Identifier
Values
[1] => [1] => [1] => ([],1) => 1
[1,2] => [1,2] => [1,2] => ([(0,1)],2) => 1
[2,1] => [2,1] => [1,2] => ([(0,1)],2) => 1
[1,2,3] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 1
[1,3,2] => [3,1,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[2,1,3] => [2,1,3] => [1,2,3] => ([(0,2),(2,1)],3) => 1
[2,3,1] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3) => 1
[3,1,2] => [1,3,2] => [1,2,3] => ([(0,2),(2,1)],3) => 1
[3,2,1] => [3,2,1] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[1,2,4,3] => [4,1,2,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[1,3,2,4] => [3,1,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 2
[1,3,4,2] => [3,4,1,2] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 2
[1,4,2,3] => [1,4,2,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[1,4,3,2] => [4,3,1,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 2
[2,1,3,4] => [2,1,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[2,1,4,3] => [2,4,1,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[2,3,1,4] => [2,3,1,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[2,3,4,1] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[2,4,1,3] => [4,2,1,3] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[2,4,3,1] => [4,2,3,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 2
[3,1,2,4] => [1,3,2,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[3,1,4,2] => [1,3,4,2] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[3,2,1,4] => [3,2,1,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 2
[3,2,4,1] => [3,2,4,1] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 2
[3,4,1,2] => [3,1,4,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 2
[3,4,2,1] => [3,4,2,1] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 2
[4,1,2,3] => [1,2,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[4,1,3,2] => [4,1,3,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 2
[4,2,1,3] => [2,1,4,3] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 1
[4,2,3,1] => [2,4,3,1] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[4,3,1,2] => [1,4,3,2] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 1
[4,3,2,1] => [4,3,2,1] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 2
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,1,2,4,5] => [1,3,2,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,1,4,2,5] => [1,3,4,2,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[3,1,4,5,2] => [1,3,4,5,2] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,1,2,3,5] => [1,2,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,1,2,5,3] => [1,2,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,2,1,3,5] => [2,1,4,3,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[4,2,1,5,3] => [2,1,4,5,3] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,1,2,3,4] => [1,2,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,2,1,3,4] => [2,1,3,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,2,3,1,4] => [2,3,1,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[5,3,1,2,4] => [1,3,2,5,4] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[2,1,3,4,5,6] => [2,1,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[2,3,1,4,5,6] => [2,3,1,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[2,3,4,1,5,6] => [2,3,4,1,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[2,3,4,5,1,6] => [2,3,4,5,1,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[2,3,4,5,6,1] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[3,1,2,4,5,6] => [1,3,2,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[3,1,4,2,5,6] => [1,3,4,2,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[3,1,4,5,2,6] => [1,3,4,5,2,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[3,1,4,5,6,2] => [1,3,4,5,6,2] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[4,1,2,3,5,6] => [1,2,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[4,1,2,5,3,6] => [1,2,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[4,1,2,5,6,3] => [1,2,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[4,2,1,3,5,6] => [2,1,4,3,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[4,2,1,5,3,6] => [2,1,4,5,3,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[4,2,1,5,6,3] => [2,1,4,5,6,3] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[5,1,2,3,4,6] => [1,2,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[5,1,2,3,6,4] => [1,2,3,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[5,2,1,3,4,6] => [2,1,3,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[5,2,1,3,6,4] => [2,1,3,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[5,2,3,1,4,6] => [2,3,1,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[5,2,3,1,6,4] => [2,3,1,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[5,3,1,2,4,6] => [1,3,2,5,4,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[5,3,1,2,6,4] => [1,3,2,5,6,4] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[6,1,2,3,4,5] => [1,2,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[6,2,1,3,4,5] => [2,1,3,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[6,2,3,1,4,5] => [2,3,1,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[6,2,3,4,1,5] => [2,3,4,1,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[6,3,1,2,4,5] => [1,3,2,4,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[6,3,1,4,2,5] => [1,3,4,2,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[6,4,1,2,3,5] => [1,2,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[6,4,2,1,3,5] => [2,1,4,3,6,5] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
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 leading coefficient of the Poincare polynomial of the poset cone.
For a poset $P$ on $\{1,\dots,n\}$, let $\mathcal K_P = \{\vec x\in\mathbb R^n| x_i < x_j \text{ for } i < _P j\}$. Furthermore let $\mathcal L(\mathcal A)$ be the intersection lattice of the braid arrangement $A_{n-1}$ and let $\mathcal L^{int} = \{ X \in \mathcal L(\mathcal A) | X \cap \mathcal K_P \neq \emptyset \}$.
Then the Poincare polynomial of the poset cone is $Poin(t) = \sum_{X\in\mathcal L^{int}} |\mu(0, X)| t^{codim X}$.
This statistic records its leading coefficient.
Map
pattern poset
Description
The pattern poset of a permutation.
This is the poset of all non-empty permutations that occur in the given permutation as a pattern, ordered by pattern containment.
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.
Map
inverse Foata bijection
Description
The inverse of Foata's bijection.
See Mp00067Foata bijection.