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Identifier

Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001874: Permutations ⟶ ℤ

Values

[1] => [1] => [1] => [1] => 0

[1,2] => [1,2] => [1,2] => [1,2] => 0

[2,1] => [2,1] => [2,1] => [2,1] => 1

[1,2,3] => [1,2,3] => [1,2,3] => [1,2,3] => 0

[1,3,2] => [1,3,2] => [1,3,2] => [1,3,2] => 1

[2,1,3] => [2,1,3] => [2,1,3] => [2,1,3] => 1

[2,3,1] => [1,3,2] => [1,3,2] => [1,3,2] => 1

[3,1,2] => [3,1,2] => [3,1,2] => [3,2,1] => 3

[3,2,1] => [3,2,1] => [2,3,1] => [3,1,2] => 1

[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0

[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1

[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1

[1,3,4,2] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1

[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => [1,4,3,2] => 3

[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => [1,4,2,3] => 1

[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1

[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2

[2,3,1,4] => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1

[2,3,4,1] => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1

[2,4,1,3] => [2,4,1,3] => [4,2,1,3] => [2,4,3,1] => 3

[2,4,3,1] => [1,4,3,2] => [1,3,4,2] => [1,4,2,3] => 1

[3,1,2,4] => [3,1,2,4] => [3,1,2,4] => [3,2,1,4] => 3

[3,1,4,2] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2

[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => [3,1,2,4] => 1

[3,2,4,1] => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2

[3,4,1,2] => [2,4,1,3] => [4,2,1,3] => [2,4,3,1] => 3

[3,4,2,1] => [1,4,3,2] => [1,3,4,2] => [1,4,2,3] => 1

[4,1,2,3] => [4,1,2,3] => [4,1,2,3] => [4,3,2,1] => 6

[4,1,3,2] => [4,1,3,2] => [3,1,4,2] => [4,2,1,3] => 3

[4,2,1,3] => [4,2,1,3] => [2,4,1,3] => [4,3,1,2] => 3

[4,2,3,1] => [4,1,3,2] => [3,1,4,2] => [4,2,1,3] => 3

[4,3,1,2] => [4,3,1,2] => [3,4,2,1] => [4,1,3,2] => 3

[4,3,2,1] => [4,3,2,1] => [3,4,1,2] => [3,1,4,2] => 2

[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0

[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1

[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1

[1,2,4,5,3] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1

[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,4,3] => 3

[1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,5,3,4] => 1

[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1

[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2

[1,3,4,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1

[1,3,4,5,2] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1

[1,3,5,2,4] => [1,3,5,2,4] => [1,5,3,2,4] => [1,3,5,4,2] => 3

[1,3,5,4,2] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,5,3,4] => 1

[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => [1,4,3,2,5] => 3

[1,4,2,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2

[1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => 1

[1,4,3,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2

[1,4,5,2,3] => [1,3,5,2,4] => [1,5,3,2,4] => [1,3,5,4,2] => 3

[1,4,5,3,2] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,5,3,4] => 1

[1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => [1,5,4,3,2] => 6

[1,5,2,4,3] => [1,5,2,4,3] => [1,4,2,5,3] => [1,5,3,2,4] => 3

[1,5,3,2,4] => [1,5,3,2,4] => [1,3,5,2,4] => [1,5,4,2,3] => 3

[1,5,3,4,2] => [1,5,2,4,3] => [1,4,2,5,3] => [1,5,3,2,4] => 3

[1,5,4,2,3] => [1,5,4,2,3] => [1,4,5,3,2] => [1,5,2,4,3] => 3

[1,5,4,3,2] => [1,5,4,3,2] => [1,4,5,2,3] => [1,4,2,5,3] => 2

[2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1

[2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2

[2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2

[2,1,4,5,3] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2

[2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,4,3] => 4

[2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => [2,1,5,3,4] => 2

[2,3,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1

[2,3,1,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2

[2,3,4,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1

[2,3,4,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1

[2,3,5,1,4] => [1,3,5,2,4] => [1,5,3,2,4] => [1,3,5,4,2] => 3

[2,3,5,4,1] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,5,3,4] => 1

[2,4,1,3,5] => [2,4,1,3,5] => [4,2,1,3,5] => [2,4,3,1,5] => 3

[2,4,1,5,3] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2

[2,4,3,1,5] => [1,4,3,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => 1

[2,4,3,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2

[2,4,5,1,3] => [1,3,5,2,4] => [1,5,3,2,4] => [1,3,5,4,2] => 3

[2,4,5,3,1] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,5,3,4] => 1

[2,5,1,3,4] => [2,5,1,3,4] => [5,2,1,3,4] => [2,5,4,3,1] => 6

[2,5,1,4,3] => [2,5,1,4,3] => [4,2,1,5,3] => [2,5,3,1,4] => 3

[2,5,3,1,4] => [1,5,3,2,4] => [1,3,5,2,4] => [1,5,4,2,3] => 3

[2,5,3,4,1] => [1,5,2,4,3] => [1,4,2,5,3] => [1,5,3,2,4] => 3

[2,5,4,1,3] => [2,5,4,1,3] => [4,2,5,3,1] => [2,5,1,4,3] => 4

[2,5,4,3,1] => [1,5,4,3,2] => [1,4,5,2,3] => [1,4,2,5,3] => 2

[3,1,2,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 3

[3,1,2,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => [3,2,1,5,4] => 4

[3,1,4,2,5] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2

[3,1,4,5,2] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2

[3,1,5,2,4] => [3,1,5,2,4] => [5,1,3,2,4] => [3,5,4,2,1] => 6

[3,1,5,4,2] => [2,1,5,4,3] => [2,1,4,5,3] => [2,1,5,3,4] => 2

[3,2,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => [3,1,2,4,5] => 1

[3,2,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => [3,1,2,5,4] => 2

[3,2,4,1,5] => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2

[3,2,4,5,1] => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2

[3,2,5,1,4] => [3,2,5,1,4] => [2,5,3,1,4] => [3,5,4,1,2] => 4

[3,2,5,4,1] => [2,1,5,4,3] => [2,1,4,5,3] => [2,1,5,3,4] => 2

[3,4,1,2,5] => [2,4,1,3,5] => [4,2,1,3,5] => [2,4,3,1,5] => 3

[3,4,1,5,2] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2

[3,4,2,1,5] => [1,4,3,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => 1

[3,4,2,5,1] => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2

[3,4,5,1,2] => [1,3,5,2,4] => [1,5,3,2,4] => [1,3,5,4,2] => 3

[3,4,5,2,1] => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,5,3,4] => 1

[3,5,1,2,4] => [3,5,1,2,4] => [5,3,2,1,4] => [3,2,5,4,1] => 4

[3,5,1,4,2] => [2,5,1,4,3] => [4,2,1,5,3] => [2,5,3,1,4] => 3

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Description

Lusztig's a-function for the symmetric group.
Let $x$ be a permutation corresponding to the pair of tableaux $(P(x),Q(x))$
by the Robinson-Schensted correspondence and
$\operatorname{shape}(Q(x)')=( \lambda_1,...,\lambda_k)$
where $Q(x)'$ is the transposed tableau.
Then $a(x)=\sum\limits_{i=1}^{k}{\binom{\lambda_i}{2}}$.
See exercise 10 on page 198 in the book by Björner and Brenti "Combinatorics of Coxeter Groups" for equivalent characterisations and properties.

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.

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..

Map

Corteel

Description

Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.