Identifier
Values
[.,.] => [1] => 1
[.,[.,.]] => [2,1] => 1
[[.,.],.] => [1,2] => 2
[.,[.,[.,.]]] => [3,2,1] => 1
[.,[[.,.],.]] => [2,3,1] => 1
[[.,.],[.,.]] => [1,3,2] => 2
[[.,[.,.]],.] => [2,1,3] => 2
[[[.,.],.],.] => [1,2,3] => 3
[.,[.,[.,[.,.]]]] => [4,3,2,1] => 1
[.,[.,[[.,.],.]]] => [3,4,2,1] => 1
[.,[[.,.],[.,.]]] => [2,4,3,1] => 1
[.,[[.,[.,.]],.]] => [3,2,4,1] => 1
[.,[[[.,.],.],.]] => [2,3,4,1] => 1
[[.,.],[.,[.,.]]] => [1,4,3,2] => 2
[[.,.],[[.,.],.]] => [1,3,4,2] => 2
[[.,[.,.]],[.,.]] => [2,1,4,3] => 2
[[[.,.],.],[.,.]] => [1,2,4,3] => 3
[[.,[.,[.,.]]],.] => [3,2,1,4] => 2
[[.,[[.,.],.]],.] => [2,3,1,4] => 2
[[[.,.],[.,.]],.] => [1,3,2,4] => 3
[[[.,[.,.]],.],.] => [2,1,3,4] => 3
[[[[.,.],.],.],.] => [1,2,3,4] => 4
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => 1
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => 1
[.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => 1
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => 1
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => 1
[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => 1
[.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => 1
[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => 1
[.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => 1
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => 1
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => 1
[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => 1
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => 1
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => 1
[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => 2
[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => 2
[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => 2
[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => 2
[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => 2
[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => 2
[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => 2
[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => 3
[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => 3
[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => 2
[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => 2
[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => 3
[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => 3
[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => 4
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => 2
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => 2
[[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => 2
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => 2
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => 2
[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => 3
[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => 3
[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => 3
[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => 4
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => 3
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => 3
[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => 4
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => 4
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => 5
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => 1
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => 1
[.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => 1
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => 1
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => 1
[.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => 1
[.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => 1
[.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => 1
[.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => 1
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => 1
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => 1
[.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => 1
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => 1
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => 1
[.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => 1
[.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => 1
[.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => 1
[.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => 1
[.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => 1
[.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => 1
[.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => 1
[.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => 1
[.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => 1
[.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => 1
[.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => 1
[.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => 1
[.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => 1
[.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => 1
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => 1
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => 1
[.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => 1
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => 1
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => 1
[.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => 1
[.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => 1
[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => 1
[.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => 1
>>> Load all 196 entries. <<<
[.,[[[.,[.,[.,.]]],.],.]] => [4,3,2,5,6,1] => 1
[.,[[[.,[[.,.],.]],.],.]] => [3,4,2,5,6,1] => 1
[.,[[[[.,.],[.,.]],.],.]] => [2,4,3,5,6,1] => 1
[.,[[[[.,[.,.]],.],.],.]] => [3,2,4,5,6,1] => 1
[.,[[[[[.,.],.],.],.],.]] => [2,3,4,5,6,1] => 1
[[.,.],[.,[.,[.,[.,.]]]]] => [1,6,5,4,3,2] => 2
[[.,.],[.,[.,[[.,.],.]]]] => [1,5,6,4,3,2] => 2
[[.,.],[.,[[.,.],[.,.]]]] => [1,4,6,5,3,2] => 2
[[.,.],[.,[[.,[.,.]],.]]] => [1,5,4,6,3,2] => 2
[[.,.],[.,[[[.,.],.],.]]] => [1,4,5,6,3,2] => 2
[[.,.],[[.,.],[.,[.,.]]]] => [1,3,6,5,4,2] => 2
[[.,.],[[.,.],[[.,.],.]]] => [1,3,5,6,4,2] => 2
[[.,.],[[.,[.,.]],[.,.]]] => [1,4,3,6,5,2] => 2
[[.,.],[[[.,.],.],[.,.]]] => [1,3,4,6,5,2] => 2
[[.,.],[[.,[.,[.,.]]],.]] => [1,5,4,3,6,2] => 2
[[.,.],[[.,[[.,.],.]],.]] => [1,4,5,3,6,2] => 2
[[.,.],[[[.,.],[.,.]],.]] => [1,3,5,4,6,2] => 2
[[.,.],[[[.,[.,.]],.],.]] => [1,4,3,5,6,2] => 2
[[.,.],[[[[.,.],.],.],.]] => [1,3,4,5,6,2] => 2
[[.,[.,.]],[.,[.,[.,.]]]] => [2,1,6,5,4,3] => 2
[[.,[.,.]],[.,[[.,.],.]]] => [2,1,5,6,4,3] => 2
[[.,[.,.]],[[.,.],[.,.]]] => [2,1,4,6,5,3] => 2
[[.,[.,.]],[[.,[.,.]],.]] => [2,1,5,4,6,3] => 2
[[.,[.,.]],[[[.,.],.],.]] => [2,1,4,5,6,3] => 2
[[[.,.],.],[.,[.,[.,.]]]] => [1,2,6,5,4,3] => 3
[[[.,.],.],[.,[[.,.],.]]] => [1,2,5,6,4,3] => 3
[[[.,.],.],[[.,.],[.,.]]] => [1,2,4,6,5,3] => 3
[[[.,.],.],[[.,[.,.]],.]] => [1,2,5,4,6,3] => 3
[[[.,.],.],[[[.,.],.],.]] => [1,2,4,5,6,3] => 3
[[.,[.,[.,.]]],[.,[.,.]]] => [3,2,1,6,5,4] => 2
[[.,[.,[.,.]]],[[.,.],.]] => [3,2,1,5,6,4] => 2
[[.,[[.,.],.]],[.,[.,.]]] => [2,3,1,6,5,4] => 2
[[.,[[.,.],.]],[[.,.],.]] => [2,3,1,5,6,4] => 2
[[[.,.],[.,.]],[.,[.,.]]] => [1,3,2,6,5,4] => 3
[[[.,.],[.,.]],[[.,.],.]] => [1,3,2,5,6,4] => 3
[[[.,[.,.]],.],[.,[.,.]]] => [2,1,3,6,5,4] => 3
[[[.,[.,.]],.],[[.,.],.]] => [2,1,3,5,6,4] => 3
[[[[.,.],.],.],[.,[.,.]]] => [1,2,3,6,5,4] => 4
[[[[.,.],.],.],[[.,.],.]] => [1,2,3,5,6,4] => 4
[[.,[.,[.,[.,.]]]],[.,.]] => [4,3,2,1,6,5] => 2
[[.,[.,[[.,.],.]]],[.,.]] => [3,4,2,1,6,5] => 2
[[.,[[.,.],[.,.]]],[.,.]] => [2,4,3,1,6,5] => 2
[[.,[[.,[.,.]],.]],[.,.]] => [3,2,4,1,6,5] => 2
[[.,[[[.,.],.],.]],[.,.]] => [2,3,4,1,6,5] => 2
[[[.,.],[.,[.,.]]],[.,.]] => [1,4,3,2,6,5] => 3
[[[.,.],[[.,.],.]],[.,.]] => [1,3,4,2,6,5] => 3
[[[.,[.,.]],[.,.]],[.,.]] => [2,1,4,3,6,5] => 3
[[[[.,.],.],[.,.]],[.,.]] => [1,2,4,3,6,5] => 4
[[[.,[.,[.,.]]],.],[.,.]] => [3,2,1,4,6,5] => 3
[[[.,[[.,.],.]],.],[.,.]] => [2,3,1,4,6,5] => 3
[[[[.,.],[.,.]],.],[.,.]] => [1,3,2,4,6,5] => 4
[[[[.,[.,.]],.],.],[.,.]] => [2,1,3,4,6,5] => 4
[[[[[.,.],.],.],.],[.,.]] => [1,2,3,4,6,5] => 5
[[.,[.,[.,[.,[.,.]]]]],.] => [5,4,3,2,1,6] => 2
[[.,[.,[.,[[.,.],.]]]],.] => [4,5,3,2,1,6] => 2
[[.,[.,[[.,.],[.,.]]]],.] => [3,5,4,2,1,6] => 2
[[.,[.,[[.,[.,.]],.]]],.] => [4,3,5,2,1,6] => 2
[[.,[.,[[[.,.],.],.]]],.] => [3,4,5,2,1,6] => 2
[[.,[[.,.],[.,[.,.]]]],.] => [2,5,4,3,1,6] => 2
[[.,[[.,.],[[.,.],.]]],.] => [2,4,5,3,1,6] => 2
[[.,[[.,[.,.]],[.,.]]],.] => [3,2,5,4,1,6] => 2
[[.,[[[.,.],.],[.,.]]],.] => [2,3,5,4,1,6] => 2
[[.,[[.,[.,[.,.]]],.]],.] => [4,3,2,5,1,6] => 2
[[.,[[.,[[.,.],.]],.]],.] => [3,4,2,5,1,6] => 2
[[.,[[[.,.],[.,.]],.]],.] => [2,4,3,5,1,6] => 2
[[.,[[[.,[.,.]],.],.]],.] => [3,2,4,5,1,6] => 2
[[.,[[[[.,.],.],.],.]],.] => [2,3,4,5,1,6] => 2
[[[.,.],[.,[.,[.,.]]]],.] => [1,5,4,3,2,6] => 3
[[[.,.],[.,[[.,.],.]]],.] => [1,4,5,3,2,6] => 3
[[[.,.],[[.,.],[.,.]]],.] => [1,3,5,4,2,6] => 3
[[[.,.],[[.,[.,.]],.]],.] => [1,4,3,5,2,6] => 3
[[[.,.],[[[.,.],.],.]],.] => [1,3,4,5,2,6] => 3
[[[.,[.,.]],[.,[.,.]]],.] => [2,1,5,4,3,6] => 3
[[[.,[.,.]],[[.,.],.]],.] => [2,1,4,5,3,6] => 3
[[[[.,.],.],[.,[.,.]]],.] => [1,2,5,4,3,6] => 4
[[[[.,.],.],[[.,.],.]],.] => [1,2,4,5,3,6] => 4
[[[.,[.,[.,.]]],[.,.]],.] => [3,2,1,5,4,6] => 3
[[[.,[[.,.],.]],[.,.]],.] => [2,3,1,5,4,6] => 3
[[[[.,.],[.,.]],[.,.]],.] => [1,3,2,5,4,6] => 4
[[[[.,[.,.]],.],[.,.]],.] => [2,1,3,5,4,6] => 4
[[[[[.,.],.],.],[.,.]],.] => [1,2,3,5,4,6] => 5
[[[.,[.,[.,[.,.]]]],.],.] => [4,3,2,1,5,6] => 3
[[[.,[.,[[.,.],.]]],.],.] => [3,4,2,1,5,6] => 3
[[[.,[[.,.],[.,.]]],.],.] => [2,4,3,1,5,6] => 3
[[[.,[[.,[.,.]],.]],.],.] => [3,2,4,1,5,6] => 3
[[[.,[[[.,.],.],.]],.],.] => [2,3,4,1,5,6] => 3
[[[[.,.],[.,[.,.]]],.],.] => [1,4,3,2,5,6] => 4
[[[[.,.],[[.,.],.]],.],.] => [1,3,4,2,5,6] => 4
[[[[.,[.,.]],[.,.]],.],.] => [2,1,4,3,5,6] => 4
[[[[[.,.],.],[.,.]],.],.] => [1,2,4,3,5,6] => 5
[[[[.,[.,[.,.]]],.],.],.] => [3,2,1,4,5,6] => 4
[[[[.,[[.,.],.]],.],.],.] => [2,3,1,4,5,6] => 4
[[[[[.,.],[.,.]],.],.],.] => [1,3,2,4,5,6] => 5
[[[[[.,[.,.]],.],.],.],.] => [2,1,3,4,5,6] => 5
[[[[[[.,.],.],.],.],.],.] => [1,2,3,4,5,6] => 6
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
click to show known generating functions       
Description
The decomposition (or block) number of a permutation.
For $\pi \in \mathcal{S}_n$, this is given by
$$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$$
This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum.
This is one plus St000234The number of global ascents of a permutation..
Map
to 312-avoiding permutation
Description
Return a 312-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 minimal element of this Sylvester class.