Identifier
-
Mp00119:
Dyck paths
—to 321-avoiding permutation (Krattenthaler)⟶
Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤ
Values
[1,0] => [1] => [1] => 1
[1,0,1,0] => [1,2] => [2] => 1
[1,1,0,0] => [2,1] => [2] => 1
[1,0,1,0,1,0] => [1,2,3] => [3] => 1
[1,0,1,1,0,0] => [1,3,2] => [1,2] => 2
[1,1,0,0,1,0] => [2,1,3] => [3] => 1
[1,1,0,1,0,0] => [2,3,1] => [3] => 1
[1,1,1,0,0,0] => [3,1,2] => [3] => 1
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [4] => 1
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [2,2] => 2
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3] => 2
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,3] => 2
[1,0,1,1,1,0,0,0] => [1,4,2,3] => [1,3] => 2
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [4] => 1
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,2] => 2
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [4] => 1
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [4] => 1
[1,1,0,1,1,0,0,0] => [2,4,1,3] => [4] => 1
[1,1,1,0,0,0,1,0] => [3,1,2,4] => [4] => 1
[1,1,1,0,0,1,0,0] => [3,1,4,2] => [2,2] => 2
[1,1,1,0,1,0,0,0] => [3,4,1,2] => [4] => 1
[1,1,1,1,0,0,0,0] => [4,1,2,3] => [4] => 1
[1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [5] => 1
[1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [3,2] => 2
[1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [2,3] => 2
[1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [2,3] => 2
[1,0,1,0,1,1,1,0,0,0] => [1,2,5,3,4] => [2,3] => 2
[1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [1,4] => 2
[1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [1,2,2] => 2
[1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [1,4] => 2
[1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,4] => 2
[1,0,1,1,0,1,1,0,0,0] => [1,3,5,2,4] => [1,4] => 2
[1,0,1,1,1,0,0,0,1,0] => [1,4,2,3,5] => [1,4] => 2
[1,0,1,1,1,0,0,1,0,0] => [1,4,2,5,3] => [1,2,2] => 2
[1,0,1,1,1,0,1,0,0,0] => [1,4,5,2,3] => [1,4] => 2
[1,0,1,1,1,1,0,0,0,0] => [1,5,2,3,4] => [1,4] => 2
[1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [5] => 1
[1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [3,2] => 2
[1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [2,3] => 2
[1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [2,3] => 2
[1,1,0,0,1,1,1,0,0,0] => [2,1,5,3,4] => [2,3] => 2
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [5] => 1
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [3,2] => 2
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [5] => 1
[1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [5] => 1
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4] => [5] => 1
[1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,5] => [5] => 1
[1,1,0,1,1,0,0,1,0,0] => [2,4,1,5,3] => [3,2] => 2
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,1,3] => [5] => 1
[1,1,0,1,1,1,0,0,0,0] => [2,5,1,3,4] => [5] => 1
[1,1,1,0,0,0,1,0,1,0] => [3,1,2,4,5] => [5] => 1
[1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => [3,2] => 2
[1,1,1,0,0,1,0,0,1,0] => [3,1,4,2,5] => [2,3] => 2
[1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => [2,3] => 2
[1,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4] => [2,3] => 2
[1,1,1,0,1,0,0,0,1,0] => [3,4,1,2,5] => [5] => 1
[1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => [3,2] => 2
[1,1,1,0,1,0,1,0,0,0] => [3,4,5,1,2] => [5] => 1
[1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => [5] => 1
[1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => [5] => 1
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => [3,2] => 2
[1,1,1,1,0,0,1,0,0,0] => [4,1,5,2,3] => [2,3] => 2
[1,1,1,1,0,1,0,0,0,0] => [4,5,1,2,3] => [5] => 1
[1,1,1,1,1,0,0,0,0,0] => [5,1,2,3,4] => [5] => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [6] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [4,2] => 2
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,2,3,5,4,6] => [3,3] => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [3,3] => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,4,5] => [3,3] => 2
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,2,4,3,5,6] => [2,4] => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,2,4,3,6,5] => [2,2,2] => 2
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,2,4,5,3,6] => [2,4] => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [2,4] => 2
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,4,6,3,5] => [2,4] => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,2,5,3,4,6] => [2,4] => 2
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,3,6,4] => [2,2,2] => 2
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,5,6,3,4] => [2,4] => 2
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,3,4,5] => [2,4] => 2
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [1,5] => 2
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [1,3,2] => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [1,2,3] => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,3,2,5,6,4] => [1,2,3] => 2
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,4,5] => [1,2,3] => 2
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [1,5] => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [1,3,2] => 2
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [1,5] => 2
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [1,5] => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,3,4,6,2,5] => [1,5] => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,2,4,6] => [1,5] => 2
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,3,5,2,6,4] => [1,3,2] => 2
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,2,4] => [1,5] => 2
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,2,4,5] => [1,5] => 2
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,4,2,3,5,6] => [1,5] => 2
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,2,3,6,5] => [1,3,2] => 2
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,2,5,3,6] => [1,2,3] => 2
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,6,3] => [1,2,3] => 2
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,2,6,3,5] => [1,2,3] => 2
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,5,2,3,6] => [1,5] => 2
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,2,6,3] => [1,3,2] => 2
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,2,3] => [1,5] => 2
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,2,3,5] => [1,5] => 2
>>> 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 global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Map
to 321-avoiding permutation (Krattenthaler)
Description
Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
Draw the path of semilength $n$ in an $n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.
Map
DEX composition
Description
The DEX composition of a permutation.
Let $\pi$ be a permutation in $\mathfrak S_n$. Let $\bar\pi$ be the word in the ordered set $\bar 1 < \dots < \bar n < 1 \dots < n$ obtained from $\pi$ by replacing every excedance $\pi(i) > i$ by $\overline{\pi(i)}$. Then the DEX set of $\pi$ is the set of indices $1 \leq i < n$ such that $\bar\pi(i) > \bar\pi(i+1)$. Finally, the DEX composition $c_1, \dots, c_k$ of $n$ corresponds to the DEX subset $\{c_1, c_1 + c_2, \dots, c_1 + \dots + c_{k-1}\}$.
The (quasi)symmetric function
$$ \sum_{\pi\in\mathfrak S_{\lambda, j}} F_{DEX(\pi)}, $$
where the sum is over the set of permutations of cycle type $\lambda$ with $j$ excedances, is the Eulerian quasisymmetric function.
Let $\pi$ be a permutation in $\mathfrak S_n$. Let $\bar\pi$ be the word in the ordered set $\bar 1 < \dots < \bar n < 1 \dots < n$ obtained from $\pi$ by replacing every excedance $\pi(i) > i$ by $\overline{\pi(i)}$. Then the DEX set of $\pi$ is the set of indices $1 \leq i < n$ such that $\bar\pi(i) > \bar\pi(i+1)$. Finally, the DEX composition $c_1, \dots, c_k$ of $n$ corresponds to the DEX subset $\{c_1, c_1 + c_2, \dots, c_1 + \dots + c_{k-1}\}$.
The (quasi)symmetric function
$$ \sum_{\pi\in\mathfrak S_{\lambda, j}} F_{DEX(\pi)}, $$
where the sum is over the set of permutations of cycle type $\lambda$ with $j$ excedances, is the Eulerian quasisymmetric function.
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!