Identifier
Values
[1,0] => [1] => [1] => ([(0,1)],2) => 1
[1,0,1,0] => [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,1,0,0] => [2,1] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7) => 1
[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) => 1
[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) => 1
[1,1,0,1,0,0] => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7) => 1
[1,1,1,0,0,0] => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 1
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,4,3] => ([(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
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9) => 1
[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) => 1
[1,0,1,1,1,0,0,0] => [1,4,3,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8) => 1
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3,4] => ([(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
[1,1,0,0,1,1,0,0] => [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) => 1
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [3,2,1,4] => ([(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) => 1
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9) => 1
[1,1,0,1,1,0,0,0] => [2,4,3,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8) => 1
[1,1,1,0,0,0,1,0] => [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8) => 1
[1,1,1,0,0,1,0,0] => [3,2,4,1] => [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) => 1
[1,1,1,0,1,0,0,0] => [3,4,2,1] => [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
[1,1,1,1,0,0,0,0] => [4,3,2,1] => [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) => 1
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8) => 1
[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 1
[1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8) => 1
[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [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) => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => [1,5,3,6,2,4] => ([(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) => 1
[1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,4,3,6,2] => [1,4,6,2,5,3] => ([(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) => 1
[1,1,0,1,0,1,1,1,0,0,0,0] => [2,3,6,5,4,1] => [5,2,3,6,1,4] => ([(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) => 1
[1,1,0,1,1,0,1,0,0,1,0,0] => [2,4,5,3,6,1] => [6,2,4,1,5,3] => ([(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) => 1
[1,1,0,1,1,1,0,0,0,0,1,0] => [2,5,4,3,1,6] => [4,2,5,1,3,6] => ([(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) => 1
[1,1,0,1,1,1,0,0,0,1,0,0] => [2,5,4,3,6,1] => [4,2,6,1,5,3] => ([(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) => 1
[1,1,0,1,1,1,0,0,1,0,0,0] => [2,5,4,6,3,1] => [5,2,6,4,1,3] => ([(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) => 1
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,5,6,4,3,1] => [5,2,4,6,1,3] => ([(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) => 1
[1,1,0,1,1,1,1,0,0,0,0,0] => [2,6,5,4,3,1] => [4,2,5,6,1,3] => ([(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) => 1
[1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,5,6,4,1] => [2,6,3,5,1,4] => ([(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) => 1
[1,1,1,0,0,1,1,1,0,0,0,0] => [3,2,6,5,4,1] => [2,5,3,6,1,4] => ([(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) => 1
[1,1,1,0,1,0,0,1,1,0,0,0] => [3,4,2,6,5,1] => [5,3,1,4,6,2] => ([(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) => 1
[1,1,1,1,0,0,0,1,0,0,1,0] => [4,3,2,5,1,6] => [3,5,1,4,2,6] => ([(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) => 1
[1,1,1,1,0,0,0,1,0,1,0,0] => [4,3,2,5,6,1] => [3,6,1,4,5,2] => ([(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) => 1
[1,1,1,1,0,0,0,1,1,0,0,0] => [4,3,2,6,5,1] => [3,5,1,4,6,2] => ([(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) => 1
[1,1,1,1,0,0,1,0,0,1,0,0] => [4,3,5,2,6,1] => [4,6,3,1,5,2] => ([(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) => 1
[1,1,1,1,0,1,0,0,0,1,0,0] => [4,5,3,2,6,1] => [4,3,6,1,5,2] => ([(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) => 1
[1,1,1,1,1,0,0,0,0,1,0,0] => [5,4,3,2,6,1] => [3,4,6,1,5,2] => ([(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) => 1
[1,1,0,1,1,0,1,0,0,1,1,0,0,0] => [2,4,5,3,7,6,1] => [6,2,4,1,5,7,3] => ([(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) => 1
[1,1,0,1,1,1,0,0,0,1,1,0,0,0] => [2,5,4,3,7,6,1] => [4,2,6,1,5,7,3] => ([(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) => 1
[1,1,0,1,1,1,0,0,1,0,0,1,0,0] => [2,5,4,6,3,7,1] => [5,2,7,4,1,6,3] => ([(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) => 1
[1,1,0,1,1,1,0,0,1,1,0,0,0,0] => [2,5,4,7,6,3,1] => [5,2,7,4,6,1,3] => ([(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) => 1
[1,1,0,1,1,1,0,1,0,0,0,1,0,0] => [2,5,6,4,3,7,1] => [5,2,4,7,1,6,3] => ([(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) => 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0] => [2,5,6,4,7,3,1] => [6,2,4,7,5,1,3] => ([(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) => 1
[1,1,0,1,1,1,1,0,0,0,0,1,0,0] => [2,6,5,4,3,7,1] => [4,2,5,7,1,6,3] => ([(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) => 1
[1,1,1,0,0,1,1,0,1,0,0,1,0,0] => [3,2,5,6,4,7,1] => [2,7,3,5,1,6,4] => ([(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) => 1
[1,1,1,0,0,1,1,1,0,0,0,1,0,0] => [3,2,6,5,4,7,1] => [2,5,3,7,1,6,4] => ([(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) => 1
[1,1,1,0,0,1,1,1,0,0,1,0,0,0] => [3,2,6,5,7,4,1] => [2,6,3,7,5,1,4] => ([(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) => 1
[1,1,1,0,0,1,1,1,0,1,0,0,0,0] => [3,2,6,7,5,4,1] => [2,6,3,5,7,1,4] => ([(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) => 1
[1,1,1,0,1,0,0,1,1,1,0,0,0,0] => [3,4,2,7,6,5,1] => [6,3,1,4,7,2,5] => ([(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) => 1
[1,1,1,1,0,0,0,1,1,0,1,0,0,0] => [4,3,2,6,7,5,1] => [3,7,1,4,6,2,5] => ([(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) => 1
[1,1,1,1,0,0,0,1,1,1,0,0,0,0] => [4,3,2,7,6,5,1] => [3,6,1,4,7,2,5] => ([(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) => 1
[1,1,1,1,0,0,1,0,0,1,1,0,0,0] => [4,3,5,2,7,6,1] => [4,6,3,1,5,7,2] => ([(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) => 1
[1,1,1,1,0,0,1,1,0,0,0,1,0,0] => [4,3,6,5,2,7,1] => [4,7,3,5,1,6,2] => ([(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) => 1
[1,1,1,1,0,1,0,0,1,0,0,1,0,0] => [4,5,3,6,2,7,1] => [5,3,7,4,1,6,2] => ([(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) => 1
[1,1,1,1,1,0,0,0,1,0,0,1,0,0] => [5,4,3,6,2,7,1] => [3,5,7,4,1,6,2] => ([(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) => 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 number of shortest chains of small intervals from the bottom to the top in a lattice.
An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.
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
to 312-avoiding permutation
Description
Sends a Dyck path to the 312-avoiding permutation according to Bandlow-Killpatrick.
This map is defined in [1] and sends the area (St000012The area of a Dyck path.) to the inversion number (St000018The number of inversions of a permutation.).
Map
Corteel
Description
Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.