Identifier
Values
[1,0] => [1,0] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[1,0,1,0] => [1,1,0,0] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 0
[1,1,0,0] => [1,0,1,0] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 0
[1,0,1,0,1,0] => [1,1,0,0,1,0] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,0,1,1,0,0] => [1,1,0,1,0,0] => [4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9) => 1
[1,1,0,0,1,0] => [1,1,1,0,0,0] => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9) => 0
[1,1,0,1,0,0] => [1,0,1,1,0,0] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,1,0,0,0] => [1,0,1,0,1,0] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9) => 0
[1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,0] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [4,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8) => 0
[1,0,1,1,0,0,1,0] => [1,1,0,0,1,0,1,0] => [2,5,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8) => 0
[1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,0] => [2,5,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8) => 0
[1,1,0,1,0,0,1,0] => [1,1,1,0,0,0,1,0] => [2,3,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8) => 0
[1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,0] => [3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0] => [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8) => 0
[1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0] => [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8) => 0
[1,0,1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 0
[1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [4,3,1,6,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9) => 0
[1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9) => 0
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => [2,4,1,5,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9) => 0
[1,0,1,1,0,1,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 0
[1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 0
[1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => [2,5,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9) => 0
[1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => [2,3,5,1,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9) => 0
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 0
[1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 0
[1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => [3,1,6,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9) => 0
[1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9) => 0
[1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => [3,1,4,6,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 0
[1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => [4,1,2,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9) => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => [2,4,1,6,3,7,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 0
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,1,1,0,0,0] => [2,6,1,5,3,7,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 0
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => [6,3,1,5,2,7,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 0
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => [6,4,1,5,2,7,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 0
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => [2,4,1,5,7,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 0
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0,1,0] => [2,5,1,3,7,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 0
[1,1,0,0,1,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,0,1,1,0,0] => [2,6,4,1,3,7,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 0
[1,1,0,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => [2,7,4,1,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 0
[1,1,0,1,0,0,1,0,1,1,0,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => [2,7,4,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 0
[1,1,0,1,0,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => [3,1,5,2,7,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 0
[1,1,0,1,0,1,0,1,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0,1,0] => [5,1,4,2,7,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 0
[1,1,0,1,1,0,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => [3,1,6,2,4,7,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 0
[1,1,1,0,0,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0,1,0] => [3,1,7,5,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 0
[1,1,1,0,1,0,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => [3,1,4,6,2,7,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 0
[] => [] => [1] => ([(0,1)],2) => 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
Description
The number of elements which do not have a complement in the lattice.
A complement of an element $x$ in a lattice is an element $y$ such that the meet of $x$ and $y$ is the bottom element and their join is the top element.
Map
lattice of intervals
Description
The lattice of intervals of a permutation.
An interval of a permutation $\pi$ is a possibly empty interval of values that appear in consecutive positions of $\pi$. The lattice of intervals of $\pi$ has as elements the intervals of $\pi$, ordered by set inclusion.
Map
Elizalde-Deutsch bijection
Description
The Elizalde-Deutsch bijection on Dyck paths.
.Let $n$ be the length of the Dyck path. Consider the steps $1,n,2,n-1,\dots$ of $D$. When considering the $i$-th step its corresponding matching step has not yet been read, let the $i$-th step of the image of $D$ be an up step, otherwise let it be a down step.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.