Identifier
-
Mp00023:
Dyck paths
—to non-crossing permutation⟶
Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤ (values match St001206The maximal dimension of an indecomposable projective eAe-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module eA.)
Values
[1,0,1,0] => [1,2] => [1,2] => [1,0,1,0] => 2
[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => [1,0,1,0,1,0] => 2
[1,1,0,0,1,0] => [2,1,3] => [2,1,3] => [1,1,0,0,1,0] => 2
[1,1,0,1,0,0] => [2,3,1] => [2,3,1] => [1,1,0,1,0,0] => 2
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0] => 2
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0] => 2
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0] => 2
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0] => 2
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0] => 2
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0] => 3
[1,1,1,0,0,0,1,0] => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0] => 2
[1,1,1,0,0,1,0,0] => [3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0] => 2
[1,1,1,0,1,0,0,0] => [4,2,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0] => 2
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => 2
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0] => 2
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0] => 2
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0] => 2
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0] => 3
[1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0] => 2
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => 2
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0] => 2
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => 2
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0] => 3
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => 3
[1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0] => 2
[1,1,0,1,1,0,1,0,0,0] => [2,5,3,4,1] => [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0] => 2
[1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => 2
[1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0] => 2
[1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0] => 3
[1,1,1,0,1,0,0,0,1,0] => [4,2,3,1,5] => [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0] => 2
[1,1,1,0,1,0,0,1,0,0] => [4,2,3,5,1] => [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0] => 2
[1,1,1,0,1,0,1,0,0,0] => [5,2,3,4,1] => [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0] => 3
[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0] => 2
[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0] => 2
[1,1,1,1,0,0,1,0,0,0] => [5,3,2,4,1] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0] => 2
[1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => [3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0] => 2
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 2
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [5,1,2,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [4,1,2,5,3,6] => [1,1,1,1,0,0,0,1,0,0,1,0] => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0] => 3
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,4,3,6] => [5,4,1,2,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => [5,4,1,2,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0] => 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] => [1,1,1,0,0,0,1,0,1,0,1,0] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [5,3,1,2,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [5,3,1,2,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0] => 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] => [1,1,1,0,0,1,0,0,1,0,1,0] => 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] => [1,1,1,0,0,1,0,1,0,0,1,0] => 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] => [1,1,1,0,0,1,0,1,0,1,0,0] => 3
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => [5,3,1,4,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 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] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,3,2,5,6] => [4,3,1,2,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => 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] => [1,1,1,1,0,0,0,1,0,0,1,0] => 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] => [1,1,1,1,0,0,0,1,0,1,0,0] => 3
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,5,3,4,2,6] => [5,1,3,4,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,5,3,4,6,2] => [5,1,3,4,6,2] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,6,3,4,5,2] => [3,1,6,4,5,2] => [1,1,1,0,0,1,1,1,0,0,0,0] => 2
[1,0,1,1,1,1,0,0,0,0,1,0] => [1,5,4,3,2,6] => [5,4,3,1,2,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,4,3,6,2] => [5,4,3,1,6,2] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
[1,0,1,1,1,1,0,0,1,0,0,0] => [1,6,4,3,5,2] => [4,6,3,1,5,2] => [1,1,1,1,0,1,1,0,0,0,0,0] => 2
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0] => 2
[1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,3,5,4,6] => [5,2,1,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,0,0,1,0,1,1,0,1,0,0] => [2,1,3,5,6,4] => [5,2,1,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
[1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => [4,2,1,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
[1,1,0,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => [4,2,1,5,3,6] => [1,1,1,1,0,0,0,1,0,0,1,0] => 2
[1,1,0,0,1,1,0,1,0,1,0,0] => [2,1,4,5,6,3] => [4,2,1,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0] => 3
[1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,5,4,3,6] => [5,4,2,1,3,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,0,0,1,1,1,0,0,1,0,0] => [2,1,5,4,6,3] => [5,4,2,1,6,3] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
[1,1,0,1,0,0,1,0,1,0,1,0] => [2,3,1,4,5,6] => [2,3,1,4,5,6] => [1,1,0,1,0,0,1,0,1,0,1,0] => 2
[1,1,0,1,0,0,1,1,0,0,1,0] => [2,3,1,5,4,6] => [5,2,3,1,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,0,1,0,0,1,1,0,1,0,0] => [2,3,1,5,6,4] => [5,2,3,1,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
[1,1,0,1,0,1,0,0,1,0,1,0] => [2,3,4,1,5,6] => [2,3,4,1,5,6] => [1,1,0,1,0,1,0,0,1,0,1,0] => 3
[1,1,0,1,0,1,0,1,0,0,1,0] => [2,3,4,5,1,6] => [2,3,4,5,1,6] => [1,1,0,1,0,1,0,1,0,0,1,0] => 3
[1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => [2,3,4,5,6,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 3
[1,1,0,1,0,1,1,0,0,0,1,0] => [2,3,5,4,1,6] => [5,2,3,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,0,1,0,1,1,0,0,1,0,0] => [2,3,5,4,6,1] => [5,2,3,4,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
[1,1,0,1,0,1,1,0,1,0,0,0] => [2,3,6,4,5,1] => [2,6,3,4,5,1] => [1,1,0,1,1,1,1,0,0,0,0,0] => 2
[1,1,0,1,1,0,0,0,1,0,1,0] => [2,4,3,1,5,6] => [4,2,3,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0] => 2
[1,1,0,1,1,0,0,1,0,0,1,0] => [2,4,3,5,1,6] => [4,2,3,5,1,6] => [1,1,1,1,0,0,0,1,0,0,1,0] => 2
[1,1,0,1,1,0,0,1,0,1,0,0] => [2,4,3,5,6,1] => [4,2,3,5,6,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => 3
[1,1,0,1,1,0,1,0,0,0,1,0] => [2,5,3,4,1,6] => [2,5,3,4,1,6] => [1,1,0,1,1,1,0,0,0,0,1,0] => 2
[1,1,0,1,1,0,1,0,0,1,0,0] => [2,5,3,4,6,1] => [2,5,3,4,6,1] => [1,1,0,1,1,1,0,0,0,1,0,0] => 2
[1,1,0,1,1,0,1,0,1,0,0,0] => [2,6,3,4,5,1] => [2,3,6,4,5,1] => [1,1,0,1,0,1,1,1,0,0,0,0] => 3
[1,1,0,1,1,1,0,0,0,0,1,0] => [2,5,4,3,1,6] => [5,4,2,3,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,0,1,1,1,0,0,0,1,0,0] => [2,5,4,3,6,1] => [5,4,2,3,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
[1,1,0,1,1,1,0,0,1,0,0,0] => [2,6,4,3,5,1] => [4,6,2,3,5,1] => [1,1,1,1,0,1,1,0,0,0,0,0] => 2
[1,1,1,0,0,0,1,0,1,0,1,0] => [3,2,1,4,5,6] => [3,2,1,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0] => 2
[1,1,1,0,0,0,1,1,0,0,1,0] => [3,2,1,5,4,6] => [5,3,2,1,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,1,0,0,0,1,1,0,1,0,0] => [3,2,1,5,6,4] => [5,3,2,1,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
[1,1,1,0,0,1,0,0,1,0,1,0] => [3,2,4,1,5,6] => [3,2,4,1,5,6] => [1,1,1,0,0,1,0,0,1,0,1,0] => 2
[1,1,1,0,0,1,0,1,0,0,1,0] => [3,2,4,5,1,6] => [3,2,4,5,1,6] => [1,1,1,0,0,1,0,1,0,0,1,0] => 3
[1,1,1,0,0,1,0,1,0,1,0,0] => [3,2,4,5,6,1] => [3,2,4,5,6,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => 3
[1,1,1,0,0,1,1,0,0,0,1,0] => [3,2,5,4,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,1,1,0,0,1,1,0,0,1,0,0] => [3,2,5,4,6,1] => [5,3,2,4,6,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2
[1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,6,4,5,1] => [3,6,2,4,5,1] => [1,1,1,0,1,1,1,0,0,0,0,0] => 2
[1,1,1,0,1,0,0,0,1,0,1,0] => [4,2,3,1,5,6] => [2,4,3,1,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0] => 2
>>> Load all 125 entries. <<<
search for individual values
searching the database for the individual values of this statistic
Description
The number of simple modules in the algebra eAe with projective dimension at most 1 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA.
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
to non-crossing permutation
Description
Sends a Dyck path D with valley at positions {(i1,j1),…,(ik,jk)} to the unique non-crossing permutation π having descents {i1,…,ik} and whose inverse has descents {j1,…,jk}.
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to n(n−1) minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to n(n−1) minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..
Map
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let (c1,…,ck) be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are c1,c1+c2,…,c1+⋯+ck.
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 (c1,…,ck) be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are c1,c1+c2,…,c1+⋯+ck.
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.
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