Identifier
-
Mp00201:
Dyck paths
—Ringel⟶
Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000619: Permutations ⟶ ℤ
Values
[1,0] => [2,1] => [2,1] => [2,1] => 1
[1,0,1,0] => [3,1,2] => [3,2,1] => [3,2,1] => 2
[1,1,0,0] => [2,3,1] => [3,1,2] => [1,3,2] => 2
[1,0,1,0,1,0] => [4,1,2,3] => [4,3,2,1] => [4,3,2,1] => 3
[1,0,1,1,0,0] => [3,1,4,2] => [4,2,1,3] => [2,4,1,3] => 2
[1,1,0,0,1,0] => [2,4,1,3] => [4,3,1,2] => [1,4,3,2] => 3
[1,1,0,1,0,0] => [4,3,1,2] => [4,2,3,1] => [2,4,3,1] => 2
[1,1,1,0,0,0] => [2,3,4,1] => [4,1,2,3] => [1,2,4,3] => 2
[1,0,1,0,1,0,1,0] => [5,1,2,3,4] => [5,4,3,2,1] => [5,4,3,2,1] => 4
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [5,3,2,1,4] => [3,5,2,1,4] => 3
[1,0,1,1,0,0,1,0] => [3,1,5,2,4] => [5,4,2,1,3] => [2,5,4,1,3] => 3
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [5,3,4,2,1] => [3,5,4,2,1] => 3
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [5,2,1,3,4] => [2,1,5,3,4] => 3
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [5,4,3,1,2] => [1,5,4,3,2] => 4
[1,1,0,0,1,1,0,0] => [2,4,1,5,3] => [5,3,1,2,4] => [1,3,5,2,4] => 2
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [5,4,2,3,1] => [2,5,4,3,1] => 3
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [4,2,5,3,1] => [4,5,2,3,1] => 2
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [5,2,3,1,4] => [2,3,5,1,4] => 2
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [5,4,1,2,3] => [1,2,5,4,3] => 3
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [5,3,4,1,2] => [1,3,5,4,2] => 3
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [5,2,3,4,1] => [2,3,5,4,1] => 2
[1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5,1,2,3,4] => [1,2,3,5,4] => 2
[1,0,1,0,1,0,1,0,1,0] => [6,1,2,3,4,5] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 5
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [6,4,3,2,1,5] => [4,6,3,2,1,5] => 4
[1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => [6,5,3,2,1,4] => [3,6,5,2,1,4] => 4
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [6,4,5,3,2,1] => [4,6,5,3,2,1] => 4
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [6,3,2,1,4,5] => [3,2,6,1,4,5] => 3
[1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => [6,5,4,2,1,3] => [2,6,5,4,1,3] => 4
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [6,4,2,1,3,5] => [2,4,6,1,3,5] => 2
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [6,5,3,4,2,1] => [3,6,5,4,2,1] => 4
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [5,3,6,4,2,1] => [5,6,3,4,2,1] => 3
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [6,3,4,2,1,5] => [3,4,6,2,1,5] => 3
[1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => [6,5,2,1,3,4] => [2,1,6,5,3,4] => 4
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [6,4,5,2,1,3] => [2,4,6,5,1,3] => 3
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [6,3,4,5,2,1] => [3,4,6,5,2,1] => 3
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [6,2,1,3,4,5] => [2,1,3,6,4,5] => 3
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [6,5,4,3,1,2] => [1,6,5,4,3,2] => 5
[1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [6,4,3,1,2,5] => [1,4,6,3,2,5] => 3
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [6,5,3,1,2,4] => [1,3,6,5,2,4] => 3
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [6,4,5,3,1,2] => [1,4,6,5,3,2] => 4
[1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => [6,3,1,2,4,5] => [1,3,2,6,4,5] => 3
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [6,5,4,2,3,1] => [2,6,5,4,3,1] => 4
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [6,4,2,3,1,5] => [2,4,6,3,1,5] => 3
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [4,2,6,5,3,1] => [6,4,5,2,3,1] => 3
[1,1,0,1,0,1,0,1,0,0] => [5,6,1,2,3,4] => [5,3,1,6,4,2] => [5,6,3,4,1,2] => 2
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [4,2,6,3,1,5] => [4,2,6,3,1,5] => 4
[1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => [6,5,2,3,1,4] => [2,3,6,5,1,4] => 3
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [6,4,5,2,3,1] => [2,4,6,5,3,1] => 3
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [5,2,4,6,3,1] => [5,2,4,6,3,1] => 3
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [6,2,3,1,4,5] => [2,3,1,6,4,5] => 3
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [6,5,4,1,2,3] => [1,2,6,5,4,3] => 4
[1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => [6,4,1,2,3,5] => [1,2,4,6,3,5] => 2
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [6,5,3,4,1,2] => [1,3,6,5,4,2] => 4
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [5,3,6,4,1,2] => [1,5,6,3,4,2] => 3
[1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => [6,3,4,1,2,5] => [1,3,4,6,2,5] => 2
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [6,5,2,3,4,1] => [2,3,6,5,4,1] => 3
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [5,2,3,6,4,1] => [5,2,6,3,4,1] => 3
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [5,2,6,3,4,1] => [2,5,6,3,4,1] => 2
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [6,2,3,4,1,5] => [2,3,4,6,1,5] => 2
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [6,5,1,2,3,4] => [1,2,3,6,5,4] => 3
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [6,4,5,1,2,3] => [1,2,4,6,5,3] => 3
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [6,3,4,5,1,2] => [1,3,4,6,5,2] => 3
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [6,2,3,4,5,1] => [2,3,4,6,5,1] => 2
[1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6,1,2,3,4,5] => [1,2,3,4,6,5] => 2
[1,1,0,0,1,0,1,1,1,0,0,0] => [2,5,1,3,6,7,4] => [7,4,3,1,2,5,6] => [1,4,3,7,2,5,6] => 3
[1,1,0,0,1,1,0,0,1,0,1,0] => [2,4,1,7,3,5,6] => [7,6,5,3,1,2,4] => [1,3,7,6,5,2,4] => 4
[1,1,0,0,1,1,0,0,1,1,0,0] => [2,4,1,6,3,7,5] => [7,5,3,1,2,4,6] => [1,3,5,7,2,4,6] => 2
[1,1,0,0,1,1,0,1,1,0,0,0] => [2,6,1,5,3,7,4] => [7,4,5,3,1,2,6] => [1,4,5,7,3,2,6] => 3
[1,1,0,0,1,1,1,0,0,0,1,0] => [2,4,1,5,7,3,6] => [7,6,3,1,2,4,5] => [1,3,2,7,6,4,5] => 4
[1,1,0,0,1,1,1,0,0,1,0,0] => [2,4,1,7,6,3,5] => [7,5,6,3,1,2,4] => [1,3,5,7,6,2,4] => 3
[1,1,0,0,1,1,1,0,1,0,0,0] => [2,7,1,5,6,3,4] => [7,4,5,6,3,1,2] => [1,4,5,7,6,3,2] => 4
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,4,1,5,6,7,3] => [7,3,1,2,4,5,6] => [1,3,2,4,7,5,6] => 3
[1,1,1,0,0,0,1,0,1,0,1,0] => [2,3,7,1,4,5,6] => [7,6,5,4,1,2,3] => [1,2,7,6,5,4,3] => 5
[1,1,1,0,0,0,1,0,1,1,0,0] => [2,3,6,1,4,7,5] => [7,5,4,1,2,3,6] => [1,2,5,7,4,3,6] => 3
[1,1,1,0,0,0,1,1,0,0,1,0] => [2,3,5,1,7,4,6] => [7,6,4,1,2,3,5] => [1,2,4,7,6,3,5] => 3
[1,1,1,0,0,0,1,1,0,1,0,0] => [2,3,7,1,6,4,5] => [7,5,6,4,1,2,3] => [1,2,5,7,6,4,3] => 4
[1,1,1,0,0,0,1,1,1,0,0,0] => [2,3,5,1,6,7,4] => [7,4,1,2,3,5,6] => [1,2,4,3,7,5,6] => 3
[1,1,1,0,0,1,0,0,1,0,1,0] => [2,7,4,1,3,5,6] => [7,6,5,3,4,1,2] => [1,3,7,6,5,4,2] => 5
[1,1,1,0,0,1,0,0,1,1,0,0] => [2,6,4,1,3,7,5] => [7,5,3,4,1,2,6] => [1,3,5,7,4,2,6] => 3
[1,1,1,0,0,1,1,0,0,0,1,0] => [2,5,4,1,7,3,6] => [7,6,3,4,1,2,5] => [1,3,4,7,6,2,5] => 3
[1,1,1,0,0,1,1,0,0,1,0,0] => [2,7,4,1,6,3,5] => [7,5,6,3,4,1,2] => [1,3,5,7,6,4,2] => 4
[1,1,1,0,0,1,1,1,0,0,0,0] => [2,5,4,1,6,7,3] => [7,3,4,1,2,5,6] => [1,3,4,2,7,5,6] => 3
[1,1,1,1,0,0,0,0,1,0,1,0] => [2,3,4,7,1,5,6] => [7,6,5,1,2,3,4] => [1,2,3,7,6,5,4] => 4
[1,1,1,1,0,0,0,0,1,1,0,0] => [2,3,4,6,1,7,5] => [7,5,1,2,3,4,6] => [1,2,3,5,7,4,6] => 2
[1,1,1,1,0,0,0,1,0,0,1,0] => [2,3,7,5,1,4,6] => [7,6,4,5,1,2,3] => [1,2,4,7,6,5,3] => 4
[1,1,1,1,0,0,0,1,0,1,0,0] => [2,3,7,6,1,4,5] => [6,4,7,5,1,2,3] => [1,2,6,7,4,5,3] => 3
[1,1,1,1,0,0,0,1,1,0,0,0] => [2,3,6,5,1,7,4] => [7,4,5,1,2,3,6] => [1,2,4,5,7,3,6] => 2
[1,1,1,1,0,0,1,0,0,0,1,0] => [2,7,4,5,1,3,6] => [7,6,3,4,5,1,2] => [1,3,4,7,6,5,2] => 4
[1,1,1,1,0,0,1,0,1,0,0,0] => [2,7,6,5,1,3,4] => [6,3,7,4,5,1,2] => [1,3,6,7,4,5,2] => 3
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,6,4,5,1,7,3] => [7,3,4,5,1,2,6] => [1,3,4,5,7,2,6] => 2
[1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => [7,6,1,2,3,4,5] => [1,2,3,4,7,6,5] => 3
[1,1,1,1,1,0,0,0,0,1,0,0] => [2,3,4,7,6,1,5] => [7,5,6,1,2,3,4] => [1,2,3,5,7,6,4] => 3
[1,1,1,1,1,0,0,0,1,0,0,0] => [2,3,7,5,6,1,4] => [7,4,5,6,1,2,3] => [1,2,4,5,7,6,3] => 3
[1,1,1,1,1,0,0,1,0,0,0,0] => [2,7,4,5,6,1,3] => [7,3,4,5,6,1,2] => [1,3,4,5,7,6,2] => 3
[1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 2
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Description
The number of cyclic descents of a permutation.
For a permutation π of {1,…,n}, this is given by the number of indices 1≤i≤n such that π(i)>π(i+1) where we set π(n+1)=π(1).
For a permutation π of {1,…,n}, this is given by the number of indices 1≤i≤n such that π(i)>π(i+1) where we set π(n+1)=π(1).
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
Foata bijection
Description
Sends a permutation to its image under the Foata bijection.
The Foata bijection ϕ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word w1w2...wn, compute the image inductively by starting with ϕ(w1)=w1.
At the i-th step, if ϕ(w1w2...wi)=v1v2...vi, define ϕ(w1w2...wiwi+1) by placing wi+1 on the end of the word v1v2...vi and breaking the word up into blocks as follows.
To compute ϕ([1,4,2,5,3]), the sequence of words is
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).
The Foata bijection ϕ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word w1w2...wn, compute the image inductively by starting with ϕ(w1)=w1.
At the i-th step, if ϕ(w1w2...wi)=v1v2...vi, define ϕ(w1w2...wiwi+1) by placing wi+1 on the end of the word v1v2...vi and breaking the word up into blocks as follows.
- If wi+1≥vi, place a vertical line to the right of each vk for which wi+1≥vk.
- If wi+1<vi, place a vertical line to the right of each vk for which wi+1<vk.
To compute ϕ([1,4,2,5,3]), the sequence of words is
- 1
- |1|4→14
- |14|2→412
- |4|1|2|5→4125
- |4|125|3→45123.
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).
Map
inverse first fundamental transformation
Description
Let σ=(i11⋯i1k1)⋯(iℓ1⋯iℓkℓ) be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps σ to the permutation [i11,…,i1k1,…,iℓ1,…,iℓkℓ] in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.
Maps σ to the permutation [i11,…,i1k1,…,iℓ1,…,iℓkℓ] in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.
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