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Your data matches 21 different statistics following compositions of up to 3 maps.
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Matching statistic: St000099
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(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 1
[2,1] => [1] => [1] => 1
[1,2,3] => [1,2] => [1,2] => 1
[1,3,2] => [1,2] => [1,2] => 1
[2,1,3] => [2,1] => [1,2] => 1
[2,3,1] => [2,1] => [1,2] => 1
[3,1,2] => [1,2] => [1,2] => 1
[3,2,1] => [2,1] => [1,2] => 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 2
[1,3,4,2] => [1,3,2] => [1,3,2] => 2
[1,4,2,3] => [1,2,3] => [1,2,3] => 1
[1,4,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3,4] => [2,1,3] => [1,3,2] => 2
[2,1,4,3] => [2,1,3] => [1,3,2] => 2
[2,3,1,4] => [2,3,1] => [1,2,3] => 1
[2,3,4,1] => [2,3,1] => [1,2,3] => 1
[2,4,1,3] => [2,1,3] => [1,3,2] => 2
[2,4,3,1] => [2,3,1] => [1,2,3] => 1
[3,1,2,4] => [3,1,2] => [1,2,3] => 1
[3,1,4,2] => [3,1,2] => [1,2,3] => 1
[3,2,1,4] => [3,2,1] => [1,2,3] => 1
[3,2,4,1] => [3,2,1] => [1,2,3] => 1
[3,4,1,2] => [3,1,2] => [1,2,3] => 1
[3,4,2,1] => [3,2,1] => [1,2,3] => 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 1
[4,1,3,2] => [1,3,2] => [1,3,2] => 2
[4,2,1,3] => [2,1,3] => [1,3,2] => 2
[4,2,3,1] => [2,3,1] => [1,2,3] => 1
[4,3,1,2] => [3,1,2] => [1,2,3] => 1
[4,3,2,1] => [3,2,1] => [1,2,3] => 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 2
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2,5] => [1,3,4,2] => [1,3,4,2] => 2
[1,3,4,5,2] => [1,3,4,2] => [1,3,4,2] => 2
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => 2
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2,5] => [1,4,3,2] => [1,4,2,3] => 2
[1,4,3,5,2] => [1,4,3,2] => [1,4,2,3] => 2
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,5,3,2] => [1,4,3,2] => [1,4,2,3] => 2
Description
The number of valleys of a permutation, including the boundary.
The number of valleys excluding the boundary is [[St000353]].
Matching statistic: St000325
Mp00252: Permutations —restriction⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 1
[2,1] => [1] => [1] => 1
[1,2,3] => [1,2] => [1,2] => 1
[1,3,2] => [1,2] => [1,2] => 1
[2,1,3] => [2,1] => [1,2] => 1
[2,3,1] => [2,1] => [1,2] => 1
[3,1,2] => [1,2] => [1,2] => 1
[3,2,1] => [2,1] => [1,2] => 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 2
[1,3,4,2] => [1,3,2] => [1,3,2] => 2
[1,4,2,3] => [1,2,3] => [1,2,3] => 1
[1,4,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3,4] => [2,1,3] => [1,3,2] => 2
[2,1,4,3] => [2,1,3] => [1,3,2] => 2
[2,3,1,4] => [2,3,1] => [1,2,3] => 1
[2,3,4,1] => [2,3,1] => [1,2,3] => 1
[2,4,1,3] => [2,1,3] => [1,3,2] => 2
[2,4,3,1] => [2,3,1] => [1,2,3] => 1
[3,1,2,4] => [3,1,2] => [1,2,3] => 1
[3,1,4,2] => [3,1,2] => [1,2,3] => 1
[3,2,1,4] => [3,2,1] => [1,2,3] => 1
[3,2,4,1] => [3,2,1] => [1,2,3] => 1
[3,4,1,2] => [3,1,2] => [1,2,3] => 1
[3,4,2,1] => [3,2,1] => [1,2,3] => 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 1
[4,1,3,2] => [1,3,2] => [1,3,2] => 2
[4,2,1,3] => [2,1,3] => [1,3,2] => 2
[4,2,3,1] => [2,3,1] => [1,2,3] => 1
[4,3,1,2] => [3,1,2] => [1,2,3] => 1
[4,3,2,1] => [3,2,1] => [1,2,3] => 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 2
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2,5] => [1,3,4,2] => [1,3,4,2] => 2
[1,3,4,5,2] => [1,3,4,2] => [1,3,4,2] => 2
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => 2
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2,5] => [1,4,3,2] => [1,4,2,3] => 2
[1,4,3,5,2] => [1,4,3,2] => [1,4,2,3] => 2
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,5,3,2] => [1,4,3,2] => [1,4,2,3] => 2
Description
The width of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The width of the tree is given by the number of leaves of this tree.
Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]].
See also [[St000308]] for the height of this tree.
Matching statistic: St000470
Mp00252: Permutations —restriction⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 1
[2,1] => [1] => [1] => 1
[1,2,3] => [1,2] => [1,2] => 1
[1,3,2] => [1,2] => [1,2] => 1
[2,1,3] => [2,1] => [1,2] => 1
[2,3,1] => [2,1] => [1,2] => 1
[3,1,2] => [1,2] => [1,2] => 1
[3,2,1] => [2,1] => [1,2] => 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 2
[1,3,4,2] => [1,3,2] => [1,3,2] => 2
[1,4,2,3] => [1,2,3] => [1,2,3] => 1
[1,4,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3,4] => [2,1,3] => [1,3,2] => 2
[2,1,4,3] => [2,1,3] => [1,3,2] => 2
[2,3,1,4] => [2,3,1] => [1,2,3] => 1
[2,3,4,1] => [2,3,1] => [1,2,3] => 1
[2,4,1,3] => [2,1,3] => [1,3,2] => 2
[2,4,3,1] => [2,3,1] => [1,2,3] => 1
[3,1,2,4] => [3,1,2] => [1,2,3] => 1
[3,1,4,2] => [3,1,2] => [1,2,3] => 1
[3,2,1,4] => [3,2,1] => [1,2,3] => 1
[3,2,4,1] => [3,2,1] => [1,2,3] => 1
[3,4,1,2] => [3,1,2] => [1,2,3] => 1
[3,4,2,1] => [3,2,1] => [1,2,3] => 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 1
[4,1,3,2] => [1,3,2] => [1,3,2] => 2
[4,2,1,3] => [2,1,3] => [1,3,2] => 2
[4,2,3,1] => [2,3,1] => [1,2,3] => 1
[4,3,1,2] => [3,1,2] => [1,2,3] => 1
[4,3,2,1] => [3,2,1] => [1,2,3] => 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 2
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 2
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2,5] => [1,3,4,2] => [1,3,4,2] => 2
[1,3,4,5,2] => [1,3,4,2] => [1,3,4,2] => 2
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => 2
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,3,2,5] => [1,4,3,2] => [1,4,2,3] => 2
[1,4,3,5,2] => [1,4,3,2] => [1,4,2,3] => 2
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,5,3,2] => [1,4,3,2] => [1,4,2,3] => 2
Description
The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.
Matching statistic: St000021
Mp00252: Permutations —restriction⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 0 = 1 - 1
[2,1] => [1] => [1] => 0 = 1 - 1
[1,2,3] => [1,2] => [1,2] => 0 = 1 - 1
[1,3,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1,3] => [2,1] => [1,2] => 0 = 1 - 1
[2,3,1] => [2,1] => [1,2] => 0 = 1 - 1
[3,1,2] => [1,2] => [1,2] => 0 = 1 - 1
[3,2,1] => [2,1] => [1,2] => 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,3,4,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,3,4] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[2,3,4,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[2,4,1,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[3,1,2,4] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,1,4,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,2,1,4] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,2,4,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,4,1,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,4,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[4,2,1,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[4,3,1,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[4,3,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,5,2] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000023
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 0 = 1 - 1
[2,1] => [1] => [1] => 0 = 1 - 1
[1,2,3] => [1,2] => [1,2] => 0 = 1 - 1
[1,3,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1,3] => [2,1] => [1,2] => 0 = 1 - 1
[2,3,1] => [2,1] => [1,2] => 0 = 1 - 1
[3,1,2] => [1,2] => [1,2] => 0 = 1 - 1
[3,2,1] => [2,1] => [1,2] => 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,3,4,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,3,4] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[2,3,4,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[2,4,1,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[3,1,2,4] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,1,4,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,2,1,4] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,2,4,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,4,1,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,4,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[4,2,1,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[4,3,1,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[4,3,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,5,2] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
Description
The number of inner peaks of a permutation.
The number of peaks including the boundary is [[St000092]].
Matching statistic: St000035
Mp00252: Permutations —restriction⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 0 = 1 - 1
[2,1] => [1] => [1] => 0 = 1 - 1
[1,2,3] => [1,2] => [1,2] => 0 = 1 - 1
[1,3,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1,3] => [2,1] => [1,2] => 0 = 1 - 1
[2,3,1] => [2,1] => [1,2] => 0 = 1 - 1
[3,1,2] => [1,2] => [1,2] => 0 = 1 - 1
[3,2,1] => [2,1] => [1,2] => 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,3,4,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,3,4] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[2,3,4,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[2,4,1,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[3,1,2,4] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,1,4,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,2,1,4] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,2,4,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,4,1,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,4,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[4,2,1,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[4,3,1,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[4,3,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,5,2] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
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: St000884
Mp00252: Permutations —restriction⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St000884: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 0 = 1 - 1
[2,1] => [1] => [1] => 0 = 1 - 1
[1,2,3] => [1,2] => [1,2] => 0 = 1 - 1
[1,3,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1,3] => [2,1] => [1,2] => 0 = 1 - 1
[2,3,1] => [2,1] => [1,2] => 0 = 1 - 1
[3,1,2] => [1,2] => [1,2] => 0 = 1 - 1
[3,2,1] => [2,1] => [1,2] => 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,3,4,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,3,4] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[2,3,4,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[2,4,1,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[3,1,2,4] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,1,4,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,2,1,4] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,2,4,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,4,1,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,4,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[4,2,1,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[4,3,1,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[4,3,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,5,2] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
Description
The number of isolated descents of a permutation.
A descent $i$ is isolated if neither $i+1$ nor $i-1$ are descents. If a permutation has only isolated descents, then it is called primitive in [1].
Matching statistic: St001729
Mp00252: Permutations —restriction⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001729: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St001729: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 0 = 1 - 1
[2,1] => [1] => [1] => 0 = 1 - 1
[1,2,3] => [1,2] => [1,2] => 0 = 1 - 1
[1,3,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1,3] => [2,1] => [1,2] => 0 = 1 - 1
[2,3,1] => [2,1] => [1,2] => 0 = 1 - 1
[3,1,2] => [1,2] => [1,2] => 0 = 1 - 1
[3,2,1] => [2,1] => [1,2] => 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,3,4,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,3,4] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[2,3,4,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[2,4,1,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[3,1,2,4] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,1,4,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,2,1,4] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,2,4,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,4,1,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,4,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[4,2,1,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[4,3,1,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[4,3,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,5,2] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
Description
The number of visible descents of a permutation.
A visible descent of a permutation $\pi$ is a position $i$ such that $\pi(i+1) \leq \min(i, \pi(i))$.
Matching statistic: St001928
Mp00252: Permutations —restriction⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => 0 = 1 - 1
[2,1] => [1] => [1] => 0 = 1 - 1
[1,2,3] => [1,2] => [1,2] => 0 = 1 - 1
[1,3,2] => [1,2] => [1,2] => 0 = 1 - 1
[2,1,3] => [2,1] => [1,2] => 0 = 1 - 1
[2,3,1] => [2,1] => [1,2] => 0 = 1 - 1
[3,1,2] => [1,2] => [1,2] => 0 = 1 - 1
[3,2,1] => [2,1] => [1,2] => 0 = 1 - 1
[1,2,3,4] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,3,4,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,4,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[2,1,3,4] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[2,3,4,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[2,4,1,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[3,1,2,4] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,1,4,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,2,1,4] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,2,4,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[3,4,1,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[3,4,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[4,1,2,3] => [1,2,3] => [1,2,3] => 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [1,3,2] => 1 = 2 - 1
[4,2,1,3] => [2,1,3] => [1,3,2] => 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [1,2,3] => 0 = 1 - 1
[4,3,1,2] => [3,1,2] => [1,2,3] => 0 = 1 - 1
[4,3,2,1] => [3,2,1] => [1,2,3] => 0 = 1 - 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,4,2,5] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,3,4,5,2] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,2,5] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,4,3,5,2] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 1
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => 1 = 2 - 1
[1,4,5,3,2] => [1,4,3,2] => [1,4,2,3] => 1 = 2 - 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
Mp00252: Permutations —restriction⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00223: Permutations —runsort⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => [1] => 1
[2,1] => [1] => [1] => [1] => 1
[1,2,3] => [1,2] => [1,2] => [2,1] => 1
[1,3,2] => [1,2] => [1,2] => [2,1] => 1
[2,1,3] => [2,1] => [1,2] => [2,1] => 1
[2,3,1] => [2,1] => [1,2] => [2,1] => 1
[3,1,2] => [1,2] => [1,2] => [2,1] => 1
[3,2,1] => [2,1] => [1,2] => [2,1] => 1
[1,2,3,4] => [1,2,3] => [1,2,3] => [3,2,1] => 1
[1,2,4,3] => [1,2,3] => [1,2,3] => [3,2,1] => 1
[1,3,2,4] => [1,3,2] => [1,3,2] => [3,1,2] => 2
[1,3,4,2] => [1,3,2] => [1,3,2] => [3,1,2] => 2
[1,4,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 1
[1,4,3,2] => [1,3,2] => [1,3,2] => [3,1,2] => 2
[2,1,3,4] => [2,1,3] => [1,3,2] => [3,1,2] => 2
[2,1,4,3] => [2,1,3] => [1,3,2] => [3,1,2] => 2
[2,3,1,4] => [2,3,1] => [1,2,3] => [3,2,1] => 1
[2,3,4,1] => [2,3,1] => [1,2,3] => [3,2,1] => 1
[2,4,1,3] => [2,1,3] => [1,3,2] => [3,1,2] => 2
[2,4,3,1] => [2,3,1] => [1,2,3] => [3,2,1] => 1
[3,1,2,4] => [3,1,2] => [1,2,3] => [3,2,1] => 1
[3,1,4,2] => [3,1,2] => [1,2,3] => [3,2,1] => 1
[3,2,1,4] => [3,2,1] => [1,2,3] => [3,2,1] => 1
[3,2,4,1] => [3,2,1] => [1,2,3] => [3,2,1] => 1
[3,4,1,2] => [3,1,2] => [1,2,3] => [3,2,1] => 1
[3,4,2,1] => [3,2,1] => [1,2,3] => [3,2,1] => 1
[4,1,2,3] => [1,2,3] => [1,2,3] => [3,2,1] => 1
[4,1,3,2] => [1,3,2] => [1,3,2] => [3,1,2] => 2
[4,2,1,3] => [2,1,3] => [1,3,2] => [3,1,2] => 2
[4,2,3,1] => [2,3,1] => [1,2,3] => [3,2,1] => 1
[4,3,1,2] => [3,1,2] => [1,2,3] => [3,2,1] => 1
[4,3,2,1] => [3,2,1] => [1,2,3] => [3,2,1] => 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 2
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 2
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 1
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 2
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 2
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 2
[1,3,4,2,5] => [1,3,4,2] => [1,3,4,2] => [4,2,1,3] => 2
[1,3,4,5,2] => [1,3,4,2] => [1,3,4,2] => [4,2,1,3] => 2
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 2
[1,3,5,4,2] => [1,3,4,2] => [1,3,4,2] => [4,2,1,3] => 2
[1,4,2,3,5] => [1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 2
[1,4,2,5,3] => [1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 2
[1,4,3,2,5] => [1,4,3,2] => [1,4,2,3] => [4,1,3,2] => 2
[1,4,3,5,2] => [1,4,3,2] => [1,4,2,3] => [4,1,3,2] => 2
[1,4,5,2,3] => [1,4,2,3] => [1,4,2,3] => [4,1,3,2] => 2
[1,4,5,3,2] => [1,4,3,2] => [1,4,2,3] => [4,1,3,2] => 2
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 11 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000245The number of ascents of a permutation. St000834The number of right outer peaks of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000619The number of cyclic descents of a permutation. St000353The number of inner valleys of a permutation. St000354The number of recoils of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset.
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