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Your data matches 60 different statistics following compositions of up to 3 maps.
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Matching statistic: St000522
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000522: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000522: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [[]]
=> 1 = 0 + 1
[.,[.,.]]
=> [[[]]]
=> 1 = 0 + 1
[[.,.],.]
=> [[],[]]
=> 1 = 0 + 1
[.,[.,[.,.]]]
=> [[[[]]]]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [[[],[]]]
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [[],[[]]]
=> 2 = 1 + 1
[[.,[.,.]],.]
=> [[[]],[]]
=> 2 = 1 + 1
[[[.,.],.],.]
=> [[],[],[]]
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [[[[],[]]]]
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [[[],[[]]]]
=> 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [[[[]],[]]]
=> 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [[[],[],[]]]
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [[],[[],[]]]
=> 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> [[[[]]],[]]
=> 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [[[],[]],[]]
=> 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [[],[[]],[]]
=> 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [[[]],[],[]]
=> 2 = 1 + 1
[[[[.,.],.],.],.]
=> [[],[],[],[]]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [[[[[[]]]]]]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [[[[[],[]]]]]
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [[[[],[[]]]]]
=> 2 = 1 + 1
[.,[.,[[.,[.,.]],.]]]
=> [[[[[]],[]]]]
=> 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [[[[],[],[]]]]
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [[[],[[[]]]]]
=> 2 = 1 + 1
[.,[[.,.],[[.,.],.]]]
=> [[[],[[],[]]]]
=> 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
=> [[[[]],[[]]]]
=> 2 = 1 + 1
[.,[[[.,.],.],[.,.]]]
=> [[[],[],[[]]]]
=> 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [[[[[]]],[]]]
=> 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [[[[],[]],[]]]
=> 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [[[],[[]],[]]]
=> 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [[[[]],[],[]]]
=> 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [[[],[],[],[]]]
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [[],[[[[]]]]]
=> 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [[],[[[],[]]]]
=> 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [[],[[],[[]]]]
=> 3 = 2 + 1
[[.,.],[[.,[.,.]],.]]
=> [[],[[[]],[]]]
=> 3 = 2 + 1
[[.,.],[[[.,.],.],.]]
=> [[],[[],[],[]]]
=> 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [[[]],[[[]]]]
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [[[]],[[],[]]]
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [[],[],[[[]]]]
=> 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [[],[],[[],[]]]
=> 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> [[[[]]],[[]]]
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [[[],[]],[[]]]
=> 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [[],[[]],[[]]]
=> 3 = 2 + 1
[[[.,[.,.]],.],[.,.]]
=> [[[]],[],[[]]]
=> 3 = 2 + 1
[[[[.,.],.],.],[.,.]]
=> [[],[],[],[[]]]
=> 2 = 1 + 1
Description
The number of 1-protected nodes of a rooted tree.
This is the number of nodes with minimal distance one to a leaf.
Matching statistic: St000023
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1] => 0
[.,[.,.]]
=> [2,1] => [2,1] => [1,2] => 0
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [1,3,2] => 1
[.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => [1,3,2] => 1
[[.,.],[.,.]]
=> [1,3,2] => [2,3,1] => [1,2,3] => 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [1,2,3] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [1,4,2,3] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,3,1,2] => [1,4,2,3] => 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,2,3,1] => [1,4,2,3] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,2,1,3] => [1,4,3,2] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [3,4,2,1] => [1,3,2,4] => 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,1,3] => [1,2,4,3] => 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,2,4,1] => [1,3,4,2] => 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [2,3,4,1] => [1,2,3,4] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [1,3,2,4] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1,2,4] => [1,3,2,4] => 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [2,3,1,4] => [1,2,3,4] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [1,2,3,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,5,2,4,3] => 2
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,1,2] => [1,5,2,4,3] => 2
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,4,2,3,1] => [1,5,2,4,3] => 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,2,1,3] => [1,5,3,2,4] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,4,1,2,3] => [1,5,3,2,4] => 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,3,4,2,1] => [1,5,2,3,4] => 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,2,4,1,3] => [1,5,3,4,2] => 2
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,3,2,4,1] => [1,5,2,3,4] => 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,2,3,4,1] => [1,5,2,3,4] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,2,1,4] => [1,5,4,2,3] => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,3,1,2,4] => [1,5,4,2,3] => 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,2,3,1,4] => [1,5,4,2,3] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,2,1,3,4] => [1,5,4,3,2] => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,5,4,3,2] => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [4,5,3,2,1] => [1,4,2,5,3] => 2
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [3,5,4,1,2] => [1,3,4,2,5] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [3,5,2,4,1] => [1,3,2,5,4] => 2
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [3,5,2,1,4] => [1,3,2,5,4] => 2
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [2,5,1,3,4] => [1,2,5,4,3] => 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [4,3,5,2,1] => [1,4,2,3,5] => 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [3,2,5,1,4] => [1,3,5,4,2] => 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [3,4,5,2,1] => [1,3,5,2,4] => 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [2,3,5,1,4] => [1,2,3,5,4] => 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [4,3,2,5,1] => [1,4,5,2,3] => 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [4,2,3,5,1] => [1,4,5,2,3] => 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [3,4,2,5,1] => [1,3,2,4,5] => 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [3,2,4,5,1] => [1,3,4,5,2] => 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [2,3,4,5,1] => [1,2,3,4,5] => 0
Description
The number of inner peaks of a permutation.
The number of peaks including the boundary is [[St000092]].
Matching statistic: St000035
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St000035: 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
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [] => 0
[.,[.,.]]
=> [2,1] => [2,1] => [1] => 0
[[.,.],.]
=> [1,2] => [1,2] => [1] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,1,2] => [1,2] => 0
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => [2,1] => 1
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [2,1] => 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,1,2,3] => [1,2,3] => 0
[.,[.,[[.,.],.]]]
=> [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] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,2,3,1] => [2,3,1] => 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,2,3] => [1,2,3] => 0
[[.,.],[[.,.],.]]
=> [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] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1,2,4] => [3,1,2] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => [3,2,1] => 1
[[[.,.],[.,.]],.]
=> [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] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,1,2,4,3] => [1,2,4,3] => 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,1,3,2,4] => [1,3,2,4] => 1
[.,[.,[[.,[.,.]],.]]]
=> [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] => 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,2,1,4,3] => [2,1,4,3] => 2
[.,[[.,[.,.]],[.,.]]]
=> [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] => 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,4,3,2,1] => [4,3,2,1] => 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,2,4,3,1] => [2,4,3,1] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,3,2,4,1] => [3,2,4,1] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,2,3,4,1] => [2,3,4,1] => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,2,3,4] => [1,2,3,4] => 0
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,2,4,3] => [1,2,4,3] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,3,2,4] => [1,3,2,4] => 1
[[.,.],[[.,[.,.]],.]]
=> [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] => 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,4,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,3,4] => [1,2,3,4] => 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,4,3] => 1
[[.,[.,[.,.]]],[.,.]]
=> [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] => 1
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,4] => 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,4] => 0
Description
The number of left outer peaks of a permutation.
A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$.
In other words, it is a peak in the word $[0,w_1,..., w_n]$.
This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.
Matching statistic: St000337
(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
St000337: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000337: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [] => [] => 0
[.,[.,.]]
=> [2,1] => [1] => [1] => 0
[[.,.],.]
=> [1,2] => [1] => [1] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [2,1] => [1,2] => 0
[.,[[.,.],.]]
=> [2,3,1] => [2,1] => [1,2] => 0
[[.,.],[.,.]]
=> [3,1,2] => [1,2] => [2,1] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1] => [1,2] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2] => [2,1] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,1] => [1,2,3] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,1] => [1,2,3] => 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,1] => [2,1,3] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,1] => [1,2,3] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,1] => [2,1,3] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,2] => [1,3,2] => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,2] => [1,3,2] => 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,1,3] => [2,3,1] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,3] => [3,2,1] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1] => [1,2,3] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1] => [2,1,3] => 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2] => [1,3,2] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3] => [2,3,1] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3] => [3,2,1] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,4,2,1] => [2,1,3,4] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,2,1] => [2,1,3,4] => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,3,1] => [1,3,2,4] => 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,1] => [2,3,1,4] => 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,1] => [3,2,1,4] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,1] => [1,2,3,4] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,1] => [2,1,3,4] => 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,1] => [1,3,2,4] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,1] => [2,3,1,4] => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,1] => [3,2,1,4] => 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,3,1,2] => [1,2,4,3] => 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,4,1,2] => [2,1,4,3] => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,1,2] => [1,2,4,3] => 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,1,2] => [2,1,4,3] => 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,1,3] => [1,3,4,2] => 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,1,3] => [1,3,4,2] => 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,2,3] => [1,4,3,2] => 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,2,3] => [1,4,3,2] => 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,1,4] => [3,2,4,1] => 2
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,2,4] => [2,4,3,1] => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 2
Description
The lec statistic, the sum of the inversion numbers of the hook factors of a permutation.
For a permutation $\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines $$ \textrm{lec} \, \sigma = \sum_{1 \leq i \leq k} \textrm{inv} \, \tau_{i} \, ,$$ where $\textrm{inv} \, \tau_{i}$ is the number of inversions of $\tau_{i}$.
Matching statistic: St000994
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1] => 0
[.,[.,.]]
=> [2,1] => [2,1] => [1,2] => 0
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [1,3,2] => 1
[.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => [1,3,2] => 1
[[.,.],[.,.]]
=> [3,1,2] => [1,3,2] => [1,2,3] => 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [1,2,3] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [1,4,2,3] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,3,1,2] => [1,4,2,3] => 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [4,1,3,2] => [1,4,2,3] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,2,1,3] => [1,4,3,2] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,4,3,2] => [1,2,4,3] => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,4,2,3] => [1,2,4,3] => 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,4,2] => [1,3,4,2] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,4,3] => [1,2,3,4] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [1,3,2,4] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1,2,4] => [1,3,2,4] => 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,3,2,4] => [1,2,3,4] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [1,2,3,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,5,2,4,3] => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,1,2] => [1,5,2,4,3] => 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [5,4,1,3,2] => [1,5,2,4,3] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,2,1,3] => [1,5,3,2,4] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,4,1,2,3] => [1,5,3,2,4] => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [5,1,4,3,2] => [1,5,2,3,4] => 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [5,1,4,2,3] => [1,5,3,4,2] => 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [5,3,1,4,2] => [1,5,2,3,4] => 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [5,1,2,4,3] => [1,5,3,2,4] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,2,1,4] => [1,5,4,2,3] => 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,3,1,2,4] => [1,5,4,2,3] => 2
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [5,1,3,2,4] => [1,5,4,2,3] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,2,1,3,4] => [1,5,4,3,2] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,5,4,3,2] => 2
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,5,4,3,2] => [1,2,5,3,4] => 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,5,4,2,3] => [1,2,5,3,4] => 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,5,2,4,3] => [1,2,5,3,4] => 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,5,3,2,4] => [1,2,5,4,3] => 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,5,2,3,4] => [1,2,5,4,3] => 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,1,5,3,2] => [1,4,3,5,2] => 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,1,5,2,3] => [1,4,2,3,5] => 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,5,4,3] => [1,2,3,5,4] => 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,2,5,3,4] => [1,2,3,5,4] => 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,3,1,5,2] => [1,4,5,2,3] => 2
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,1,2,5,3] => [1,4,5,3,2] => 2
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,4,2,5,3] => [1,2,4,5,3] => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [3,1,2,5,4] => [1,3,2,4,5] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,5,4] => [1,2,3,4,5] => 0
Description
The number of cycle peaks and the number of cycle valleys of a permutation.
A '''cycle peak''' of a permutation $\pi$ is an index $i$ such that $\pi^{-1}(i) < i > \pi(i)$. Analogously, a '''cycle valley''' is an index $i$ such that $\pi^{-1}(i) > i < \pi(i)$.
Clearly, every cycle of $\pi$ contains as many peaks as valleys.
Matching statistic: St001489
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001489: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001489: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1] => 0
[.,[.,.]]
=> [2,1] => [2,1] => [1,2] => 0
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [1,2,3] => 0
[.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => [1,2,3] => 0
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => [1,3,2] => 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,3,1,2] => [1,2,3,4] => 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,1,3,2] => [1,3,2,4] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,2,1,3] => [1,3,2,4] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,1,2,3] => [1,2,3,4] => 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,3,2] => [1,4,2,3] => 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => [1,4,2,3] => 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => [1,4,2,3] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1,2,4] => [1,2,4,3] => 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => [1,3,4,2] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,1,2] => [1,2,3,4,5] => 0
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,4,1,3,2] => [1,3,2,4,5] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,2,1,3] => [1,3,2,4,5] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,4,1,2,3] => [1,2,3,4,5] => 0
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,1,4,3,2] => [1,4,2,3,5] => 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,1,4,2,3] => [1,4,2,3,5] => 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,2,1,4,3] => [1,4,2,3,5] => 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,1,2,4,3] => [1,2,4,3,5] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,2,1,4] => [1,4,2,3,5] => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,3,1,2,4] => [1,2,4,3,5] => 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,1,3,2,4] => [1,3,2,4,5] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,2,1,3,4] => [1,3,4,2,5] => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,2,3,4,5] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,5,2,3,4] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,4,2,3] => [1,5,2,3,4] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,2,4,3] => [1,5,2,4,3] => 2
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,3,2,4] => [1,5,2,4,3] => 2
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,4,3] => [1,5,2,3,4] => 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,3,4] => [1,5,2,3,4] => 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,4,3] => [1,2,5,3,4] => 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,2,1,5,4] => [1,5,2,3,4] => 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,1,2,5,4] => [1,2,5,3,4] => 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,2,4] => 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
Description
The maximum of the number of descents and the number of inverse descents.
This is, the maximum of [[St000021]] and [[St000354]].
Matching statistic: St001839
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
St001839: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
St001839: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => {{1}}
=> {{1}}
=> 0
[.,[.,.]]
=> [2,1] => {{1,2}}
=> {{1,2}}
=> 0
[[.,.],.]
=> [1,2] => {{1},{2}}
=> {{1},{2}}
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => {{1,3},{2}}
=> {{1,3},{2}}
=> 1
[.,[[.,.],.]]
=> [2,3,1] => {{1,2,3}}
=> {{1,2,3}}
=> 0
[[.,.],[.,.]]
=> [3,1,2] => {{1,3},{2}}
=> {{1,3},{2}}
=> 1
[[.,[.,.]],.]
=> [2,1,3] => {{1,2},{3}}
=> {{1},{2,3}}
=> 0
[[[.,.],.],.]
=> [1,2,3] => {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> {{1,5},{2,4},{3}}
=> 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => {{1,4},{2,5},{3}}
=> {{1,4},{2,5},{3}}
=> 2
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => {{1,5},{2,3,4}}
=> {{1,5},{2,3,4}}
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => {{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => {{1,3,5},{2,4}}
=> {{1,3,5},{2,4}}
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => {{1,5},{2,4},{3}}
=> {{1,5},{2,4},{3}}
=> 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => {{1,4},{2,5},{3}}
=> {{1,4},{2,5},{3}}
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> {{1,5},{2},{3,4}}
=> 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => {{1,4,5},{2,3}}
=> {{1,2,5},{3,4}}
=> 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => {{1,3},{2,4,5}}
=> {{1,2,4},{3,5}}
=> 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => {{1,4,5},{2},{3}}
=> {{1,2,5},{3},{4}}
=> 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => {{1,3,4,5},{2}}
=> {{1,2,3,5},{4}}
=> 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => {{1,5},{2,4},{3}}
=> {{1,5},{2,4},{3}}
=> 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => {{1,4},{2,5},{3}}
=> {{1,4},{2,5},{3}}
=> 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => {{1,5},{2,3,4}}
=> {{1,5},{2,3,4}}
=> 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => {{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => {{1,3,5},{2,4}}
=> {{1,3,5},{2,4}}
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => {{1,5},{2,4},{3}}
=> {{1,5},{2,4},{3}}
=> 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => {{1,4},{2,5},{3}}
=> {{1,4},{2,5},{3}}
=> 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => {{1,5},{2,4},{3}}
=> {{1,5},{2,4},{3}}
=> 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => {{1,4},{2,5},{3}}
=> {{1,4},{2,5},{3}}
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> {{1,5},{2},{3,4}}
=> 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => {{1,5},{2,3},{4}}
=> {{1,5},{2},{3,4}}
=> 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> 1
Description
The number of excedances of a set partition.
The Mahonian representation of a set partition $\{B_1,\dots,B_k\}$ of $\{1,\dots,n\}$ is the restricted growth word $w_1 \dots w_n$ obtained by sorting the blocks of the set partition according to their maximal element, and setting $w_i$ to the index of the block containing $i$.
Let $\bar w$ be the nondecreasing rearrangement of $w$.
The word $w$ has an excedance at position $i$ if $w_i > \bar w_i$.
Matching statistic: St001840
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St001840: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St001840: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => {{1}}
=> 0
[.,[.,.]]
=> [2,1] => [2,1] => {{1,2}}
=> 0
[[.,.],.]
=> [1,2] => [1,2] => {{1},{2}}
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => {{1,3},{2}}
=> 1
[.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => {{1,3},{2}}
=> 1
[[.,.],[.,.]]
=> [3,1,2] => [1,3,2] => {{1},{2,3}}
=> 0
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,3,1,2] => {{1,4},{2,3}}
=> 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [4,1,3,2] => {{1,4},{2},{3}}
=> 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,2,1,3] => {{1,4},{2},{3}}
=> 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,1,2,3] => {{1,4},{2},{3}}
=> 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,1,4,2] => {{1,3,4},{2}}
=> 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 2
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,1,2] => {{1,5},{2,4},{3}}
=> 2
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [5,4,1,3,2] => {{1,5},{2,4},{3}}
=> 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,2,1,3] => {{1,5},{2,4},{3}}
=> 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,4,1,2,3] => {{1,5},{2,4},{3}}
=> 2
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [5,1,4,3,2] => {{1,5},{2},{3,4}}
=> 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [5,1,4,2,3] => {{1,5},{2},{3,4}}
=> 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [5,3,1,4,2] => {{1,5},{2,3},{4}}
=> 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [5,1,2,4,3] => {{1,5},{2},{3},{4}}
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,3,1,2,4] => {{1,5},{2,3},{4}}
=> 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [5,1,3,2,4] => {{1,5},{2},{3},{4}}
=> 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,1,2,3,4] => {{1,5},{2},{3},{4}}
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [1,5,4,2,3] => {{1},{2,5},{3,4}}
=> 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [1,5,2,4,3] => {{1},{2,5},{3},{4}}
=> 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [1,5,3,2,4] => {{1},{2,5},{3},{4}}
=> 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [1,5,2,3,4] => {{1},{2,5},{3},{4}}
=> 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,1,5,3,2] => {{1,4},{2},{3,5}}
=> 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,1,5,2,3] => {{1,4},{2},{3,5}}
=> 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,3,1,5,2] => {{1,4,5},{2,3}}
=> 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,1,2,5,3] => {{1,4,5},{2},{3}}
=> 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,4,2,5,3] => {{1},{2,4,5},{3}}
=> 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 0
Description
The number of descents of a set partition.
The Mahonian representation of a set partition $\{B_1,\dots,B_k\}$ of $\{1,\dots,n\}$ is the restricted growth word $w_1\dots w_n\}$ obtained by sorting the blocks of the set partition according to their maximal element, and setting $w_i$ to the index of the block containing $i$.
The word $w$ has a descent at position $i$ if $w_i > w_{i+1}$.
Matching statistic: St001928
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1] => 0
[.,[.,.]]
=> [2,1] => [1,2] => [1,2] => 0
[[.,.],.]
=> [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [1,2,3] => 0
[.,[[.,.],.]]
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => [1,3,2] => 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 0
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [1,2,4,3] => [1,2,4,3] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [1,2,4,3] => [1,2,4,3] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,3,4,2] => [1,4,3,2] => 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [1,4,2,3] => [1,4,2,3] => 1
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,4,2,3] => [1,4,2,3] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,4,2,3] => [1,4,2,3] => 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,4,2] => [1,4,3,2] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,4,5,2,3] => [1,5,2,4,3] => 2
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,3,5,2,4] => [1,5,3,2,4] => 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,3,4,5,2] => [1,5,3,4,2] => 2
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [1,4,5,2,3] => [1,5,2,4,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [1,5,2,3,4] => [1,5,2,3,4] => 1
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [1,3,5,2,4] => [1,5,3,2,4] => 1
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
Description
The number of non-overlapping descents in a permutation.
In other words, any maximal descending subsequence $\pi_i,\pi_{i+1},\dots,\pi_k$ contributes $\lfloor\frac{k-i+1}{2}\rfloor$ to the total count.
Matching statistic: St000092
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00326: Permutations —weak order rowmotion⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1] => 1 = 0 + 1
[.,[.,.]]
=> [2,1] => [1,2] => [2,1] => 1 = 0 + 1
[[.,.],.]
=> [1,2] => [2,1] => [1,2] => 1 = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [1,2,3] => [2,3,1] => 1 = 0 + 1
[.,[[.,.],.]]
=> [2,3,1] => [2,1,3] => [3,2,1] => 1 = 0 + 1
[[.,.],[.,.]]
=> [3,1,2] => [1,3,2] => [2,1,3] => 2 = 1 + 1
[[.,[.,.]],.]
=> [2,1,3] => [3,1,2] => [3,1,2] => 2 = 1 + 1
[[[.,.],.],.]
=> [1,2,3] => [3,2,1] => [1,3,2] => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [1,2,3,4] => [2,3,4,1] => 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,1,2,4] => [3,4,2,1] => 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,4,1,3] => [4,2,1,3] => 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,3,1,4] => [4,2,3,1] => 2 = 1 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,2,1,4] => [4,3,2,1] => 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [1,3,4,2] => [2,1,3,4] => 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,2,4] => 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [1,2,4,3] => [2,3,1,4] => 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,4,3,2] => [2,1,4,3] => 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [4,1,2,3] => [3,4,1,2] => 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [4,2,1,3] => [4,3,1,2] => 2 = 1 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [4,1,3,2] => [3,1,4,2] => 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [4,3,1,2] => [4,1,3,2] => 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [4,3,2,1] => [1,4,3,2] => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [1,2,3,4,5] => [2,3,4,5,1] => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,1,2,3,5] => [3,4,5,2,1] => 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,5,1,2,4] => [4,5,2,1,3] => 2 = 1 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [3,4,1,2,5] => [4,5,2,3,1] => 2 = 1 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [4,3,1,2,5] => [4,5,3,2,1] => 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [2,4,5,1,3] => [5,2,1,3,4] => 2 = 1 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,2,5,1,3] => [5,3,1,2,4] => 2 = 1 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [2,3,5,1,4] => [5,2,3,1,4] => 3 = 2 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,5,3,1,4] => [5,2,4,1,3] => 3 = 2 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [2,3,4,1,5] => [5,2,3,4,1] => 2 = 1 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,2,4,1,5] => [5,3,2,4,1] => 2 = 1 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [2,4,3,1,5] => [5,2,4,3,1] => 2 = 1 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [4,2,3,1,5] => [5,3,4,2,1] => 2 = 1 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [4,3,2,1,5] => [5,4,3,2,1] => 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [1,3,4,5,2] => [2,1,3,4,5] => 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,1,3,5,2] => [3,1,4,2,5] => 3 = 2 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,5,1,4,2] => [4,1,2,5,3] => 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [3,4,1,5,2] => [4,1,2,3,5] => 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [4,3,1,5,2] => [4,1,3,2,5] => 3 = 2 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [1,2,4,5,3] => [2,3,1,4,5] => 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,1,2,5,3] => [3,4,1,2,5] => 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [1,4,5,3,2] => [2,1,5,3,4] => 3 = 2 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,5,3,2] => [3,1,5,2,4] => 3 = 2 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [1,2,3,5,4] => [2,3,4,1,5] => 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,5,4,1,3] => [5,2,1,4,3] => 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [1,3,5,4,2] => [2,1,3,5,4] => 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [1,2,5,4,3] => [2,3,1,5,4] => 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,5,4,3,2] => [2,1,5,4,3] => 2 = 1 + 1
Description
The number of outer peaks of a permutation.
An outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$ or $n$ if $w_{n} > w_{n-1}$.
In other words, it is a peak in the word $[0,w_1,..., w_n,0]$.
The following 50 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000093The cardinality of a maximal independent set of vertices of a graph. St000099The number of valleys of a permutation, including the boundary. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000568The hook number of a binary tree. St000353The number of inner valleys of a permutation. St000711The number of big exceedences of a permutation. St000480The number of lower covers of a partition in dominance order. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St000668The least common multiple of the parts of the partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000660The number of rises of length at least 3 of a Dyck path. St000260The radius of a connected graph. St001470The cyclic holeyness of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001597The Frobenius rank of a skew partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000455The second largest eigenvalue of a graph if it is integral. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001060The distinguishing index of a graph. St001423The number of distinct cubes in a binary word. St001823The Stasinski-Voll length of a signed permutation. St000628The balance of a binary word. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001569The maximal modular displacement of a permutation. St000758The length of the longest staircase fitting into an integer composition. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000920The logarithmic height of a Dyck path.
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