Identifier
-
Mp00017:
Binary trees
—to 312-avoiding permutation⟶
Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000034: Permutations ⟶ ℤ
Values
[.,.] => [1] => [1] => [1] => 0
[.,[.,.]] => [2,1] => [2,1] => [2,1] => 0
[[.,.],.] => [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]] => [3,2,1] => [3,2,1] => [2,3,1] => 0
[.,[[.,.],.]] => [2,3,1] => [1,3,2] => [1,3,2] => 0
[[.,.],[.,.]] => [1,3,2] => [1,3,2] => [1,3,2] => 0
[[.,[.,.]],.] => [2,1,3] => [2,1,3] => [2,1,3] => 0
[[[.,.],.],.] => [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]] => [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 1
[.,[.,[[.,.],.]]] => [3,4,2,1] => [1,4,3,2] => [1,3,4,2] => 0
[.,[[.,.],[.,.]]] => [2,4,3,1] => [1,4,3,2] => [1,3,4,2] => 0
[.,[[.,[.,.]],.]] => [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 0
[.,[[[.,.],.],.]] => [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 0
[[.,.],[.,[.,.]]] => [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 0
[[.,.],[[.,.],.]] => [1,3,4,2] => [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,.]],[.,.]] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[[[.,.],.],[.,.]] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,[.,.]]],.] => [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 0
[[.,[[.,.],.]],.] => [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,.],[.,.]],.] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,[.,.]],.],.] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [5,4,3,2,1] => [3,4,2,5,1] => 2
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [3,2,1,5,4] => [2,3,1,5,4] => 0
[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => [1,5,4,3,2] => [1,4,3,5,2] => 1
[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => 0
[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 0
[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [4,3,2,1,5] => [3,2,4,1,5] => 1
[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [1,4,3,2,5] => [1,3,4,2,5] => 0
[[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => [1,4,3,2,5] => [1,3,4,2,5] => 0
[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => 0
[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 0
[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 0
[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 0
[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 0
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [4,3,5,2,6,1] => 3
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [1,6,5,4,3,2] => [1,4,5,3,6,2] => 2
[.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => [1,6,5,4,3,2] => [1,4,5,3,6,2] => 2
[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [2,1,6,5,4,3] => [2,1,5,4,6,3] => 1
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [1,2,6,5,4,3] => [1,2,5,4,6,3] => 1
[.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => [1,6,5,4,3,2] => [1,4,5,3,6,2] => 2
[.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => [1,2,6,5,4,3] => [1,2,5,4,6,3] => 1
[.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => [2,1,6,5,4,3] => [2,1,5,4,6,3] => 1
[.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => [1,2,6,5,4,3] => [1,2,5,4,6,3] => 1
[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [3,2,1,6,5,4] => [2,3,1,5,6,4] => 0
[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [1,3,2,6,5,4] => [1,3,2,5,6,4] => 0
[.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => [1,3,2,6,5,4] => [1,3,2,5,6,4] => 0
[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [2,1,3,6,5,4] => [2,1,3,5,6,4] => 0
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [1,2,3,6,5,4] => [1,2,3,5,6,4] => 0
[.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => [1,6,5,4,3,2] => [1,4,5,3,6,2] => 2
[.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => [1,2,6,5,4,3] => [1,2,5,4,6,3] => 1
[.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => [1,2,6,5,4,3] => [1,2,5,4,6,3] => 1
[.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => [1,3,2,6,5,4] => [1,3,2,5,6,4] => 0
[.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => [1,2,3,6,5,4] => [1,2,3,5,6,4] => 0
[.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => [2,1,6,5,4,3] => [2,1,5,4,6,3] => 1
[.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => [2,1,3,6,5,4] => [2,1,3,5,6,4] => 0
[.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => [1,2,6,5,4,3] => [1,2,5,4,6,3] => 1
[.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => [1,2,3,6,5,4] => [1,2,3,5,6,4] => 0
[.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => [3,2,1,6,5,4] => [2,3,1,5,6,4] => 0
[.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => [1,3,2,6,5,4] => [1,3,2,5,6,4] => 0
[.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => [1,3,2,6,5,4] => [1,3,2,5,6,4] => 0
[.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => [2,1,3,6,5,4] => [2,1,3,5,6,4] => 0
[.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => [1,2,3,6,5,4] => [1,2,3,5,6,4] => 0
[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [4,3,2,1,6,5] => [3,2,4,1,6,5] => 1
[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [1,4,3,2,6,5] => [1,3,4,2,6,5] => 0
[.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => [1,4,3,2,6,5] => [1,3,4,2,6,5] => 0
[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 0
[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => 0
[.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => [1,4,3,2,6,5] => [1,3,4,2,6,5] => 0
[.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => 0
[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => [2,1,4,3,6,5] => [2,1,4,3,6,5] => 0
[.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => [1,2,4,3,6,5] => [1,2,4,3,6,5] => 0
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Description
The maximum defect over any reduced expression for a permutation and any subexpression.
Map
inverse first fundamental transformation
Description
Let $\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.
Map
Inverse fireworks map
Description
Sends a permutation to an inverse fireworks permutation.
A permutation $\sigma$ is inverse fireworks if its inverse avoids the vincular pattern $3-12$. The inverse fireworks map sends any permutation $\sigma$ to an inverse fireworks permutation that is below $\sigma$ in left weak order and has the same Rajchgot index St001759The Rajchgot index of a permutation..
A permutation $\sigma$ is inverse fireworks if its inverse avoids the vincular pattern $3-12$. The inverse fireworks map sends any permutation $\sigma$ to an inverse fireworks permutation that is below $\sigma$ in left weak order and has the same Rajchgot index St001759The Rajchgot index 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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