Identifier
-
Mp00127:
Permutations
—left-to-right-maxima to Dyck path⟶
Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001906: Permutations ⟶ ℤ
Values
[1] => [1,0] => [1,0] => [2,1] => 0
[1,2] => [1,0,1,0] => [1,1,0,0] => [2,3,1] => 0
[2,1] => [1,1,0,0] => [1,0,1,0] => [3,1,2] => 0
[1,2,3] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => [2,3,4,1] => 0
[1,3,2] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => [2,4,1,3] => 0
[2,1,3] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => [3,1,4,2] => 0
[2,3,1] => [1,1,0,1,0,0] => [1,0,1,0,1,0] => [4,1,2,3] => 0
[3,1,2] => [1,1,1,0,0,0] => [1,1,0,1,0,0] => [4,3,1,2] => 0
[3,2,1] => [1,1,1,0,0,0] => [1,1,0,1,0,0] => [4,3,1,2] => 0
[1,2,3,4] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 0
[1,2,4,3] => [1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => 0
[1,3,2,4] => [1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => 0
[1,3,4,2] => [1,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => 0
[1,4,2,3] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 0
[1,4,3,2] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => 0
[2,1,3,4] => [1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => 0
[2,1,4,3] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => 0
[2,3,1,4] => [1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => 0
[2,3,4,1] => [1,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0] => [5,1,2,3,4] => 0
[2,4,1,3] => [1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 0
[2,4,3,1] => [1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => [5,1,4,2,3] => 0
[3,1,2,4] => [1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => 0
[3,1,4,2] => [1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 0
[3,2,1,4] => [1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => 0
[3,2,4,1] => [1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => [5,3,1,2,4] => 0
[3,4,1,2] => [1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => 1
[3,4,2,1] => [1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => [5,4,1,2,3] => 1
[4,1,2,3] => [1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 0
[4,1,3,2] => [1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 0
[4,2,1,3] => [1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 0
[4,2,3,1] => [1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 0
[4,3,1,2] => [1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 0
[4,3,2,1] => [1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0] => [5,3,4,1,2] => 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => 0
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => 0
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => 0
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 0
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => 0
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => 0
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => 0
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => 0
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => 0
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => 0
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0] => [1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => 0
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => 0
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => 0
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => 0
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => 0
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => 0
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => 0
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => 0
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => 0
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => 0
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => 0
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => 0
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => 0
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => 0
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => 0
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => 0
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => 0
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => 0
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => 0
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => 0
[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => 0
[2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => 0
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => 0
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => 0
[2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => 0
[2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => 0
[2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => 0
[2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => 1
[2,4,5,3,1] => [1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => 1
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => 0
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => 0
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => 0
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => 0
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => 0
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => 0
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => 0
[3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => 0
[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => 0
[3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => 0
[3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => 0
[3,1,5,4,2] => [1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => 0
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => 0
[3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => 0
[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => 0
[3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => 0
[3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => 0
[3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => 0
[3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => 1
[3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => 1
[3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => 1
[3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => 1
[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => 2
[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => 2
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => 1
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => 1
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Description
Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation.
Let $\pi$ be a permutation. Its total displacement St000830The total displacement of a permutation. is $D(\pi) = \sum_i |\pi(i) - i|$, and its absolute length St000216The absolute length of a permutation. is the minimal number $T(\pi)$ of transpositions whose product is $\pi$. Finally, let $I(\pi)$ be the number of inversions St000018The number of inversions of a permutation. of $\pi$.
This statistic equals $\left(D(\pi)-T(\pi)-I(\pi)\right)/2$.
Diaconis and Graham [1] proved that this statistic is always nonnegative.
Let $\pi$ be a permutation. Its total displacement St000830The total displacement of a permutation. is $D(\pi) = \sum_i |\pi(i) - i|$, and its absolute length St000216The absolute length of a permutation. is the minimal number $T(\pi)$ of transpositions whose product is $\pi$. Finally, let $I(\pi)$ be the number of inversions St000018The number of inversions of a permutation. of $\pi$.
This statistic equals $\left(D(\pi)-T(\pi)-I(\pi)\right)/2$.
Diaconis and Graham [1] proved that this statistic is always nonnegative.
Map
peaks-to-valleys
Description
Return the path that has a valley wherever the original path has a peak of height at least one.
More precisely, the height of a valley in the image is the height of the corresponding peak minus $2$.
This is also (the inverse of) rowmotion on Dyck paths regarded as order ideals in the triangular poset.
More precisely, the height of a valley in the image is the height of the corresponding peak minus $2$.
This is also (the inverse of) rowmotion on Dyck paths regarded as order ideals in the triangular poset.
Map
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
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