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Your data matches 49 different statistics following compositions of up to 3 maps.
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Matching statistic: St000203
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
St000203: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> 2 = 0 + 2
[[.,.],.]
=> 2 = 0 + 2
[.,[.,[.,.]]]
=> 3 = 1 + 2
[.,[[.,.],.]]
=> 2 = 0 + 2
[[.,.],[.,.]]
=> 3 = 1 + 2
[[.,[.,.]],.]
=> 2 = 0 + 2
[[[.,.],.],.]
=> 3 = 1 + 2
[.,[.,[.,[.,.]]]]
=> 4 = 2 + 2
[.,[.,[[.,.],.]]]
=> 3 = 1 + 2
[.,[[.,.],[.,.]]]
=> 3 = 1 + 2
[.,[[.,[.,.]],.]]
=> 2 = 0 + 2
[.,[[[.,.],.],.]]
=> 2 = 0 + 2
[[.,.],[.,[.,.]]]
=> 4 = 2 + 2
[[.,.],[[.,.],.]]
=> 3 = 1 + 2
[[.,[.,.]],[.,.]]
=> 3 = 1 + 2
[[[.,.],.],[.,.]]
=> 4 = 2 + 2
[[.,[.,[.,.]]],.]
=> 2 = 0 + 2
[[.,[[.,.],.]],.]
=> 2 = 0 + 2
[[[.,.],[.,.]],.]
=> 3 = 1 + 2
[[[.,[.,.]],.],.]
=> 3 = 1 + 2
[[[[.,.],.],.],.]
=> 4 = 2 + 2
[.,[.,[.,[.,[.,.]]]]]
=> 5 = 3 + 2
[.,[.,[.,[[.,.],.]]]]
=> 4 = 2 + 2
[.,[.,[[.,.],[.,.]]]]
=> 4 = 2 + 2
[.,[.,[[.,[.,.]],.]]]
=> 3 = 1 + 2
[.,[.,[[[.,.],.],.]]]
=> 3 = 1 + 2
[.,[[.,.],[.,[.,.]]]]
=> 4 = 2 + 2
[.,[[.,.],[[.,.],.]]]
=> 3 = 1 + 2
[.,[[.,[.,.]],[.,.]]]
=> 3 = 1 + 2
[.,[[[.,.],.],[.,.]]]
=> 3 = 1 + 2
[.,[[.,[.,[.,.]]],.]]
=> 2 = 0 + 2
[.,[[.,[[.,.],.]],.]]
=> 2 = 0 + 2
[.,[[[.,.],[.,.]],.]]
=> 2 = 0 + 2
[.,[[[.,[.,.]],.],.]]
=> 2 = 0 + 2
[.,[[[[.,.],.],.],.]]
=> 2 = 0 + 2
[[.,.],[.,[.,[.,.]]]]
=> 5 = 3 + 2
[[.,.],[.,[[.,.],.]]]
=> 4 = 2 + 2
[[.,.],[[.,.],[.,.]]]
=> 4 = 2 + 2
[[.,.],[[.,[.,.]],.]]
=> 3 = 1 + 2
[[.,.],[[[.,.],.],.]]
=> 3 = 1 + 2
[[.,[.,.]],[.,[.,.]]]
=> 4 = 2 + 2
[[.,[.,.]],[[.,.],.]]
=> 3 = 1 + 2
[[[.,.],.],[.,[.,.]]]
=> 5 = 3 + 2
[[[.,.],.],[[.,.],.]]
=> 4 = 2 + 2
[[.,[.,[.,.]]],[.,.]]
=> 3 = 1 + 2
[[.,[[.,.],.]],[.,.]]
=> 3 = 1 + 2
[[[.,.],[.,.]],[.,.]]
=> 4 = 2 + 2
[[[.,[.,.]],.],[.,.]]
=> 4 = 2 + 2
[[[[.,.],.],.],[.,.]]
=> 5 = 3 + 2
[[.,[.,[.,[.,.]]]],.]
=> 2 = 0 + 2
Description
The number of external nodes of a binary tree.
That is, the number of nodes that can be reached from the root by only left steps or only right steps, plus $1$ for the root node itself. A counting formula for the number of external node in all binary trees of size $n$ can be found in [1].
Matching statistic: St000546
(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
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
[[.,.],.]
=> [1,2] => [1] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => 1
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => 1
[[[.,.],.],.]
=> [1,2,3] => [1,2] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => 2
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => 2
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => 2
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => 3
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => 3
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => 3
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => 2
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => 2
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => 2
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => 2
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => 3
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: St000007
(load all 13 compositions to match this statistic)
(load all 13 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 = 0 + 1
[[.,.],.]
=> [1,2] => [1] => 1 = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => 2 = 1 + 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => 2 = 1 + 1
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => 1 = 0 + 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => 2 = 1 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,2] => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => 3 = 2 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => 3 = 2 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => 3 = 2 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => 1 = 0 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => 3 = 2 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => 1 = 0 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => 4 = 3 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => 4 = 3 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => 3 = 2 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => 4 = 3 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => 3 = 2 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => 3 = 2 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => 3 = 2 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => 4 = 3 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => 3 = 2 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => 3 = 2 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => 3 = 2 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => 3 = 2 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => 4 = 3 + 1
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: St000234
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000234: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [2,1] => [1] => 0
[[.,.],.]
=> [1,2] => [1,2] => [1] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,1,2] => [1,2] => 1
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => [2,1] => 0
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,1] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,1,2,3] => [1,2,3] => 2
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,1,3,2] => [1,3,2] => 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,2,1,3] => [2,1,3] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,3,2,1] => [3,2,1] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,2,3,1] => [2,3,1] => 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,2,3] => [1,2,3] => 2
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => [1,3,2] => 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [2,1,3] => 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,2,3] => 2
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1,2,4] => [3,1,2] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => [3,2,1] => 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3] => 2
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,1,2,3,4] => [1,2,3,4] => 3
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,1,2,4,3] => [1,2,4,3] => 2
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,1,3,2,4] => [1,3,2,4] => 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,1,4,3,2] => [1,4,3,2] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,1,3,4,2] => [1,3,4,2] => 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,2,1,3,4] => [2,1,3,4] => 2
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,2,1,4,3] => [2,1,4,3] => 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,3,2,1,4] => [3,2,1,4] => 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,2,3,1,4] => [2,3,1,4] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,4,2,3,1] => [4,2,3,1] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,4,3,2,1] => [4,3,2,1] => 0
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,2,4,3,1] => [2,4,3,1] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,3,2,4,1] => [3,2,4,1] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,1] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,2,3,4] => [1,2,3,4] => 3
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,2,4,3] => [1,2,4,3] => 2
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,3,2,4] => [1,3,2,4] => 2
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,4,3,2] => [1,4,3,2] => 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,2] => 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,3,4] => [2,1,3,4] => 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,3] => 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,3,4] => [1,2,3,4] => 3
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,3] => 2
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,1,2,5,4] => [3,1,2,4] => 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,4] => 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,4] => 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,4] => 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4] => 3
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,1,2,3,5] => [4,1,2,3] => 0
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: 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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St001640: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,.]]
=> [2,1] => [1] => [1] => 0
[[.,.],.]
=> [1,2] => [1] => [1] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [1,2] => 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [1,2] => 1
[[.,.],[.,.]]
=> [1,3,2] => [1,2] => [2,1] => 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [1,2] => 1
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [2,1] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [1,2,3] => 2
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [1,2,3] => 2
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [2,3,1] => [3,1,2] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [1,2,3] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [3,1,2] => 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,3,2] => [2,1,3] => 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,2] => [2,1,3] => 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,3] => [3,2,1] => 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,3] => [2,3,1] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [1,2,3] => 2
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [3,1,2] => 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2] => [2,1,3] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [3,2,1] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [2,3,1] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 3
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [1,2,3,4] => 3
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [3,4,2,1] => [4,1,2,3] => 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [1,2,3,4] => 3
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [4,1,2,3] => 2
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [2,4,3,1] => [3,1,2,4] => 2
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [2,4,3,1] => [3,1,2,4] => 2
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [3,2,4,1] => [4,3,1,2] => 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [2,3,4,1] => [3,4,1,2] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [1,2,3,4] => 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [4,1,2,3] => 2
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [2,4,3,1] => [3,1,2,4] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [4,3,1,2] => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [3,4,1,2] => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,4,3,2] => [2,1,3,4] => 2
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,4,3,2] => [2,1,3,4] => 2
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,3,4,2] => [2,4,1,3] => 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,4,3,2] => [2,1,3,4] => 2
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,3,4,2] => [2,4,1,3] => 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,4,3] => [3,2,1,4] => 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,4,3] => [3,2,1,4] => 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,4,3] => [2,3,1,4] => 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,4,3] => [2,3,1,4] => 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,4] => [4,3,2,1] => 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [2,3,1,4] => [3,4,2,1] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,4] => [2,4,3,1] => 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,4] => [3,2,4,1] => 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,4] => [2,3,4,1] => 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => [1,2,3,4] => 3
Description
The number of ascent tops in the permutation such that all smaller elements appear before.
Matching statistic: St000056
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ 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] => [2,1] => [1] => 1 = 0 + 1
[[.,.],.]
=> [1,2] => [1,2] => [1] => 1 = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [3,1,2] => [1,2] => 2 = 1 + 1
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => [2,1] => 1 = 0 + 1
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,2] => 2 = 1 + 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,1] => 1 = 0 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2] => 2 = 1 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,1,2,3] => [1,2,3] => 3 = 2 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,1,3,2] => [1,3,2] => 2 = 1 + 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,2,1,3] => [2,1,3] => 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,3,2,1] => [3,2,1] => 1 = 0 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,2,3,1] => [2,3,1] => 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,2,3] => [1,2,3] => 3 = 2 + 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => [1,3,2] => 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [2,1,3] => 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,2,3] => 3 = 2 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1,2,4] => [3,1,2] => 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => [3,2,1] => 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2] => 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3] => 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3] => 3 = 2 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,1,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,1,3,2,4] => [1,3,2,4] => 3 = 2 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,1,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,1,3,4,2] => [1,3,4,2] => 2 = 1 + 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,2,1,3,4] => [2,1,3,4] => 3 = 2 + 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,2,1,4,3] => [2,1,4,3] => 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,3,2,1,4] => [3,2,1,4] => 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,2,3,1,4] => [2,3,1,4] => 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,4,2,3,1] => [4,2,3,1] => 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,2,4,3,1] => [2,4,3,1] => 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,3,2,4,1] => [3,2,4,1] => 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,1] => 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,3,2,4] => [1,3,2,4] => 3 = 2 + 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,3,4,2] => 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,3,4] => [2,1,3,4] => 3 = 2 + 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,3] => 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,3,4] => [1,2,3,4] => 4 = 3 + 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,3] => 3 = 2 + 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,1,2,5,4] => [3,1,2,4] => 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,4] => 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,4] => 3 = 2 + 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,4] => 3 = 2 + 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4] => 4 = 3 + 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,1,2,3,5] => [4,1,2,3] => 1 = 0 + 1
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 = 0 + 1
[[.,.],.]
=> [1,2] => [1] => ([],1)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => ([],2)
=> 2 = 1 + 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => ([],2)
=> 2 = 1 + 1
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => ([],2)
=> 2 = 1 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => ([],3)
=> 3 = 2 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => ([],3)
=> 3 = 2 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => ([(1,2)],3)
=> 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => ([],3)
=> 3 = 2 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => ([(1,2)],3)
=> 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => ([],3)
=> 3 = 2 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => ([(1,2)],3)
=> 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => ([],4)
=> 4 = 3 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => ([],4)
=> 4 = 3 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
=> 3 = 2 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => ([],4)
=> 4 = 3 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => ([(2,3)],4)
=> 3 = 2 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> 3 = 2 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => ([(2,3)],4)
=> 3 = 2 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => ([],4)
=> 4 = 3 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => ([(2,3)],4)
=> 3 = 2 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => ([(2,3)],4)
=> 3 = 2 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 3 = 2 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 3 = 2 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => ([(2,3)],4)
=> 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => ([(1,3),(2,3)],4)
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => ([],4)
=> 4 = 3 + 1
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 = 0 + 1
[[.,.],.]
=> [1,2] => [1] => ([],1)
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => ([],2)
=> 1 = 0 + 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,2] => ([],2)
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1 = 0 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => ([],3)
=> 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => ([(1,2)],3)
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => ([],3)
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => ([],4)
=> 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
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 = 0 + 1
[[.,.],.]
=> [1,2] => [2,1] => [1] => 1 = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [1,2] => 2 = 1 + 1
[.,[[.,.],.]]
=> [2,3,1] => [1,3,2] => [1,2] => 2 = 1 + 1
[[.,.],[.,.]]
=> [3,1,2] => [2,1,3] => [2,1] => 1 = 0 + 1
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => [1,2] => 2 = 1 + 1
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => [2,1] => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3] => 3 = 2 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,4,3] => [1,2,3] => 3 = 2 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [1,3,2,4] => [1,3,2] => 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,4,2,3] => [1,2,3] => 3 = 2 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,4,3,2] => [1,3,2] => 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [2,1,3,4] => [2,1,3] => 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [2,1,4,3] => [2,1,3] => 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,2,4] => [3,1,2] => 1 = 0 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [3,2,1,4] => [3,2,1] => 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => [1,2,3] => 3 = 2 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,1,3,2] => [1,3,2] => 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [4,2,1,3] => [2,1,3] => 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => [3,1,2] => 1 = 0 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => [3,2,1] => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4] => 4 = 3 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,5,4] => [1,2,3,4] => 4 = 3 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [1,2,4,3,5] => [1,2,4,3] => 3 = 2 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,5,3,4] => [1,2,3,4] => 4 = 3 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,3] => 3 = 2 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [1,3,2,4,5] => [1,3,2,4] => 3 = 2 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [1,3,2,5,4] => [1,3,2,4] => 3 = 2 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [1,4,2,3,5] => [1,4,2,3] => 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [1,4,3,2,5] => [1,4,3,2] => 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,5,2,3,4] => [1,2,3,4] => 4 = 3 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,5,2,4,3] => [1,2,4,3] => 3 = 2 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [1,5,3,2,4] => [1,3,2,4] => 3 = 2 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,5,4,2,3] => [1,4,2,3] => 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,5,4,3,2] => [1,4,3,2] => 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [2,1,3,4,5] => [2,1,3,4] => 3 = 2 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [2,1,3,5,4] => [2,1,3,4] => 3 = 2 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [2,1,4,3,5] => [2,1,4,3] => 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [2,1,5,3,4] => [2,1,3,4] => 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [2,1,5,4,3] => [2,1,4,3] => 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [3,1,2,4,5] => [3,1,2,4] => 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [3,1,2,5,4] => [3,1,2,4] => 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,2,1,4,5] => [3,2,1,4] => 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,4] => 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,1,2,3,5] => [4,1,2,3] => 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,1,3,2,5] => [4,1,3,2] => 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [4,2,1,3,5] => [4,2,1,3] => 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [4,3,1,2,5] => [4,3,1,2] => 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1] => 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [5,1,2,3,4] => [1,2,3,4] => 4 = 3 + 1
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 = 0 + 1
[[.,.],.]
=> [1,2] => [1] => [1] => 1 = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [1,2] => 2 = 1 + 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [1,2] => 2 = 1 + 1
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => [2,1] => 1 = 0 + 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [1,2] => 2 = 1 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [2,1] => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [1,2,3] => 3 = 2 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [1,2,3] => 3 = 2 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [1,2,3] => 3 = 2 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => [1,3,2] => 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => [1,3,2] => 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => [3,2,1] => 1 = 0 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => [2,3,1] => 1 = 0 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [1,2,3] => 3 = 2 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [3,1,2] => 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => [1,3,2] => 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [3,2,1] => 1 = 0 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [2,3,1] => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 4 = 3 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [1,2,3,4] => 4 = 3 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => [4,1,2,3] => 3 = 2 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [1,2,3,4] => 4 = 3 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [4,1,2,3] => 3 = 2 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => [1,4,2,3] => 3 = 2 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => [1,4,2,3] => 3 = 2 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => [4,3,1,2] => 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => [3,4,1,2] => 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [1,2,3,4] => 4 = 3 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [4,1,2,3] => 3 = 2 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => [1,4,2,3] => 3 = 2 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [4,3,1,2] => 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [3,4,1,2] => 2 = 1 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 3 = 2 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => [1,2,4,3] => 3 = 2 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => [4,1,3,2] => 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => [1,2,4,3] => 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => [4,1,3,2] => 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => [1,4,3,2] => 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => [1,4,3,2] => 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => [1,3,4,2] => 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => [1,3,4,2] => 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => [4,3,2,1] => 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => [3,4,2,1] => 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => [4,2,3,1] => 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => [3,2,4,1] => 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [4,3,2,1] => [1,2,3,4] => 4 = 3 + 1
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
The following 39 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000908The length of the shortest maximal antichain in a poset. 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. St000989The number of final rises of a permutation. St000654The first descent of a permutation. St000914The sum of the values of the Möbius function of a poset. St000990The first ascent of a permutation. St000681The Grundy value of Chomp on Ferrers diagrams. St000815The number of semistandard Young tableaux of partition weight of given shape. St000937The number of positive values of the symmetric group character corresponding to the partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000993The multiplicity of the largest part of an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000454The largest eigenvalue of a graph if it is integral. St000181The number of connected components of the Hasse diagram for the poset. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000260The radius of a connected graph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000013The height of a Dyck path. St001557The number of inversions of the second entry of a permutation. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000455The second largest eigenvalue of a graph if it is integral. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001060The distinguishing index of a graph. St001820The size of the image of the pop stack sorting operator. St001875The number of simple modules with projective dimension at most 1. St000307The number of rowmotion orbits of a poset. St000264The girth of a graph, which is not a tree. St000632The jump number of the poset. St001570The minimal number of edges to add to make a graph Hamiltonian. St000735The last entry on the main diagonal of a standard tableau.
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