => ? = 5 - 1
[.,[.,[[[[.,.],.],[.,[.,.]]],.]]]
=> [7,6,3,4,5,8,2,1] => [7,6,3,4,5,2,1] => ? = 5 - 1
[.,[.,[[[[[.,[.,.]],.],.],.],.]]]
=> [4,3,5,6,7,8,2,1] => [4,3,5,6,7,2,1] => ? = 3 - 1
[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [8,7,6,5,4,2,3,1] => [7,6,5,4,2,3,1] => ? = 6 - 1
[.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> [7,8,6,5,4,2,3,1] => [7,6,5,4,2,3,1] => ? = 6 - 1
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [8,6,7,5,4,2,3,1] => [6,7,5,4,2,3,1] => ? = 5 - 1
[.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> [8,7,5,6,4,2,3,1] => [7,5,6,4,2,3,1] => ? = 5 - 1
[.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> [8,7,6,4,5,2,3,1] => [7,6,4,5,2,3,1] => ? = 5 - 1
[.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> [8,7,6,5,3,2,4,1] => [7,6,5,3,2,4,1] => ? = 5 - 1
[.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [8,6,5,7,3,2,4,1] => [6,5,7,3,2,4,1] => ? = 3 - 1
[.,[[.,[.,.]],[[[[.,.],.],.],.]]]
=> [5,6,7,8,3,2,4,1] => [5,6,7,3,2,4,1] => ? = 3 - 1
[.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]
=> [8,7,6,5,2,3,4,1] => [7,6,5,2,3,4,1] => ? = 5 - 1
[.,[[[[.,[.,.]],.],.],[[.,.],.]]]
=> [7,8,3,2,4,5,6,1] => [7,3,2,4,5,6,1] => ? = 3 - 1
[.,[[.,[.,[.,[.,[[.,.],.]]]]],.]]
=> [6,7,5,4,3,2,8,1] => [6,7,5,4,3,2,1] => ? = 6 - 1
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [6,5,7,4,3,2,8,1] => [6,5,7,4,3,2,1] => ? = 5 - 1
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [6,5,4,3,7,2,8,1] => [6,5,4,3,7,2,1] => ? = 3 - 1
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [5,4,6,3,7,2,8,1] => [5,4,6,3,7,2,1] => ? = 3 - 1
[.,[[.,[[[.,.],[.,[.,.]]],.]],.]]
=> [6,5,3,4,7,2,8,1] => [6,5,3,4,7,2,1] => ? = 3 - 1
[.,[[[.,.],[.,[.,[.,[.,.]]]]],.]]
=> [7,6,5,4,2,3,8,1] => [7,6,5,4,2,3,1] => ? = 6 - 1
[.,[[[.,[.,.]],[.,[.,[.,.]]]],.]]
=> [7,6,5,3,2,4,8,1] => [7,6,5,3,2,4,1] => ? = 5 - 1
[.,[[[[.,.],.],[.,[.,[.,.]]]],.]]
=> [7,6,5,2,3,4,8,1] => [7,6,5,2,3,4,1] => ? = 5 - 1
[.,[[[[.,[.,[.,[.,.]]]],.],.],.]]
=> [5,4,3,2,6,7,8,1] => [5,4,3,2,6,7,1] => ? = 2 - 1
[[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> [8,6,7,5,4,3,1,2] => [6,7,5,4,3,1,2] => ? = 5 - 1
[[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> [8,7,5,6,4,3,1,2] => [7,5,6,4,3,1,2] => ? = 5 - 1
[[.,.],[.,[.,[[[.,[.,.]],.],.]]]]
=> [6,5,7,8,4,3,1,2] => [6,5,7,4,3,1,2] => ? = 4 - 1
[[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
=> [8,7,6,4,5,3,1,2] => [7,6,4,5,3,1,2] => ? = 5 - 1
[[.,.],[.,[[.,.],[.,[[.,.],.]]]]]
=> [7,8,6,4,5,3,1,2] => [7,6,4,5,3,1,2] => ? = 5 - 1
[[.,.],[[.,[.,[.,.]]],[[.,.],.]]]
=> [7,8,5,4,3,6,1,2] => [7,5,4,3,6,1,2] => ? = 3 - 1
[[.,.],[[[.,[.,.]],.],[[.,.],.]]]
=> [7,8,4,3,5,6,1,2] => [7,4,3,5,6,1,2] => ? = 3 - 1
[[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> [8,4,3,5,6,7,1,2] => [4,3,5,6,7,1,2] => ? = 2 - 1
Description
The number of global descents of a permutation.
The global descents are the integers in the set
$$C(\pi)=\{i\in [n-1] : \forall 1 \leq j \leq i < k \leq n :\quad \pi(j) > \pi(k)\}.$$
In particular, if $i\in C(\pi)$ then $i$ is a descent.
For the number of global ascents, see [[St000234]].
Matching statistic: St000654
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000654: Permutations ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000654: Permutations ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1,2] => [1] => ? = 1
[[.,.],.]
=> [1,2] => [2,1] => [1] => ? = 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [1,2] => 2
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [1,2] => 2
[[.,.],[.,.]]
=> [1,3,2] => [2,3,1] => [2,1] => 1
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => [1,2] => 2
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => [2,1] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3] => 3
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => [1,2,3] => 3
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,3,4,2] => [1,3,2] => 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => [1,2,3] => 3
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => [1,3,2] => 2
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,3,4,1] => [2,3,1] => 2
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,3,1] => [2,3,1] => 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,4,1,2] => [3,1,2] => 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [3,4,2,1] => [3,2,1] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => [1,2,3] => 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => [1,3,2] => 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [4,2,3,1] => [2,3,1] => 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => [3,1,2] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => [3,2,1] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4] => 4
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => [1,2,3,4] => 4
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,2,4,5,3] => [1,2,4,3] => 3
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => [1,2,3,4] => 4
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,3] => 3
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,3,4,5,2] => [1,3,4,2] => 3
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,3,5,4,2] => [1,3,4,2] => 3
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,4,5,2,3] => [1,4,2,3] => 2
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,4,5,3,2] => [1,4,3,2] => 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => [1,2,3,4] => 4
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => [1,2,4,3] => 3
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,5,3,4,2] => [1,3,4,2] => 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => [1,4,2,3] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,4,3,2] => 2
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [2,3,4,5,1] => [2,3,4,1] => 3
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [2,3,5,4,1] => [2,3,4,1] => 3
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [2,4,5,3,1] => [2,4,3,1] => 2
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [2,5,3,4,1] => [2,3,4,1] => 3
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [2,5,4,3,1] => [2,4,3,1] => 2
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [3,4,5,1,2] => [3,4,1,2] => 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [3,5,4,1,2] => [3,4,1,2] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [3,4,5,2,1] => [3,4,2,1] => 2
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [3,5,4,2,1] => [3,4,2,1] => 2
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [4,5,1,2,3] => [4,1,2,3] => 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [4,5,1,3,2] => [4,1,3,2] => 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [4,5,2,3,1] => [4,2,3,1] => 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [4,5,3,1,2] => [4,3,1,2] => 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [4,5,3,2,1] => [4,3,2,1] => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [5,1,2,3,4] => [1,2,3,4] => 4
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [5,1,2,4,3] => [1,2,4,3] => 3
[[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [5,1,3,4,2] => [1,3,4,2] => 3
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [5,7,8,6,4,3,2,1] => [1,2,3,4,6,8,7,5] => ? => ? = 6
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [6,5,7,8,4,3,2,1] => [1,2,3,4,8,7,5,6] => ? => ? = 5
[.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> [5,4,7,6,8,3,2,1] => [1,2,3,8,6,7,4,5] => ? => ? = 5
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [3,7,8,6,5,4,2,1] => [1,2,4,5,6,8,7,3] => ? => ? = 6
[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [3,6,8,7,5,4,2,1] => [1,2,4,5,7,8,6,3] => ? => ? = 5
[.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [5,4,3,8,7,6,2,1] => [1,2,6,7,8,3,4,5] => ? => ? = 4
[.,[.,[[.,[.,[.,[.,[.,.]]]]],.]]]
=> [7,6,5,4,3,8,2,1] => [1,2,8,3,4,5,6,7] => ? => ? = 7
[.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]
=> [3,7,6,5,4,8,2,1] => [1,2,8,4,5,6,7,3] => ? => ? = 6
[.,[.,[[[.,[.,.]],[.,[.,.]]],.]]]
=> [4,3,7,6,5,8,2,1] => [1,2,8,5,6,7,3,4] => ? => ? = 5
[.,[.,[[[[.,.],.],[.,[.,.]]],.]]]
=> [3,4,7,6,5,8,2,1] => [1,2,8,5,6,7,4,3] => ? => ? = 5
[.,[.,[[[[[.,[.,.]],.],.],.],.]]]
=> [4,3,5,6,7,8,2,1] => [1,2,8,7,6,5,3,4] => ? => ? = 3
[.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> [2,5,8,7,6,4,3,1] => [1,3,4,6,7,8,5,2] => ? => ? = 5
[.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [3,2,6,5,8,7,4,1] => [1,4,7,8,5,6,2,3] => ? => ? = 3
[.,[[.,[.,.]],[[[[.,.],.],.],.]]]
=> [3,2,5,6,7,8,4,1] => [1,4,8,7,6,5,2,3] => ? => ? = 3
[.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]
=> [2,3,8,7,6,5,4,1] => [1,4,5,6,7,8,3,2] => ? => ? = 5
[.,[[[[.,[.,.]],.],.],[[.,.],.]]]
=> [3,2,4,5,7,8,6,1] => [1,6,8,7,5,4,2,3] => ? => ? = 3
[.,[[.,[[[.,.],[.,[.,.]]],.]],.]]
=> [3,6,5,4,7,2,8,1] => [1,8,2,7,4,5,6,3] => ? => ? = 3
[.,[[[.,.],[.,[.,[.,[.,.]]]]],.]]
=> [2,7,6,5,4,3,8,1] => [1,8,3,4,5,6,7,2] => ? => ? = 6
[.,[[[.,[.,.]],[.,[.,[.,.]]]],.]]
=> [3,2,7,6,5,4,8,1] => [1,8,4,5,6,7,2,3] => ? => ? = 5
[.,[[[[.,.],.],[.,[.,[.,.]]]],.]]
=> [2,3,7,6,5,4,8,1] => [1,8,4,5,6,7,3,2] => ? => ? = 5
[.,[[[[.,[.,[.,[.,.]]]],.],.],.]]
=> [5,4,3,2,6,7,8,1] => [1,8,7,6,2,3,4,5] => [1,7,6,2,3,4,5] => ? = 2
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [4,3,2,5,6,7,8,1] => [1,8,7,6,5,2,3,4] => ? => ? = 2
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,1] => [1,8,7,6,5,4,2,3] => [1,7,6,5,4,2,3] => ? = 2
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,8,7,6,5,4,3,2] => [2,3,4,5,6,7,8,1] => [2,3,4,5,6,7,1] => ? = 6
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,7,8,6,5,4,3,2] => [2,3,4,5,6,8,7,1] => [2,3,4,5,6,7,1] => ? = 6
[[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,6,8,7,5,4,3,2] => [2,3,4,5,7,8,6,1] => ? => ? = 5
[[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,5,8,7,6,4,3,2] => [2,3,4,6,7,8,5,1] => ? => ? = 5
[[.,.],[.,[.,[[[.,[.,.]],.],.]]]]
=> [1,6,5,7,8,4,3,2] => [2,3,4,8,7,5,6,1] => [2,3,4,7,5,6,1] => ? = 4
[[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,4,8,7,6,5,3,2] => [2,3,5,6,7,8,4,1] => [2,3,5,6,7,4,1] => ? = 5
[[.,.],[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,4,7,8,6,5,3,2] => [2,3,5,6,8,7,4,1] => [2,3,5,6,7,4,1] => ? = 5
[[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,3,8,7,6,5,4,2] => [2,4,5,6,7,8,3,1] => ? => ? = 5
[[.,.],[[.,.],[[.,[.,[.,.]]],.]]]
=> [1,3,7,6,5,8,4,2] => [2,4,8,5,6,7,3,1] => ? => ? = 5
[[.,.],[[.,.],[[[.,[.,.]],.],.]]]
=> [1,3,6,5,7,8,4,2] => [2,4,8,7,5,6,3,1] => ? => ? = 3
[[.,.],[[.,[.,[.,.]]],[[.,.],.]]]
=> [1,5,4,3,7,8,6,2] => [2,6,8,7,3,4,5,1] => ? => ? = 3
[[.,.],[[[.,[.,.]],.],[[.,.],.]]]
=> [1,4,3,5,7,8,6,2] => [2,6,8,7,5,3,4,1] => ? => ? = 3
[[.,.],[[[[.,[.,.]],.],.],[.,.]]]
=> [1,4,3,5,6,8,7,2] => [2,7,8,6,5,3,4,1] => ? => ? = 2
[[.,.],[[.,[.,[.,[.,[.,.]]]]],.]]
=> [1,7,6,5,4,3,8,2] => [2,8,3,4,5,6,7,1] => [2,3,4,5,6,7,1] => ? = 6
[[.,.],[[.,[.,[[.,[.,.]],.]]],.]]
=> [1,6,5,7,4,3,8,2] => [2,8,3,4,7,5,6,1] => ? => ? = 4
[[.,.],[[.,[[[.,[.,.]],.],.]],.]]
=> [1,5,4,6,7,3,8,2] => [2,8,3,7,6,4,5,1] => ? => ? = 3
[[.,.],[[[.,[[.,[.,.]],.]],.],.]]
=> [1,5,4,6,3,7,8,2] => [2,8,7,3,6,4,5,1] => ? => ? = 2
[[.,.],[[[[.,[.,.]],[.,.]],.],.]]
=> [1,4,3,6,5,7,8,2] => [2,8,7,5,6,3,4,1] => ? => ? = 2
[[.,.],[[[[.,[.,[.,.]]],.],.],.]]
=> [1,5,4,3,6,7,8,2] => [2,8,7,6,3,4,5,1] => ? => ? = 2
[[.,.],[[[[[.,[.,.]],.],.],.],.]]
=> [1,4,3,5,6,7,8,2] => [2,8,7,6,5,3,4,1] => [2,7,6,5,3,4,1] => ? = 2
[[.,[.,.]],[.,[.,[.,[.,[.,.]]]]]]
=> [2,1,8,7,6,5,4,3] => [3,4,5,6,7,8,1,2] => [3,4,5,6,7,1,2] => ? = 5
[[.,[.,.]],[.,[.,[[.,[.,.]],.]]]]
=> [2,1,7,6,8,5,4,3] => [3,4,5,8,6,7,1,2] => ? => ? = 5
[[.,[.,.]],[.,[[[[.,.],.],.],.]]]
=> [2,1,5,6,7,8,4,3] => [3,4,8,7,6,5,1,2] => [3,4,7,6,5,1,2] => ? = 3
[[.,[.,.]],[[[[[.,.],.],.],.],.]]
=> [2,1,4,5,6,7,8,3] => [3,8,7,6,5,4,1,2] => [3,7,6,5,4,1,2] => ? = 2
[[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
=> [1,2,8,7,6,5,4,3] => [3,4,5,6,7,8,2,1] => ? => ? = 5
Description
The first descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the smallest index $0 < i \leq n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(n+1)=0$.
Matching statistic: St000989
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000989: Permutations ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000989: Permutations ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [1,1,0,0]
=> [1,2] => [1] => ? = 1 - 1
[[.,.],.]
=> [1,0,1,0]
=> [2,1] => [1] => ? = 1 - 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,2] => 1 = 2 - 1
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> [3,1,2] => [1,2] => 1 = 2 - 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> [2,1,3] => [2,1] => 0 = 1 - 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,3,2] => [1,2] => 1 = 2 - 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [2,3,1] => [2,1] => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3] => 2 = 3 - 1
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,2,3] => 2 = 3 - 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,2,3] => 2 = 3 - 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,2] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3] => 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,1,3] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2] => 0 = 1 - 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,3,1] => 0 = 1 - 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,3] => 2 = 3 - 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,1,2] => 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,3] => 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,3,2] => 0 = 1 - 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,3,1] => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4] => 3 = 4 - 1
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1,2,3] => 2 = 3 - 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [1,2,3,4] => 3 = 4 - 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [4,1,2,3] => 2 = 3 - 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4] => 2 = 3 - 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,1,2,4] => 2 = 3 - 1
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [1,4,2,3] => 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,4,1,2] => 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [1,2,3,4] => 3 = 4 - 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [4,1,2,3] => 2 = 3 - 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,1,2,4] => 2 = 3 - 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [1,4,2,3] => 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,4,1,2] => 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4] => 2 = 3 - 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,3,4] => 2 = 3 - 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,4,1,3] => 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,3,4] => 2 = 3 - 1
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [2,4,1,3] => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4] => 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,3,2,4] => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4] => 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [2,3,1,4] => 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [1,2,4,3] => 0 = 1 - 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,1,4,2] => 0 = 1 - 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3] => 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,3,4,2] => 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,3,4,1] => 0 = 1 - 1
[[.,[.,[.,[.,.]]]],.]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [1,2,3,4] => 3 = 4 - 1
[[.,[.,[[.,.],.]]],.]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [4,1,2,3] => 2 = 3 - 1
[[.,[[.,.],[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,4] => 2 = 3 - 1
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [7,8,1,2,3,4,5,6] => [7,1,2,3,4,5,6] => ? = 6 - 1
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [6,8,1,2,3,4,5,7] => ? => ? = 6 - 1
[.,[.,[.,[.,[[.,[.,[.,.]]],.]]]]]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,2,8,3,4,5,6,7] => ? => ? = 7 - 1
[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [6,1,8,2,3,4,5,7] => [6,1,2,3,4,5,7] => ? = 6 - 1
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
=> [1,7,8,2,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 5 - 1
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [6,7,8,1,2,3,4,5] => [6,7,1,2,3,4,5] => ? = 5 - 1
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [5,1,2,3,4,6,7,8] => [5,1,2,3,4,6,7] => ? = 6 - 1
[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> [5,8,1,2,3,4,6,7] => ? => ? = 6 - 1
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [5,7,1,2,3,4,6,8] => [5,7,1,2,3,4,6] => ? = 5 - 1
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> [1,7,2,8,3,4,5,6] => [1,7,2,3,4,5,6] => ? = 5 - 1
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [1,1,1,1,0,1,1,1,0,0,0,1,0,0,0,0]
=> [5,1,2,8,3,4,6,7] => [5,1,2,3,4,6,7] => ? = 6 - 1
[.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> [1,1,1,1,1,0,0,1,1,0,0,1,0,0,0,0]
=> [1,6,2,8,3,4,5,7] => [1,6,2,3,4,5,7] => ? = 5 - 1
[.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> [1,1,1,1,0,1,0,1,1,0,0,1,0,0,0,0]
=> [5,6,1,8,2,3,4,7] => ? => ? = 5 - 1
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [5,6,7,8,1,2,3,4] => [5,6,7,1,2,3,4] => ? = 4 - 1
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [4,1,2,3,5,6,7,8] => [4,1,2,3,5,6,7] => ? = 6 - 1
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> [4,8,1,2,3,5,6,7] => [4,1,2,3,5,6,7] => ? = 6 - 1
[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> [4,7,1,2,3,5,6,8] => [4,7,1,2,3,5,6] => ? = 5 - 1
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [1,1,1,0,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,6,1,2,3,5,7,8] => [4,6,1,2,3,5,7] => ? = 5 - 1
[.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [1,2,6,3,4,5,7,8] => ? => ? = 4 - 1
[.,[.,[[.,[[[.,[.,.]],.],.]],.]]]
=> [1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0]
=> [1,6,7,2,8,3,4,5] => [1,6,7,2,3,4,5] => ? = 4 - 1
[.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]
=> [1,1,1,0,1,1,1,1,0,0,0,0,1,0,0,0]
=> [4,1,2,3,8,5,6,7] => [4,1,2,3,5,6,7] => ? = 6 - 1
[.,[.,[[[.,[.,.]],[.,[.,.]]],.]]]
=> [1,1,1,1,0,0,1,1,1,0,0,0,1,0,0,0]
=> [1,5,2,3,8,4,6,7] => ? => ? = 5 - 1
[.,[.,[[[[.,.],.],[.,[.,.]]],.]]]
=> [1,1,1,0,1,0,1,1,1,0,0,0,1,0,0,0]
=> [4,5,1,2,8,3,6,7] => [4,5,1,2,3,6,7] => ? = 5 - 1
[.,[.,[[[[[.,[.,.]],.],.],.],.]]]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [1,5,6,7,8,2,3,4] => [1,5,6,7,2,3,4] => ? = 3 - 1
[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [3,1,2,4,5,6,7,8] => [3,1,2,4,5,6,7] => ? = 6 - 1
[.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> [1,1,0,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [3,8,1,2,4,5,6,7] => ? => ? = 6 - 1
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [1,1,0,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [3,7,1,2,4,5,6,8] => [3,7,1,2,4,5,6] => ? = 5 - 1
[.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> [1,1,0,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [3,6,1,2,4,5,7,8] => ? => ? = 5 - 1
[.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [3,5,1,2,4,6,7,8] => ? => ? = 5 - 1
[.,[[.,[.,.]],[[[[.,.],.],.],.]]]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,4,6,7,8,2,3,5] => ? => ? = 3 - 1
[.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]
=> [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [3,4,1,2,5,6,7,8] => [3,4,1,2,5,6,7] => ? = 5 - 1
[.,[[[[.,[.,.]],.],.],[[.,.],.]]]
=> [1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,4,5,6,8,2,3,7] => ? => ? = 3 - 1
[.,[[.,[.,[.,[.,[[.,.],.]]]]],.]]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [7,1,2,3,4,8,5,6] => [7,1,2,3,4,5,6] => ? = 6 - 1
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,7,2,3,4,8,5,6] => [1,7,2,3,4,5,6] => ? = 5 - 1
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,6,2,7,3,8,4,5] => [1,6,2,7,3,4,5] => ? = 3 - 1
[.,[[.,[[[.,.],[.,[.,.]]],.]],.]]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,1,0,0]
=> [4,1,2,7,3,8,5,6] => ? => ? = 3 - 1
[.,[[[.,.],[.,[.,[.,[.,.]]]]],.]]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [3,1,2,4,5,8,6,7] => [3,1,2,4,5,6,7] => ? = 6 - 1
[.,[[[[.,.],.],[.,[.,[.,.]]]],.]]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [3,4,1,2,5,8,6,7] => ? => ? = 5 - 1
[.,[[[[.,[.,[.,[.,.]]]],.],.],.]]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> [1,2,3,6,7,8,4,5] => ? => ? = 2 - 1
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7] => ? = 6 - 1
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 6 - 1
[[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [2,7,1,3,4,5,6,8] => [2,7,1,3,4,5,6] => ? = 5 - 1
[[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [2,6,1,3,4,5,7,8] => [2,6,1,3,4,5,7] => ? = 5 - 1
[[.,.],[.,[.,[[[.,[.,.]],.],.]]]]
=> [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [2,1,7,8,3,4,5,6] => [2,1,7,3,4,5,6] => ? = 4 - 1
[[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> [2,5,1,3,4,6,7,8] => ? => ? = 5 - 1
[[.,.],[.,[[.,.],[.,[[.,.],.]]]]]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [2,5,8,1,3,4,6,7] => ? => ? = 5 - 1
[[.,.],[[.,.],[.,[.,[.,[.,.]]]]]]
=> [1,0,1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,4,1,3,5,6,7,8] => ? => ? = 5 - 1
[[.,.],[[.,.],[[.,[.,[.,.]]],.]]]
=> [1,0,1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,4,1,3,8,5,6,7] => [2,4,1,3,5,6,7] => ? = 5 - 1
Description
The number of final rises of a permutation.
For a permutation $\pi$ of length $n$, this is the maximal $k$ such that
$$\pi(n-k) \leq \pi(n-k+1) \leq \cdots \leq \pi(n-1) \leq \pi(n).$$
Equivalently, this is $n-1$ minus the position of the last descent [[St000653]].
Matching statistic: St001640
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1] => [1] => 0 = 1 - 1
[[.,.],.]
=> [1,2] => [1] => [1] => 0 = 1 - 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [1,2] => 1 = 2 - 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [1,2] => 1 = 2 - 1
[[.,.],[.,.]]
=> [1,3,2] => [1,2] => [2,1] => 0 = 1 - 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [1,2] => 1 = 2 - 1
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [2,1] => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [1,2,3] => 2 = 3 - 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [1,2,3] => 2 = 3 - 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,3,1] => [3,1,2] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [1,2,3] => 2 = 3 - 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [3,1,2] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,3,2] => [2,1,3] => 1 = 2 - 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,2] => [2,1,3] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,3] => [3,2,1] => 0 = 1 - 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,3] => [2,3,1] => 0 = 1 - 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [1,2,3] => 2 = 3 - 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [3,1,2] => 1 = 2 - 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2] => [2,1,3] => 1 = 2 - 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [3,2,1] => 0 = 1 - 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [2,3,1] => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 3 = 4 - 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [1,2,3,4] => 3 = 4 - 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [3,4,2,1] => [4,1,2,3] => 2 = 3 - 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [1,2,3,4] => 3 = 4 - 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [4,1,2,3] => 2 = 3 - 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,4,3,1] => [3,1,2,4] => 2 = 3 - 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,4,3,1] => [3,1,2,4] => 2 = 3 - 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,4,1] => [4,3,1,2] => 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [2,3,4,1] => [3,4,1,2] => 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [1,2,3,4] => 3 = 4 - 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [4,1,2,3] => 2 = 3 - 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,4,3,1] => [3,1,2,4] => 2 = 3 - 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [4,3,1,2] => 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [3,4,1,2] => 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,4,3,2] => [2,1,3,4] => 2 = 3 - 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,4,3,2] => [2,1,3,4] => 2 = 3 - 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,3,4,2] => [2,4,1,3] => 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,4,3,2] => [2,1,3,4] => 2 = 3 - 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,3,4,2] => [2,4,1,3] => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,4,3] => [3,2,1,4] => 1 = 2 - 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,4,3] => [3,2,1,4] => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,4,3] => [2,3,1,4] => 1 = 2 - 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,4,3] => [2,3,1,4] => 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,4] => [4,3,2,1] => 0 = 1 - 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,3,1,4] => [3,4,2,1] => 0 = 1 - 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,4] => [2,4,3,1] => 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,4] => [3,2,4,1] => 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,4] => [2,3,4,1] => 0 = 1 - 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => [1,2,3,4] => 3 = 4 - 1
[.,[.,[.,[.,[.,[[[.,.],.],.]]]]]]
=> [6,7,8,5,4,3,2,1] => [6,7,5,4,3,2,1] => [7,1,2,3,4,5,6] => ? = 6 - 1
[.,[.,[.,[.,[[.,.],[[.,.],.]]]]]]
=> [5,7,8,6,4,3,2,1] => ? => ? => ? = 6 - 1
[.,[.,[.,[.,[[[.,.],[.,.]],.]]]]]
=> [5,7,6,8,4,3,2,1] => ? => ? => ? = 6 - 1
[.,[.,[.,[.,[[[.,[.,.]],.],.]]]]]
=> [6,5,7,8,4,3,2,1] => [6,5,7,4,3,2,1] => [7,6,1,2,3,4,5] => ? = 5 - 1
[.,[.,[.,[.,[[[[.,.],.],.],.]]]]]
=> [5,6,7,8,4,3,2,1] => [5,6,7,4,3,2,1] => [6,7,1,2,3,4,5] => ? = 5 - 1
[.,[.,[.,[[.,.],[.,[.,[.,.]]]]]]]
=> [4,8,7,6,5,3,2,1] => [4,7,6,5,3,2,1] => [5,1,2,3,4,6,7] => ? = 6 - 1
[.,[.,[.,[[.,.],[.,[[.,.],.]]]]]]
=> [4,7,8,6,5,3,2,1] => ? => ? => ? = 6 - 1
[.,[.,[.,[[.,.],[[.,.],[.,.]]]]]]
=> [4,6,8,7,5,3,2,1] => ? => ? => ? = 5 - 1
[.,[.,[.,[[.,[[.,[.,.]],.]],.]]]]
=> [6,5,7,4,8,3,2,1] => [6,5,7,4,3,2,1] => [7,6,1,2,3,4,5] => ? = 5 - 1
[.,[.,[.,[[[.,.],[.,[.,.]]],.]]]]
=> [4,7,6,5,8,3,2,1] => [4,7,6,5,3,2,1] => [5,1,2,3,4,6,7] => ? = 6 - 1
[.,[.,[.,[[[.,[.,.]],[.,.]],.]]]]
=> [5,4,7,6,8,3,2,1] => ? => ? => ? = 5 - 1
[.,[.,[.,[[[[.,.],.],[.,.]],.]]]]
=> [4,5,7,6,8,3,2,1] => ? => ? => ? = 5 - 1
[.,[.,[.,[[[[[.,.],.],.],.],.]]]]
=> [4,5,6,7,8,3,2,1] => [4,5,6,7,3,2,1] => [5,6,7,1,2,3,4] => ? = 4 - 1
[.,[.,[[.,.],[.,[.,[.,[.,.]]]]]]]
=> [3,8,7,6,5,4,2,1] => ? => ? => ? = 6 - 1
[.,[.,[[.,.],[.,[.,[[.,.],.]]]]]]
=> [3,7,8,6,5,4,2,1] => ? => ? => ? = 6 - 1
[.,[.,[[.,.],[.,[[.,.],[.,.]]]]]]
=> [3,6,8,7,5,4,2,1] => ? => ? => ? = 5 - 1
[.,[.,[[.,.],[[.,.],[.,[.,.]]]]]]
=> [3,5,8,7,6,4,2,1] => [3,5,7,6,4,2,1] => [4,6,1,2,3,5,7] => ? = 5 - 1
[.,[.,[[.,[.,[.,.]]],[.,[.,.]]]]]
=> [5,4,3,8,7,6,2,1] => ? => ? => ? = 4 - 1
[.,[.,[[.,[[[.,[.,.]],.],.]],.]]]
=> [5,4,6,7,3,8,2,1] => [5,4,6,7,3,2,1] => [6,5,7,1,2,3,4] => ? = 4 - 1
[.,[.,[[[.,.],[.,[.,[.,.]]]],.]]]
=> [3,7,6,5,4,8,2,1] => ? => ? => ? = 6 - 1
[.,[.,[[[.,[.,.]],[.,[.,.]]],.]]]
=> [4,3,7,6,5,8,2,1] => ? => ? => ? = 5 - 1
[.,[.,[[[[.,.],.],[.,[.,.]]],.]]]
=> [3,4,7,6,5,8,2,1] => ? => ? => ? = 5 - 1
[.,[.,[[[[[.,[.,.]],.],.],.],.]]]
=> [4,3,5,6,7,8,2,1] => [4,3,5,6,7,2,1] => [5,4,6,7,1,2,3] => ? = 3 - 1
[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
=> [2,8,7,6,5,4,3,1] => ? => ? => ? = 6 - 1
[.,[[.,.],[.,[.,[.,[[.,.],.]]]]]]
=> [2,7,8,6,5,4,3,1] => [2,7,6,5,4,3,1] => [3,1,2,4,5,6,7] => ? = 6 - 1
[.,[[.,.],[.,[.,[[.,.],[.,.]]]]]]
=> [2,6,8,7,5,4,3,1] => [2,6,7,5,4,3,1] => [3,7,1,2,4,5,6] => ? = 5 - 1
[.,[[.,.],[.,[[.,.],[.,[.,.]]]]]]
=> [2,5,8,7,6,4,3,1] => ? => ? => ? = 5 - 1
[.,[[.,.],[[.,.],[.,[.,[.,.]]]]]]
=> [2,4,8,7,6,5,3,1] => [2,4,7,6,5,3,1] => [3,5,1,2,4,6,7] => ? = 5 - 1
[.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
=> [3,2,8,7,6,5,4,1] => ? => ? => ? = 5 - 1
[.,[[.,[.,.]],[[.,[.,.]],[.,.]]]]
=> [3,2,6,5,8,7,4,1] => ? => ? => ? = 3 - 1
[.,[[.,[.,.]],[[[[.,.],.],.],.]]]
=> [3,2,5,6,7,8,4,1] => ? => ? => ? = 3 - 1
[.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]
=> [2,3,8,7,6,5,4,1] => ? => ? => ? = 5 - 1
[.,[[[[.,[.,.]],.],.],[[.,.],.]]]
=> [3,2,4,5,7,8,6,1] => ? => ? => ? = 3 - 1
[.,[[.,[.,[.,[.,[[.,.],.]]]]],.]]
=> [6,7,5,4,3,2,8,1] => [6,7,5,4,3,2,1] => [7,1,2,3,4,5,6] => ? = 6 - 1
[.,[[.,[.,[.,[[.,[.,.]],.]]]],.]]
=> [6,5,7,4,3,2,8,1] => [6,5,7,4,3,2,1] => [7,6,1,2,3,4,5] => ? = 5 - 1
[.,[[.,[[.,[.,[.,[.,.]]]],.]],.]]
=> [6,5,4,3,7,2,8,1] => [6,5,4,3,7,2,1] => [7,6,5,4,1,2,3] => ? = 3 - 1
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
=> [5,4,6,3,7,2,8,1] => [5,4,6,3,7,2,1] => [6,5,7,4,1,2,3] => ? = 3 - 1
[.,[[.,[[[.,.],[.,[.,.]]],.]],.]]
=> [3,6,5,4,7,2,8,1] => ? => ? => ? = 3 - 1
[.,[[[.,.],[.,[.,[.,[.,.]]]]],.]]
=> [2,7,6,5,4,3,8,1] => ? => ? => ? = 6 - 1
[.,[[[.,[.,.]],[.,[.,[.,.]]]],.]]
=> [3,2,7,6,5,4,8,1] => ? => ? => ? = 5 - 1
[.,[[[[.,.],.],[.,[.,[.,.]]]],.]]
=> [2,3,7,6,5,4,8,1] => ? => ? => ? = 5 - 1
[.,[[[[.,[.,[.,[.,.]]]],.],.],.]]
=> [5,4,3,2,6,7,8,1] => [5,4,3,2,6,7,1] => [6,5,4,3,7,1,2] => ? = 2 - 1
[.,[[[[[.,[.,[.,.]]],.],.],.],.]]
=> [4,3,2,5,6,7,8,1] => [4,3,2,5,6,7,1] => [5,4,3,6,7,1,2] => ? = 2 - 1
[.,[[[[[[.,[.,.]],.],.],.],.],.]]
=> [3,2,4,5,6,7,8,1] => [3,2,4,5,6,7,1] => [4,3,5,6,7,1,2] => ? = 2 - 1
[[.,.],[.,[.,[.,[.,[.,[.,.]]]]]]]
=> [1,8,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => [2,1,3,4,5,6,7] => ? = 6 - 1
[[.,.],[.,[.,[.,[.,[[.,.],.]]]]]]
=> [1,7,8,6,5,4,3,2] => [1,7,6,5,4,3,2] => [2,1,3,4,5,6,7] => ? = 6 - 1
[[.,.],[.,[.,[.,[[.,.],[.,.]]]]]]
=> [1,6,8,7,5,4,3,2] => [1,6,7,5,4,3,2] => [2,7,1,3,4,5,6] => ? = 5 - 1
[[.,.],[.,[.,[[.,.],[.,[.,.]]]]]]
=> [1,5,8,7,6,4,3,2] => [1,5,7,6,4,3,2] => [2,6,1,3,4,5,7] => ? = 5 - 1
[[.,.],[.,[.,[[[.,[.,.]],.],.]]]]
=> [1,6,5,7,8,4,3,2] => ? => ? => ? = 4 - 1
[[.,.],[.,[[.,.],[.,[.,[.,.]]]]]]
=> [1,4,8,7,6,5,3,2] => [1,4,7,6,5,3,2] => [2,5,1,3,4,6,7] => ? = 5 - 1
Description
The number of ascent tops in the permutation such that all smaller elements appear before.
The following 9 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000056The decomposition (or block) number of a permutation. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000990The first ascent of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St001557The number of inversions of the second entry of a permutation. St000735The last entry on the main diagonal of a standard tableau.
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