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Your data matches 20 different statistics following compositions of up to 3 maps.
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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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
St000546: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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
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: St000056
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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
[[.,.],[.,.]]
=> [3,1,2] => [2,1,3] => [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
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => [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
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,3,4] => [2,1,3] => 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,3] => 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,2,4] => [3,1,2] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,2,1,4] => [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
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [4,2,1,3] => [2,1,3] => 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
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,4,3,5] => [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
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,3,2,4,5] => [1,3,2,4] => 3
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,3,2,5,4] => [1,3,2,4] => 3
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,4,2,3,5] => [1,4,2,3] => 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => [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
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,5,3,2,4] => [1,3,2,4] => 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
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [2,1,3,4,5] => [2,1,3,4] => 3
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,3,5,4] => [2,1,3,4] => 3
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [2,1,4,3,5] => [2,1,4,3] => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [2,1,5,3,4] => [2,1,3,4] => 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [2,1,5,4,3] => [2,1,4,3] => 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [3,1,2,4,5] => [3,1,2,4] => 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,1,2,5,4] => [3,1,2,4] => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,2,1,4,5] => [3,2,1,4] => 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,4] => 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,1,2,3,5] => [4,1,2,3] => 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,1,3,2,5] => [4,1,3,2] => 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [4,2,1,3,5] => [4,2,1,3] => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [4,3,1,2,5] => [4,3,1,2] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1] => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [5,1,2,3,4] => [1,2,3,4] => 4
Description
The decomposition (or block) number of a permutation.
For $\pi \in \mathcal{S}_n$, this is given by
$$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$$
This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum.
This is one plus [[St000234]].
Matching statistic: St000069
(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
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: St000286
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000286: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000286: Graphs ⟶ 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] => ([(0,1)],2)
=> 2
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => ([(0,1)],2)
=> 2
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => ([],2)
=> 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 2
[[[.,.],.],.]
=> [1,2,3] => [1,2] => ([],2)
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => ([],3)
=> 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => ([(1,2)],3)
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => ([],3)
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => ([],4)
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4
Description
The number of connected components of the complement of a graph.
The complement of a graph is the graph on the same vertex set with complementary edges.
Matching statistic: St000314
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●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
[[.,.],[.,.]]
=> [3,1,2] => [2,1,3] => [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
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => [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
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,3,4] => [2,1,3] => 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,3] => 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,2,4] => [3,1,2] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,2,1,4] => [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
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [4,2,1,3] => [2,1,3] => 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
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,4,3,5] => [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
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,3,2,4,5] => [1,3,2,4] => 3
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,3,2,5,4] => [1,3,2,4] => 3
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,4,2,3,5] => [1,4,2,3] => 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => [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
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,5,3,2,4] => [1,3,2,4] => 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
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [2,1,3,4,5] => [2,1,3,4] => 3
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,3,5,4] => [2,1,3,4] => 3
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [2,1,4,3,5] => [2,1,4,3] => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [2,1,5,3,4] => [2,1,3,4] => 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [2,1,5,4,3] => [2,1,4,3] => 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [3,1,2,4,5] => [3,1,2,4] => 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,1,2,5,4] => [3,1,2,4] => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,2,1,4,5] => [3,2,1,4] => 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,4] => 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,1,2,3,5] => [4,1,2,3] => 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,1,3,2,5] => [4,1,3,2] => 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [4,2,1,3,5] => [4,2,1,3] => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [4,3,1,2,5] => [4,3,1,2] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1] => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [5,1,2,3,4] => [1,2,3,4] => 4
Description
The number of left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a '''left-to-right-maximum''' if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$.
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000740: Permutations ⟶ 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] => [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
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
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: St000991
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000991: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000991: Permutations ⟶ 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] => [1,2] => 2
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [1,2] => 2
[[.,.],[.,.]]
=> [1,3,2] => [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
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [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
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,3,2] => [2,1,3] => 2
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,2] => [2,1,3] => 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,3] => [3,2,1] => 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [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
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2] => [2,1,3] => 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
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [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
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,4,3,1] => [3,1,2,4] => 3
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,4,3,1] => [3,1,2,4] => 3
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,4,1] => [4,3,1,2] => 2
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [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
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,4,3,1] => [3,1,2,4] => 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
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,4,3,2] => [2,1,3,4] => 3
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,4,3,2] => [2,1,3,4] => 3
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,3,4,2] => [2,4,1,3] => 2
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,4,3,2] => [2,1,3,4] => 3
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,3,4,2] => [2,4,1,3] => 2
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,4,3] => [3,2,1,4] => 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,4,3] => [3,2,1,4] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,4,3] => [2,3,1,4] => 2
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,4,3] => [2,3,1,4] => 2
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,4] => [4,3,2,1] => 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,3,1,4] => [3,4,2,1] => 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,4] => [2,4,3,1] => 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,4] => [3,2,4,1] => 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,4] => [2,3,4,1] => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => [1,2,3,4] => 4
Description
The number of right-to-left minima of a permutation.
For the number of left-to-right maxima, see [[St000314]].
Matching statistic: St001184
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1] => [1,0]
=> 1
[[.,.],.]
=> [1,2] => [1] => [1,0]
=> 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [1,1,0,0]
=> 2
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [1,1,0,0]
=> 2
[[.,.],[.,.]]
=> [1,3,2] => [1,2] => [1,0,1,0]
=> 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [1,1,0,0]
=> 2
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [1,0,1,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 3
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 3
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 3
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 3
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 4
Description
Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.
The following 10 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000203The number of external nodes of a binary tree. St000234The number of global ascents of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000654The first descent of a permutation. St000914The sum of the values of the Möbius function of a poset. St000990The first ascent of a permutation. St000989The number of final rises 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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