Identifier
Values
=>
Cc0010;cc-rep-0
[.,.]=>[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 [.,[[[.,[.,[.,.]]],.],.]]=>[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.