Identifier
Values
[.,.] => [1,0] => [1] => ([(0,1)],2) => 1
[.,[.,.]] => [1,1,0,0] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[[.,.],.] => [1,0,1,0] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 1
[.,[.,[.,.]]] => [1,1,1,0,0,0] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7) => 1
[.,[[.,.],.]] => [1,1,0,1,0,0] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 1
[[.,.],[.,.]] => [1,0,1,1,0,0] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 1
[[.,[.,.]],.] => [1,1,0,0,1,0] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 1
[[[.,.],.],.] => [1,0,1,0,1,0] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 1
[.,[.,[[.,.],.]]] => [1,1,1,0,1,0,0,0] => [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
[.,[[.,.],[.,.]]] => [1,1,0,1,1,0,0,0] => [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
[.,[[.,[.,.]],.]] => [1,1,1,0,0,1,0,0] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8) => 1
[.,[[[.,.],.],.]] => [1,1,0,1,0,1,0,0] => [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
[[.,.],[.,[.,.]]] => [1,0,1,1,1,0,0,0] => [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) => 1
[[.,.],[[.,.],.]] => [1,0,1,1,0,1,0,0] => [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
[[.,[.,.]],[.,.]] => [1,1,0,0,1,1,0,0] => [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
[[[.,.],.],[.,.]] => [1,0,1,0,1,1,0,0] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 1
[[.,[.,[.,.]]],.] => [1,1,1,0,0,0,1,0] => [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) => 1
[[.,[[.,.],.]],.] => [1,1,0,1,0,0,1,0] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8) => 1
[[[.,.],[.,.]],.] => [1,0,1,1,0,0,1,0] => [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
[[[.,[.,.]],.],.] => [1,1,0,0,1,0,1,0] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 1
[[[[.,.],.],.],.] => [1,0,1,0,1,0,1,0] => [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
[.,[[[.,.],.],[.,.]]] => [1,1,0,1,0,1,1,0,0,0] => [1,3,5,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8) => 1
[.,[[[.,[.,.]],.],.]] => [1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8) => 1
[[.,.],[[.,.],[.,.]]] => [1,0,1,1,0,1,1,0,0,0] => [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) => 1
[[.,[.,.]],[[.,.],.]] => [1,1,0,0,1,1,0,1,0,0] => [3,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8) => 1
[[[.,.],.],[.,[.,.]]] => [1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8) => 1
[[[.,.],.],[[.,.],.]] => [1,0,1,0,1,1,0,1,0,0] => [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) => 1
[[.,[[.,.],.]],[.,.]] => [1,1,0,1,0,0,1,1,0,0] => [3,5,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8) => 1
[[[.,.],[.,.]],[.,.]] => [1,0,1,1,0,0,1,1,0,0] => [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) => 1
[[[[.,.],.],.],[.,.]] => [1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8) => 1
[[.,[[.,[.,.]],.]],.] => [1,1,1,0,0,1,0,0,1,0] => [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) => 1
[[[.,[.,.]],[.,.]],.] => [1,1,0,0,1,1,0,0,1,0] => [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) => 1
[[[.,[.,[.,.]]],.],.] => [1,1,1,0,0,0,1,0,1,0] => [4,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8) => 1
[[[.,[[.,.],.]],.],.] => [1,1,0,1,0,0,1,0,1,0] => [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) => 1
[[[[.,[.,.]],.],.],.] => [1,1,0,0,1,0,1,0,1,0] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8) => 1
[.,[[[.,.],.],[[.,.],.]]] => [1,1,0,1,0,1,1,0,1,0,0,0] => [1,3,5,2,6,4] => ([(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
[.,[[[.,[[.,.],.]],.],.]] => [1,1,1,0,1,0,0,1,0,1,0,0] => [1,4,2,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9) => 1
[[.,.],[[.,.],[[.,.],.]]] => [1,0,1,1,0,1,1,0,1,0,0,0] => [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) => 1
[[[.,.],.],[.,[[.,.],.]]] => [1,0,1,0,1,1,1,0,1,0,0,0] => [2,4,5,1,6,3] => ([(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) => 1
[[[.,.],.],[[.,[.,.]],.]] => [1,0,1,0,1,1,1,0,0,1,0,0] => [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) => 1
[[[.,.],.],[[[.,.],.],.]] => [1,0,1,0,1,1,0,1,0,1,0,0] => [2,4,1,5,3,6] => ([(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
[[.,[[.,.],.]],[[.,.],.]] => [1,1,0,1,0,0,1,1,0,1,0,0] => [3,5,1,6,2,4] => ([(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
[[[.,.],[.,.]],[[.,.],.]] => [1,0,1,1,0,0,1,1,0,1,0,0] => [2,5,1,6,3,4] => ([(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
[[[.,[.,.]],.],[[.,.],.]] => [1,1,0,0,1,0,1,1,0,1,0,0] => [3,4,1,6,2,5] => ([(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) => 1
[[[[.,.],.],.],[.,[.,.]]] => [1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5] => ([(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) => 1
[[[[.,.],.],.],[[.,.],.]] => [1,0,1,0,1,0,1,1,0,1,0,0] => [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) => 1
[[[.,[[.,.],.]],.],[.,.]] => [1,1,0,1,0,0,1,0,1,1,0,0] => [3,5,1,2,6,4] => ([(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) => 1
[[[[.,.],[.,.]],.],[.,.]] => [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) => 1
[[.,[[.,[[.,.],.]],.]],.] => [1,1,1,0,1,0,0,1,0,0,1,0] => [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) => 1
[[[.,[[.,.],.]],[.,.]],.] => [1,1,0,1,0,0,1,1,0,0,1,0] => [3,1,5,6,2,4] => ([(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) => 1
[[[[.,[.,.]],.],[.,.]],.] => [1,1,0,0,1,0,1,1,0,0,1,0] => [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) => 1
[[[.,[.,[[.,.],.]]],.],.] => [1,1,1,0,1,0,0,0,1,0,1,0] => [4,1,6,2,3,5] => ([(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) => 1
[[[.,[[.,.],[.,.]]],.],.] => [1,1,0,1,1,0,0,0,1,0,1,0] => [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) => 1
[[[.,[[[.,.],.],.]],.],.] => [1,1,0,1,0,1,0,0,1,0,1,0] => [3,1,5,2,4,6] => ([(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
[[[[.,[.,[.,.]]],.],.],.] => [1,1,1,0,0,0,1,0,1,0,1,0] => [4,1,5,2,6,3] => ([(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) => 1
[[[[.,[[.,.],.]],.],.],.] => [1,1,0,1,0,0,1,0,1,0,1,0] => [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) => 1
[[[.,.],.],[[[.,.],[.,.]],.]] => [1,0,1,0,1,1,0,1,1,0,0,1,0,0] => [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) => 1
[[.,[[.,.],.]],[[[.,.],.],.]] => [1,1,0,1,0,0,1,1,0,1,0,1,0,0] => [3,5,1,6,2,7,4] => ([(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) => 1
[[[.,.],[.,.]],[[[.,.],.],.]] => [1,0,1,1,0,0,1,1,0,1,0,1,0,0] => [2,5,1,6,3,7,4] => ([(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) => 1
[[[[.,.],.],.],[.,[[.,.],.]]] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [2,4,6,1,7,3,5] => ([(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) => 1
[[[[.,.],.],.],[[.,.],[.,.]]] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0] => [2,4,6,1,3,7,5] => ([(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) => 1
[[[[.,.],.],.],[[[.,.],.],.]] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [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) => 1
[[.,[[[.,.],.],.]],[[.,.],.]] => [1,1,0,1,0,1,0,0,1,1,0,1,0,0] => [3,5,1,7,2,4,6] => ([(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) => 1
[[[[.,.],[[.,.],.]],.],[.,.]] => [1,0,1,1,0,1,0,0,1,0,1,1,0,0] => [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) => 1
[[[[.,[.,.]],.],[[.,.],.]],.] => [1,1,0,0,1,0,1,1,0,1,0,0,1,0] => [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) => 1
[[[.,[[[.,.],.],.]],[.,.]],.] => [1,1,0,1,0,1,0,0,1,1,0,0,1,0] => [3,1,5,7,2,4,6] => ([(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) => 1
[[[.,[[[.,.],[.,.]],.]],.],.] => [1,1,0,1,1,0,0,1,0,0,1,0,1,0] => [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) => 1
[[[[.,[.,[[.,.],.]]],.],.],.] => [1,1,1,0,1,0,0,0,1,0,1,0,1,0] => [4,1,6,2,7,3,5] => ([(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) => 1
[[[[.,[[.,[.,.]],.]],.],.],.] => [1,1,1,0,0,1,0,0,1,0,1,0,1,0] => [4,1,5,2,7,3,6] => ([(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) => 1
[[[[.,[[[.,.],.],.]],.],.],.] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0] => [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) => 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
to Tamari-corresponding Dyck path
Description
Return the Dyck path associated with a binary tree in consistency with the Tamari order on Dyck words and binary trees.
The bijection is defined recursively as follows:
  • a leaf is associated with an empty Dyck path,
  • a tree with children $l,r$ is associated with the Dyck word $T(l) 1 T(r) 0$ where $T(l)$ and $T(r)$ are the images of this bijection to $l$ and $r$.
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 321-avoiding permutation
Description
Sends a Dyck path to a 321-avoiding permutation.
This bijection defined in [3, pp. 60] and in [2, Section 3.1].
It is shown in [1] that it sends the number of centered tunnels to the number of fixed points, the number of right tunnels to the number of exceedences, and the semilength plus the height of the middle point to 2 times the length of the longest increasing subsequence.