Identifier
Values
[.,.] => [1] => ([],1) => ([],1) => 1
[[.,.],.] => [1,2] => ([(0,1)],2) => ([(0,1)],2) => 2
[[.,[.,.]],.] => [2,1,3] => ([(0,2),(1,2)],3) => ([(0,2),(1,2)],3) => 4
[[[.,.],.],.] => [1,2,3] => ([(0,2),(2,1)],3) => ([(0,2),(1,2)],3) => 4
[[.,[[.,.],.]],.] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(1,2),(2,3)],4) => 8
[[[.,.],[.,.]],.] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(1,2),(2,3)],4) => 8
[[[[.,.],.],.],.] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => ([(0,3),(1,2),(2,3)],4) => 8
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 16
[[[.,.],[[.,.],.]],.] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 16
[[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 16
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 16
[[.,[[[[.,.],.],.],.]],.] => [2,3,4,5,1,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 32
[[[.,.],[[[.,.],.],.]],.] => [3,4,5,1,2,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 32
[[[[.,.],.],[[.,.],.]],.] => [4,5,1,2,3,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 32
[[[[[.,.],.],.],[.,.]],.] => [5,1,2,3,4,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 32
[[[[[[.,.],.],.],.],.],.] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 32
[[.,[[[[[.,.],.],.],.],.]],.] => [2,3,4,5,6,1,7] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7) => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 64
[[[.,.],[[[[.,.],.],.],.]],.] => [3,4,5,6,1,2,7] => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7) => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 64
[[[[.,.],.],[[[.,.],.],.]],.] => [4,5,6,1,2,3,7] => ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7) => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 64
[[[[[.,.],.],.],[[.,.],.]],.] => [5,6,1,2,3,4,7] => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7) => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 64
[[[[[[.,.],.],.],.],[.,.]],.] => [6,1,2,3,4,5,7] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7) => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 64
[[[[[[[.,.],.],.],.],.],.],.] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 64
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 pebbling number of a connected graph.
Map
permutation poset
Description
Sends a permutation to its permutation poset.
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
Map
to graph
Description
Returns the Hasse diagram of the poset as an undirected graph.
Map
to 132-avoiding permutation
Description
Return a 132-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the maximal element of the Sylvester class.