Processing math: 100%

Identifier
Values
[1,0] => [1] => [1] => ([(0,1)],2) => 0
[1,0,1,0] => [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[1,1,0,0] => [2,1] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 0
[1,0,1,0,1,0] => [1,2,3] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 0
[1,0,1,1,0,0] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 0
[1,1,0,0,1,0] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 0
[1,1,0,1,0,0] => [2,3,1] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 0
[1,1,1,0,0,0] => [3,1,2] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 0
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,4,3,2] => ([(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
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,4,3,2] => ([(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
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,4,3,2] => ([(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
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,4,3,2] => ([(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
[1,0,1,1,1,0,0,0] => [1,4,2,3] => [1,4,3,2] => ([(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
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8) => 0
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8) => 0
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8) => 0
[1,1,0,1,1,0,0,0] => [2,4,1,3] => [2,4,1,3] => ([(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] => [3,1,2,4] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,1,0,0,1,0,0] => [3,1,4,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 0
[1,1,1,0,1,0,0,0] => [3,4,1,2] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8) => 0
[1,1,1,1,0,0,0,0] => [4,1,2,3] => [4,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8) => 0
[1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [2,5,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8) => 0
[1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [2,5,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8) => 0
[1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [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
[1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4] => [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
[1,1,0,1,1,0,0,0,1,0] => [2,4,1,3,5] => [2,5,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8) => 0
[1,1,0,1,1,0,0,1,0,0] => [2,4,1,5,3] => [2,5,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8) => 0
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,1,3] => [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
[1,1,0,1,1,1,0,0,0,0] => [2,5,1,3,4] => [2,5,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8) => 0
[1,1,1,0,0,0,1,0,1,0] => [3,1,2,4,5] => [3,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8) => 0
[1,1,1,0,0,0,1,1,0,0] => [3,1,2,5,4] => [3,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8) => 0
[1,1,1,0,0,1,0,0,1,0] => [3,1,4,2,5] => [3,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8) => 0
[1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => [3,1,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8) => 0
[1,1,1,0,0,1,1,0,0,0] => [3,1,5,2,4] => [3,1,5,4,2] => ([(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] => [3,4,1,2,5] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,1,1,0,1,0,0,1,0,0] => [3,4,1,5,2] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,1,1,0,1,1,0,0,0,0] => [3,5,1,2,4] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 0
[1,1,1,1,0,0,0,0,1,0] => [4,1,2,3,5] => [4,1,5,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8) => 0
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,5,3] => [4,1,5,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8) => 0
[1,1,1,1,0,0,1,0,0,0] => [4,1,5,2,3] => [4,1,5,3,2] => ([(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] => [3,4,1,2,5,6] => [3,6,1,5,4,2] => ([(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
[1,1,1,0,1,0,0,0,1,1,0,0] => [3,4,1,2,6,5] => [3,6,1,5,4,2] => ([(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] => [3,4,1,5,2,6] => [3,6,1,5,4,2] => ([(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
[1,1,1,0,1,0,0,1,0,1,0,0] => [3,4,1,5,6,2] => [3,6,1,5,4,2] => ([(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
[1,1,1,0,1,0,0,1,1,0,0,0] => [3,4,1,6,2,5] => [3,6,1,5,4,2] => ([(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
[1,1,1,0,1,0,1,0,0,0,1,0] => [3,4,5,1,2,6] => [3,6,5,1,4,2] => ([(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
[1,1,1,0,1,0,1,0,0,1,0,0] => [3,4,5,1,6,2] => [3,6,5,1,4,2] => ([(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
[1,1,1,0,1,0,1,1,0,0,0,0] => [3,4,6,1,2,5] => [3,6,5,1,4,2] => ([(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
[1,1,1,0,1,1,0,0,0,0,1,0] => [3,5,1,2,4,6] => [3,6,1,5,4,2] => ([(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
[1,1,1,0,1,1,0,0,0,1,0,0] => [3,5,1,2,6,4] => [3,6,1,5,4,2] => ([(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
[1,1,1,0,1,1,0,0,1,0,0,0] => [3,5,1,6,2,4] => [3,6,1,5,4,2] => ([(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
[1,1,1,0,1,1,0,1,0,0,0,0] => [3,5,6,1,2,4] => [3,6,5,1,4,2] => ([(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
[1,1,1,0,1,1,1,0,0,0,0,0] => [3,6,1,2,4,5] => [3,6,1,5,4,2] => ([(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
[1,1,1,1,0,1,0,0,0,0,1,0] => [4,5,1,2,3,6] => [4,6,1,5,3,2] => ([(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
[1,1,1,1,0,1,0,0,0,1,0,0] => [4,5,1,2,6,3] => [4,6,1,5,3,2] => ([(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
[1,1,1,1,0,1,0,0,1,0,0,0] => [4,5,1,6,2,3] => [4,6,1,5,3,2] => ([(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
[1,1,1,1,0,1,1,0,0,0,0,0] => [4,6,1,2,3,5] => [4,6,1,5,3,2] => ([(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
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 π is a possibly empty interval of values that appear in consecutive positions of π. The lattice of intervals of π has as elements the intervals of π, ordered by set inclusion.
Map
Simion-Schmidt map
Description
The Simion-Schmidt map sends any permutation to a 123-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between 132-avoiding permutations and 123-avoiding permutations, see [1, Proposition 19].
Map
to 321-avoiding permutation (Krattenthaler)
Description
Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength n in an n×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.