Identifier
Values
[1,0,1,0] => [2,1] => ([(0,1)],2) => 0
[1,1,0,0] => [1,2] => ([(0,1)],2) => 0
[1,0,1,0,1,0] => [3,2,1] => ([(0,2),(2,1)],3) => 0
[1,0,1,1,0,0] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,1,0,0,1,0] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,1,0,1,0,0] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,1,1,0,0,0] => [1,2,3] => ([(0,2),(2,1)],3) => 0
[1,0,1,0,1,0,1,0] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4) => 0
[1,0,1,0,1,1,0,0] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[1,0,1,1,0,0,1,0] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 7
[1,0,1,1,0,1,0,0] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 7
[1,0,1,1,1,0,0,0] => [2,3,4,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[1,1,0,0,1,0,1,0] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[1,1,0,0,1,1,0,0] => [3,4,1,2] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 2
[1,1,0,1,0,0,1,0] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 7
[1,1,0,1,0,1,0,0] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[1,1,0,1,1,0,0,0] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 7
[1,1,1,0,0,0,1,0] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[1,1,1,0,0,1,0,0] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7) => 7
[1,1,1,0,1,0,0,0] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 2
[1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 0
[1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
[1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 0
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
click to show known generating functions       
Description
The balance constant multiplied with the number of linear extensions of a poset.
A pair of elements $x,y$ of a poset is $\alpha$-balanced if the proportion $P(x,y)$ of linear extensions where $x$ comes before $y$ is between $\alpha$ and $1-\alpha$. The balance constant of a poset is $\max\min(P(x,y), P(y,x)).$
Kislitsyn [1] conjectured that every poset which is not a chain is $1/3$-balanced. Brightwell, Felsner and Trotter [2] show that it is at least $(1-\sqrt 5)/10$-balanced.
Olson and Sagan [3] exhibit various posets that are $1/2$-balanced.
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
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].