Identifier
-
Mp00017:
Binary trees
—to 312-avoiding permutation⟶
Permutations
St000056: Permutations ⟶ ℤ (values match St000234The number of global ascents of a permutation.)
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
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searching the database for the individual values of this statistic
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searching the database for statistics with the same generating function
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..
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.
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.
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