Identifier
-
Mp00028:
Dyck paths
—reverse⟶
Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000485: Permutations ⟶ ℤ
Values
[1,0,1,0] => [1,0,1,0] => [2,1] => [2,1] => 2
[1,1,0,0] => [1,1,0,0] => [1,2] => [1,2] => 1
[1,0,1,0,1,0] => [1,0,1,0,1,0] => [2,3,1] => [3,2,1] => 2
[1,0,1,1,0,0] => [1,1,0,0,1,0] => [1,3,2] => [1,3,2] => 2
[1,1,0,0,1,0] => [1,0,1,1,0,0] => [2,1,3] => [2,1,3] => 2
[1,1,0,1,0,0] => [1,1,0,1,0,0] => [3,1,2] => [3,1,2] => 3
[1,1,1,0,0,0] => [1,1,1,0,0,0] => [1,2,3] => [1,2,3] => 1
[1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => [2,3,4,1] => [4,2,3,1] => 2
[1,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,0] => [1,3,4,2] => [1,4,3,2] => 2
[1,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0] => [3,1,4,2] => [3,4,1,2] => 2
[1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,0] => [1,2,4,3] => [1,2,4,3] => 2
[1,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,0] => [2,3,1,4] => [3,2,1,4] => 2
[1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0] => [1,3,2,4] => [1,3,2,4] => 2
[1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0] => [2,4,1,3] => [4,2,1,3] => 3
[1,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,0] => [3,4,1,2] => [4,1,3,2] => 3
[1,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => [1,4,2,3] => [1,4,2,3] => 3
[1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0] => [2,1,3,4] => [2,1,3,4] => 2
[1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0] => [3,1,2,4] => [3,1,2,4] => 3
[1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0] => [4,1,2,3] => [4,1,2,3] => 4
[1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,1] => [5,2,3,4,1] => 2
[1,0,1,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,0,1,0] => [1,3,4,5,2] => [1,5,3,4,2] => 2
[1,0,1,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,0,1,0] => [2,1,4,5,3] => [2,1,5,4,3] => 2
[1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => [3,1,4,5,2] => [3,5,1,4,2] => 2
[1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => [1,2,4,5,3] => [1,2,5,4,3] => 2
[1,0,1,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => [2,3,1,5,4] => [3,2,1,5,4] => 2
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0,1,0] => [2,4,1,5,3] => [4,2,5,1,3] => 2
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [3,4,1,5,2] => [4,5,3,1,2] => 2
[1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3] => [1,4,5,2,3] => 2
[1,0,1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [3,1,2,5,4] => [3,1,2,5,4] => 3
[1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0,1,0] => [4,1,2,5,3] => [4,1,5,2,3] => 3
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => [1,2,3,5,4] => [1,2,3,5,4] => 2
[1,1,0,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => [2,3,4,1,5] => [4,2,3,1,5] => 2
[1,1,0,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => [1,3,4,2,5] => [1,4,3,2,5] => 2
[1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => [3,1,4,2,5] => [3,4,1,2,5] => 2
[1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,1,0,0] => [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,1,0,0] => [2,3,5,1,4] => [5,2,3,1,4] => 3
[1,1,0,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => [1,3,5,2,4] => [1,5,3,2,4] => 3
[1,1,0,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,1,0,0] => [2,4,5,1,3] => [5,2,1,4,3] => 3
[1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,1,0,0] => [3,4,5,1,2] => [5,1,3,4,2] => 3
[1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => [1,4,5,2,3] => [1,5,2,4,3] => 3
[1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => [2,1,5,3,4] => [2,1,5,3,4] => 3
[1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => [3,1,5,2,4] => [3,5,1,2,4] => 3
[1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,1,0,0] => [4,1,5,2,3] => [4,5,2,1,3] => 3
[1,1,0,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => [1,2,5,3,4] => [1,2,5,3,4] => 3
[1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,0,0] => [2,3,1,4,5] => [3,2,1,4,5] => 2
[1,1,1,0,0,0,1,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => [2,4,1,3,5] => [4,2,1,3,5] => 3
[1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [3,4,1,2,5] => [4,1,3,2,5] => 3
[1,1,1,0,0,1,1,0,0,0] => [1,1,1,0,0,1,1,0,0,0] => [1,4,2,3,5] => [1,4,2,3,5] => 3
[1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,1,0,0,0] => [2,5,1,3,4] => [5,2,1,3,4] => 4
[1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [3,5,1,2,4] => [5,1,3,2,4] => 4
[1,1,1,0,1,0,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [4,5,1,2,3] => [5,1,2,4,3] => 4
[1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,5,2,3,4] => [1,5,2,3,4] => 4
[1,1,1,1,0,0,0,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => [2,1,3,4,5] => [2,1,3,4,5] => 2
[1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => [3,1,2,4,5] => [3,1,2,4,5] => 3
[1,1,1,1,0,0,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [4,1,2,3,5] => [4,1,2,3,5] => 4
[1,1,1,1,0,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [5,1,2,3,4] => [5,1,2,3,4] => 5
[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [2,3,4,5,6,1] => [6,2,3,4,5,1] => 2
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => [1,3,4,5,6,2] => [1,6,3,4,5,2] => 2
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,4,5,6,3] => [2,1,6,4,5,3] => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => [3,1,4,5,6,2] => [3,6,1,4,5,2] => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => 2
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => [2,3,1,5,6,4] => [3,2,1,6,5,4] => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => [1,3,2,5,6,4] => [1,3,2,6,5,4] => 2
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0,1,0,1,0] => [2,4,1,5,6,3] => [4,2,6,1,5,3] => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => [3,4,1,5,6,2] => [4,6,3,1,5,2] => 2
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,0,1,0,1,0] => [1,4,2,5,6,3] => [1,4,6,2,5,3] => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,3,5,6,4] => [2,1,3,6,5,4] => 2
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0,1,0] => [3,1,2,5,6,4] => [3,1,2,6,5,4] => 3
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0,1,0,1,0] => [4,1,2,5,6,3] => [4,1,6,2,5,3] => 3
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => 2
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => [2,3,4,1,6,5] => [4,2,3,1,6,5] => 2
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0,1,0] => [1,3,4,2,6,5] => [1,4,3,2,6,5] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0,1,0] => [3,1,4,2,6,5] => [3,4,1,2,6,5] => 2
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,1,0,0,1,0] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,1,0,0,1,0] => [2,3,5,1,6,4] => [5,2,3,6,1,4] => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,1,0,0,1,0] => [1,3,5,2,6,4] => [1,5,3,6,2,4] => 2
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,1,0,0,1,0] => [2,4,5,1,6,3] => [5,2,6,4,1,3] => 2
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => [3,4,5,1,6,2] => [5,6,3,4,1,2] => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0,1,0] => [1,4,5,2,6,3] => [1,5,6,4,2,3] => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,5,3,6,4] => [2,1,5,6,3,4] => 2
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0,1,0] => [3,1,5,2,6,4] => [3,5,1,6,2,4] => 2
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,1,0,0,1,0] => [4,1,5,2,6,3] => [4,5,6,1,2,3] => 2
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,1,0,0,1,0] => [1,2,5,3,6,4] => [1,2,5,6,3,4] => 2
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => [2,3,1,4,6,5] => [3,2,1,4,6,5] => 2
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => [1,3,2,4,6,5] => [1,3,2,4,6,5] => 2
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,6,5] => [4,2,1,3,6,5] => 3
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,0,1,1,0,0,0,1,0] => [3,4,1,2,6,5] => [4,1,3,2,6,5] => 3
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,1,0,0,1,1,0,0,0,1,0] => [1,4,2,3,6,5] => [1,4,2,3,6,5] => 3
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,1,0,0,0,1,0] => [2,5,1,3,6,4] => [5,2,1,6,3,4] => 3
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0,1,0] => [3,5,1,2,6,4] => [5,1,3,6,2,4] => 3
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0,1,0] => [4,5,1,2,6,3] => [5,1,6,4,2,3] => 3
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0,1,0] => [1,5,2,3,6,4] => [1,5,2,6,3,4] => 3
[1,0,1,1,1,1,0,0,0,0,1,0] => [1,0,1,1,1,1,0,0,0,0,1,0] => [2,1,3,4,6,5] => [2,1,3,4,6,5] => 2
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Description
The length of the longest cycle of a permutation.
Map
descent views to invisible inversion bottoms
Description
Return a permutation whose multiset of invisible inversion bottoms is the multiset of descent views of the given permutation.
An invisible inversion of a permutation σ is a pair i<j such that i<σ(j)<σ(i). The element σ(j) is then an invisible inversion bottom.
A descent view in a permutation π is an element π(j) such that π(i+1)<π(j)<π(i), and additionally the smallest element in the decreasing run containing π(i) is smaller than the smallest element in the decreasing run containing π(j).
This map is a bijection χ:Sn→Sn, such that
An invisible inversion of a permutation σ is a pair i<j such that i<σ(j)<σ(i). The element σ(j) is then an invisible inversion bottom.
A descent view in a permutation π is an element π(j) such that π(i+1)<π(j)<π(i), and additionally the smallest element in the decreasing run containing π(i) is smaller than the smallest element in the decreasing run containing π(j).
This map is a bijection χ:Sn→Sn, such that
- the multiset of descent views in π is the multiset of invisible inversion bottoms in χ(π),
- the set of left-to-right maxima of π is the set of maximal elements in the cycles of χ(π),
- the set of global ascent of π is the set of global ascent of χ(π),
- the set of maximal elements in the decreasing runs of π is the set of weak deficiency positions of χ(π), and
- the set of minimal elements in the decreasing runs of π is the set of weak deficiency values of χ(π).
Map
to 321-avoiding permutation (Billey-Jockusch-Stanley)
Description
The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.
Map
reverse
Description
The reversal of a Dyck path.
This is the Dyck path obtained by reading the path backwards.
This is the Dyck path obtained by reading the path backwards.
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