Identifier
-
Mp00031:
Dyck paths
—to 312-avoiding permutation⟶
Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000080: Posets ⟶ ℤ (values match St000528The height of a poset.)
Values
[1,0] => [1] => [1] => ([],1) => 0
[1,0,1,0] => [1,2] => [1,2] => ([(0,1)],2) => 1
[1,1,0,0] => [2,1] => [2,1] => ([],2) => 0
[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 2
[1,0,1,1,0,0] => [1,3,2] => [3,1,2] => ([(1,2)],3) => 1
[1,1,0,0,1,0] => [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3) => 1
[1,1,0,1,0,0] => [2,3,1] => [2,3,1] => ([(1,2)],3) => 1
[1,1,1,0,0,0] => [3,2,1] => [3,2,1] => ([],3) => 0
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [4,1,2,3] => ([(1,2),(2,3)],4) => 2
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4) => 2
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4) => 1
[1,0,1,1,1,0,0,0] => [1,4,3,2] => [4,3,1,2] => ([(2,3)],4) => 1
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4) => 2
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [4,2,1,3] => ([(1,3),(2,3)],4) => 1
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4) => 2
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4) => 2
[1,1,0,1,1,0,0,0] => [2,4,3,1] => [4,2,3,1] => ([(2,3)],4) => 1
[1,1,1,0,0,0,1,0] => [3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4) => 1
[1,1,1,0,0,1,0,0] => [3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4) => 1
[1,1,1,0,1,0,0,0] => [3,4,2,1] => [3,4,2,1] => ([(2,3)],4) => 1
[1,1,1,1,0,0,0,0] => [4,3,2,1] => [4,3,2,1] => ([],4) => 0
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5) => 3
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => 3
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [4,1,2,5,3] => ([(0,4),(1,2),(2,3),(2,4)],5) => 2
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [5,4,1,2,3] => ([(2,3),(3,4)],5) => 2
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5) => 3
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5) => 2
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [3,1,4,2,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => 2
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [3,1,4,5,2] => ([(0,4),(1,2),(1,4),(4,3)],5) => 2
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => [5,3,1,4,2] => ([(1,4),(2,3),(2,4)],5) => 1
[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2
[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [4,3,1,5,2] => ([(0,4),(1,4),(2,3),(2,4)],5) => 1
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => [4,5,1,3,2] => ([(0,4),(1,2),(1,3)],5) => 1
[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [5,4,3,1,2] => ([(3,4)],5) => 1
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5) => 3
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5) => 2
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5) => 2
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [4,2,1,5,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 1
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => [5,4,2,1,3] => ([(2,4),(3,4)],5) => 1
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5) => 3
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5) => 2
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => 3
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5) => 3
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5) => 2
[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2
[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5) => 2
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5) => 1
[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [5,4,2,3,1] => ([(3,4)],5) => 1
[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5) => 2
[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5) => 1
[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5) => 2
[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5) => 2
[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [5,3,2,4,1] => ([(2,4),(3,4)],5) => 1
[1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5) => 2
[1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5) => 2
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => [3,4,5,2,1] => ([(2,3),(3,4)],5) => 2
[1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => [5,3,4,2,1] => ([(3,4)],5) => 1
[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5) => 1
[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5) => 1
[1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => [4,3,5,2,1] => ([(2,4),(3,4)],5) => 1
[1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => [4,5,3,2,1] => ([(3,4)],5) => 1
[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [5,4,3,2,1] => ([],5) => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 5
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6) => 4
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => 4
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [5,1,2,3,6,4] => ([(0,5),(1,3),(3,4),(4,2),(4,5)],6) => 3
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => [6,5,1,2,3,4] => ([(2,3),(3,5),(5,4)],6) => 3
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [4,1,2,3,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6) => 4
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [6,4,1,2,3,5] => ([(1,5),(2,3),(3,4),(4,5)],6) => 3
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [4,1,2,5,3,6] => ([(0,5),(1,2),(2,3),(2,5),(3,4),(5,4)],6) => 3
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [4,1,2,5,6,3] => ([(0,5),(1,4),(4,2),(4,5),(5,3)],6) => 3
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,5,3] => [6,4,1,2,5,3] => ([(1,5),(2,3),(3,4),(3,5)],6) => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => [5,4,1,2,3,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 3
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => [5,4,1,2,6,3] => ([(0,5),(1,5),(2,3),(3,4),(3,5)],6) => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,4,3] => [5,6,1,2,4,3] => ([(0,4),(1,5),(5,2),(5,3)],6) => 2
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => [6,5,4,1,2,3] => ([(3,4),(4,5)],6) => 2
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [3,1,2,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6) => 4
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [6,3,1,2,4,5] => ([(1,5),(2,3),(3,5),(5,4)],6) => 3
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [5,3,1,2,4,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6) => 3
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [5,3,1,2,6,4] => ([(0,5),(1,4),(1,5),(2,3),(3,4),(3,5)],6) => 2
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => [6,5,3,1,2,4] => ([(2,5),(3,4),(4,5)],6) => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [3,1,4,2,5,6] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6) => 3
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [6,3,1,4,2,5] => ([(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [3,1,4,5,2,6] => ([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6) => 3
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [3,1,4,5,6,2] => ([(0,5),(1,3),(1,5),(4,2),(5,4)],6) => 3
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,5,2] => [6,3,1,4,5,2] => ([(1,5),(2,3),(2,5),(5,4)],6) => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => [5,3,1,4,2,6] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,4,6,2] => [5,3,1,4,6,2] => ([(0,5),(1,4),(2,3),(2,4),(4,5)],6) => 2
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,4,2] => [5,3,1,6,4,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => [6,5,3,1,4,2] => ([(2,5),(3,4),(3,5)],6) => 1
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => [4,3,1,2,5,6] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6) => 3
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => [6,4,3,1,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6) => 2
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,3,5,2,6] => [4,3,1,5,2,6] => ([(0,5),(1,5),(2,3),(2,5),(3,4),(5,4)],6) => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => [4,3,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,5),(5,4)],6) => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => [6,4,3,1,5,2] => ([(1,5),(2,5),(3,4),(3,5)],6) => 1
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,3,2,6] => [4,5,1,3,2,6] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => 2
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => [4,5,1,3,6,2] => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6) => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => [4,5,1,6,3,2] => ([(0,4),(1,2),(1,3),(1,5),(4,5)],6) => 2
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => [6,4,5,1,3,2] => ([(1,5),(2,3),(2,4)],6) => 1
>>> Load all 196 entries. <<<
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
Description
The rank of the poset.
Map
to 312-avoiding permutation
Description
Map
permutation poset
Description
Sends a permutation to its permutation poset.
For a permutation π of length n, this poset has vertices
{(i,π(i)) : 1≤i≤n}
and the cover relation is given by (w,x)≤(y,z) if w≤y and x≤z.
For example, the permutation [3,1,5,4,2] is mapped to the poset with cover relations
{(2,1)≺(5,2), (2,1)≺(4,4), (2,1)≺(3,5), (1,3)≺(4,4), (1,3)≺(3,5)}.
For a permutation π of length n, this poset has vertices
{(i,π(i)) : 1≤i≤n}
and the cover relation is given by (w,x)≤(y,z) if w≤y and x≤z.
For example, the permutation [3,1,5,4,2] is mapped to the poset with cover relations
{(2,1)≺(5,2), (2,1)≺(4,4), (2,1)≺(3,5), (1,3)≺(4,4), (1,3)≺(3,5)}.
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.).
searching the database
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!