Identifier
Mp00045:
Integer partitions
—reading tableau⟶
Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Images
[1] => [[1]] => [1] => [1] => [1]
[2] => [[1,2]] => [1,2] => [1,2] => [1,2]
[1,1] => [[1],[2]] => [2,1] => [2,1] => [2,1]
[3] => [[1,2,3]] => [1,2,3] => [1,2,3] => [1,2,3]
[2,1] => [[1,3],[2]] => [2,1,3] => [2,1,3] => [2,1,3]
[1,1,1] => [[1],[2],[3]] => [3,2,1] => [3,2,1] => [3,2,1]
[4] => [[1,2,3,4]] => [1,2,3,4] => [1,2,3,4] => [1,2,3,4]
[3,1] => [[1,3,4],[2]] => [2,1,3,4] => [2,1,3,4] => [2,1,3,4]
[2,2] => [[1,2],[3,4]] => [3,4,1,2] => [3,1,4,2] => [3,4,1,2]
[2,1,1] => [[1,4],[2],[3]] => [3,2,1,4] => [3,2,1,4] => [3,2,1,4]
[1,1,1,1] => [[1],[2],[3],[4]] => [4,3,2,1] => [4,3,2,1] => [4,3,2,1]
[5] => [[1,2,3,4,5]] => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5]
[4,1] => [[1,3,4,5],[2]] => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5]
[3,2] => [[1,2,5],[3,4]] => [3,4,1,2,5] => [3,1,4,2,5] => [3,4,1,2,5]
[3,1,1] => [[1,4,5],[2],[3]] => [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5]
[2,2,1] => [[1,3],[2,5],[4]] => [4,2,5,1,3] => [2,4,1,5,3] => [4,2,5,1,3]
[2,1,1,1] => [[1,5],[2],[3],[4]] => [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1,5]
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1]
[6] => [[1,2,3,4,5,6]] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5,6]
[5,1] => [[1,3,4,5,6],[2]] => [2,1,3,4,5,6] => [2,1,3,4,5,6] => [2,1,3,4,5,6]
[4,2] => [[1,2,5,6],[3,4]] => [3,4,1,2,5,6] => [3,1,4,2,5,6] => [3,4,1,2,5,6]
[4,1,1] => [[1,4,5,6],[2],[3]] => [3,2,1,4,5,6] => [3,2,1,4,5,6] => [3,2,1,4,5,6]
[3,3] => [[1,2,3],[4,5,6]] => [4,5,6,1,2,3] => [4,1,5,2,6,3] => [4,5,6,1,2,3]
[3,2,1] => [[1,3,6],[2,5],[4]] => [4,2,5,1,3,6] => [2,4,1,5,3,6] => [4,2,5,1,3,6]
[3,1,1,1] => [[1,5,6],[2],[3],[4]] => [4,3,2,1,5,6] => [4,3,2,1,5,6] => [4,3,2,1,5,6]
[2,2,2] => [[1,2],[3,4],[5,6]] => [5,6,3,4,1,2] => [5,3,1,6,4,2] => [5,6,3,4,1,2]
[2,2,1,1] => [[1,4],[2,6],[3],[5]] => [5,3,2,6,1,4] => [3,2,5,1,6,4] => [5,3,2,6,1,4]
[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => [5,4,3,2,1,6] => [5,4,3,2,1,6]
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1]
[7] => [[1,2,3,4,5,6,7]] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7]
[6,1] => [[1,3,4,5,6,7],[2]] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7]
[5,2] => [[1,2,5,6,7],[3,4]] => [3,4,1,2,5,6,7] => [3,1,4,2,5,6,7] => [3,4,1,2,5,6,7]
[5,1,1] => [[1,4,5,6,7],[2],[3]] => [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7]
[4,3] => [[1,2,3,7],[4,5,6]] => [4,5,6,1,2,3,7] => [4,1,5,2,6,3,7] => [4,5,6,1,2,3,7]
[4,2,1] => [[1,3,6,7],[2,5],[4]] => [4,2,5,1,3,6,7] => [2,4,1,5,3,6,7] => [4,2,5,1,3,6,7]
[4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7]
[3,3,1] => [[1,3,4],[2,6,7],[5]] => [5,2,6,7,1,3,4] => [5,6,2,1,3,7,4] => [5,2,6,7,1,3,4]
[3,2,2] => [[1,2,7],[3,4],[5,6]] => [5,6,3,4,1,2,7] => [5,3,1,6,4,2,7] => [5,6,3,4,1,2,7]
[3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => [5,3,2,6,1,4,7] => [3,2,5,1,6,4,7] => [5,3,2,6,1,4,7]
[3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7]
[2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => [6,4,7,2,5,1,3] => [4,2,6,1,7,5,3] => [6,4,7,2,5,1,3]
[2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => [6,4,3,2,7,1,5] => [4,3,2,6,1,7,5] => [6,4,3,2,7,1,5]
[2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7]
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1]
[8] => [[1,2,3,4,5,6,7,8]] => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8]
[7,1] => [[1,3,4,5,6,7,8],[2]] => [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,8]
[6,2] => [[1,2,5,6,7,8],[3,4]] => [3,4,1,2,5,6,7,8] => [3,1,4,2,5,6,7,8] => [3,4,1,2,5,6,7,8]
[6,1,1] => [[1,4,5,6,7,8],[2],[3]] => [3,2,1,4,5,6,7,8] => [3,2,1,4,5,6,7,8] => [3,2,1,4,5,6,7,8]
[5,3] => [[1,2,3,7,8],[4,5,6]] => [4,5,6,1,2,3,7,8] => [4,1,5,2,6,3,7,8] => [4,5,6,1,2,3,7,8]
[5,2,1] => [[1,3,6,7,8],[2,5],[4]] => [4,2,5,1,3,6,7,8] => [2,4,1,5,3,6,7,8] => [4,2,5,1,3,6,7,8]
[5,1,1,1] => [[1,5,6,7,8],[2],[3],[4]] => [4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7,8] => [4,3,2,1,5,6,7,8]
[4,4] => [[1,2,3,4],[5,6,7,8]] => [5,6,7,8,1,2,3,4] => [5,1,6,2,7,3,8,4] => [5,6,7,8,1,2,3,4]
[4,3,1] => [[1,3,4,8],[2,6,7],[5]] => [5,2,6,7,1,3,4,8] => [5,6,2,1,3,7,4,8] => [5,2,6,7,1,3,4,8]
[4,2,2] => [[1,2,7,8],[3,4],[5,6]] => [5,6,3,4,1,2,7,8] => [5,3,1,6,4,2,7,8] => [5,6,3,4,1,2,7,8]
[4,2,1,1] => [[1,4,7,8],[2,6],[3],[5]] => [5,3,2,6,1,4,7,8] => [3,2,5,1,6,4,7,8] => [5,3,2,6,1,4,7,8]
[4,1,1,1,1] => [[1,6,7,8],[2],[3],[4],[5]] => [5,4,3,2,1,6,7,8] => [5,4,3,2,1,6,7,8] => [5,4,3,2,1,6,7,8]
[3,3,2] => [[1,2,5],[3,4,8],[6,7]] => [6,7,3,4,8,1,2,5] => [3,6,4,1,7,2,8,5] => [6,7,3,4,8,1,2,5]
[3,3,1,1] => [[1,4,5],[2,7,8],[3],[6]] => [6,3,2,7,8,1,4,5] => [2,6,7,3,1,4,8,5] => [6,2,7,3,8,1,4,5]
[3,2,2,1] => [[1,3,8],[2,5],[4,7],[6]] => [6,4,7,2,5,1,3,8] => [4,2,6,1,7,5,3,8] => [6,4,7,2,5,1,3,8]
[3,2,1,1,1] => [[1,5,8],[2,7],[3],[4],[6]] => [6,4,3,2,7,1,5,8] => [4,3,2,6,1,7,5,8] => [6,4,3,2,7,1,5,8]
[3,1,1,1,1,1] => [[1,7,8],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,7,8] => [6,5,4,3,2,1,7,8]
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [7,8,5,6,3,4,1,2] => [7,5,3,1,8,6,4,2] => [7,8,5,6,3,4,1,2]
[2,2,2,1,1] => [[1,4],[2,6],[3,8],[5],[7]] => [7,5,3,8,2,6,1,4] => [3,5,2,7,1,8,6,4] => [7,5,3,8,2,6,1,4]
[2,2,1,1,1,1] => [[1,6],[2,8],[3],[4],[5],[7]] => [7,5,4,3,2,8,1,6] => [5,4,3,2,7,1,8,6] => [7,5,4,3,2,8,1,6]
[2,1,1,1,1,1,1] => [[1,8],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8] => [7,6,5,4,3,2,1,8]
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1]
[9] => [[1,2,3,4,5,6,7,8,9]] => [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9]
[8,1] => [[1,3,4,5,6,7,8,9],[2]] => [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,9]
[7,2] => [[1,2,5,6,7,8,9],[3,4]] => [3,4,1,2,5,6,7,8,9] => [3,1,4,2,5,6,7,8,9] => [3,4,1,2,5,6,7,8,9]
[7,1,1] => [[1,4,5,6,7,8,9],[2],[3]] => [3,2,1,4,5,6,7,8,9] => [3,2,1,4,5,6,7,8,9] => [3,2,1,4,5,6,7,8,9]
[6,3] => [[1,2,3,7,8,9],[4,5,6]] => [4,5,6,1,2,3,7,8,9] => [4,1,5,2,6,3,7,8,9] => [4,5,6,1,2,3,7,8,9]
[6,2,1] => [[1,3,6,7,8,9],[2,5],[4]] => [4,2,5,1,3,6,7,8,9] => [2,4,1,5,3,6,7,8,9] => [4,2,5,1,3,6,7,8,9]
[6,1,1,1] => [[1,5,6,7,8,9],[2],[3],[4]] => [4,3,2,1,5,6,7,8,9] => [4,3,2,1,5,6,7,8,9] => [4,3,2,1,5,6,7,8,9]
[5,4] => [[1,2,3,4,9],[5,6,7,8]] => [5,6,7,8,1,2,3,4,9] => [5,1,6,2,7,3,8,4,9] => [5,6,7,8,1,2,3,4,9]
[5,3,1] => [[1,3,4,8,9],[2,6,7],[5]] => [5,2,6,7,1,3,4,8,9] => [5,6,2,1,3,7,4,8,9] => [5,2,6,7,1,3,4,8,9]
[5,2,2] => [[1,2,7,8,9],[3,4],[5,6]] => [5,6,3,4,1,2,7,8,9] => [5,3,1,6,4,2,7,8,9] => [5,6,3,4,1,2,7,8,9]
[5,2,1,1] => [[1,4,7,8,9],[2,6],[3],[5]] => [5,3,2,6,1,4,7,8,9] => [3,2,5,1,6,4,7,8,9] => [5,3,2,6,1,4,7,8,9]
[5,1,1,1,1] => [[1,6,7,8,9],[2],[3],[4],[5]] => [5,4,3,2,1,6,7,8,9] => [5,4,3,2,1,6,7,8,9] => [5,4,3,2,1,6,7,8,9]
[4,4,1] => [[1,3,4,5],[2,7,8,9],[6]] => [6,2,7,8,9,1,3,4,5] => [6,7,3,1,8,2,4,9,5] => [6,3,7,8,1,9,2,4,5]
[4,3,2] => [[1,2,5,9],[3,4,8],[6,7]] => [6,7,3,4,8,1,2,5,9] => [3,6,4,1,7,2,8,5,9] => [6,7,3,4,8,1,2,5,9]
[4,3,1,1] => [[1,4,5,9],[2,7,8],[3],[6]] => [6,3,2,7,8,1,4,5,9] => [2,6,7,3,1,4,8,5,9] => [6,2,7,3,8,1,4,5,9]
[4,2,2,1] => [[1,3,8,9],[2,5],[4,7],[6]] => [6,4,7,2,5,1,3,8,9] => [4,2,6,1,7,5,3,8,9] => [6,4,7,2,5,1,3,8,9]
[4,2,1,1,1] => [[1,5,8,9],[2,7],[3],[4],[6]] => [6,4,3,2,7,1,5,8,9] => [4,3,2,6,1,7,5,8,9] => [6,4,3,2,7,1,5,8,9]
[4,1,1,1,1,1] => [[1,7,8,9],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7,8,9] => [6,5,4,3,2,1,7,8,9] => [6,5,4,3,2,1,7,8,9]
[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => [7,8,9,4,5,6,1,2,3] => [7,4,1,8,5,2,9,6,3] => [7,8,9,4,5,6,1,2,3]
[3,3,2,1] => [[1,3,6],[2,5,9],[4,8],[7]] => [7,4,8,2,5,9,1,3,6] => [2,7,4,5,1,8,3,9,6] => [7,8,2,4,5,9,1,3,6]
[3,3,1,1,1] => [[1,5,6],[2,8,9],[3],[4],[7]] => [7,4,3,2,8,9,1,5,6] => [3,2,7,8,4,1,5,9,6] => [7,3,8,2,4,9,1,5,6]
[3,2,2,2] => [[1,2,9],[3,4],[5,6],[7,8]] => [7,8,5,6,3,4,1,2,9] => [7,5,3,1,8,6,4,2,9] => [7,8,5,6,3,4,1,2,9]
[3,2,2,1,1] => [[1,4,9],[2,6],[3,8],[5],[7]] => [7,5,3,8,2,6,1,4,9] => [3,5,2,7,1,8,6,4,9] => [7,5,3,8,2,6,1,4,9]
[3,2,1,1,1,1] => [[1,6,9],[2,8],[3],[4],[5],[7]] => [7,5,4,3,2,8,1,6,9] => [5,4,3,2,7,1,8,6,9] => [7,5,4,3,2,8,1,6,9]
[3,1,1,1,1,1,1] => [[1,8,9],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1,8,9] => [7,6,5,4,3,2,1,8,9] => [7,6,5,4,3,2,1,8,9]
[2,2,2,2,1] => [[1,3],[2,5],[4,7],[6,9],[8]] => [8,6,9,4,7,2,5,1,3] => [6,4,2,8,1,9,7,5,3] => [8,6,9,4,7,2,5,1,3]
[2,2,2,1,1,1] => [[1,5],[2,7],[3,9],[4],[6],[8]] => [8,6,4,3,9,2,7,1,5] => [4,3,6,2,8,1,9,7,5] => [8,6,4,3,9,2,7,1,5]
[2,2,1,1,1,1,1] => [[1,7],[2,9],[3],[4],[5],[6],[8]] => [8,6,5,4,3,2,9,1,7] => [6,5,4,3,2,8,1,9,7] => [8,6,5,4,3,2,9,1,7]
[2,1,1,1,1,1,1,1] => [[1,9],[2],[3],[4],[5],[6],[7],[8]] => [8,7,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1,9] => [8,7,6,5,4,3,2,1,9]
[1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => [9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1] => [9,8,7,6,5,4,3,2,1]
[10] => [[1,2,3,4,5,6,7,8,9,10]] => [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10]
[9,1] => [[1,3,4,5,6,7,8,9,10],[2]] => [2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9,10] => [2,1,3,4,5,6,7,8,9,10]
[8,2] => [[1,2,5,6,7,8,9,10],[3,4]] => [3,4,1,2,5,6,7,8,9,10] => [3,1,4,2,5,6,7,8,9,10] => [3,4,1,2,5,6,7,8,9,10]
[8,1,1] => [[1,4,5,6,7,8,9,10],[2],[3]] => [3,2,1,4,5,6,7,8,9,10] => [3,2,1,4,5,6,7,8,9,10] => [3,2,1,4,5,6,7,8,9,10]
[7,3] => [[1,2,3,7,8,9,10],[4,5,6]] => [4,5,6,1,2,3,7,8,9,10] => [4,1,5,2,6,3,7,8,9,10] => [4,5,6,1,2,3,7,8,9,10]
>>> Load all 143 entries. <<<Map
reading tableau
Description
Return the RSK recording tableau of the reading word of the (standard) tableau $T$ labeled down (in English convention) each column to the shape of a partition.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.
Map
Lehmer-code to major-code bijection
Description
Sends a permutation to the unique permutation such that the Lehmer code is sent to the major code.
The Lehmer code encodes the inversions of a permutation and the major code encodes its major index. In particular, the number of inversions of a permutation equals the major index of its image under this map.
* The Lehmer code of a permutation $\sigma$ is given by $L(\sigma) = l_1 \ldots l_n$ with $l_i = \# \{ j > i : \sigma_j < \sigma_i \}$. In particular, $l_i$ is the number of boxes in the $i$-th column of the Rothe diagram. For example, the Lehmer code of $\sigma = [4,3,1,5,2]$ is $32010$. The Lehmer code $L : \mathfrak{S}_n\ \tilde\longrightarrow\ S_n$ is a bijection between permutations of size $n$ and sequences $l_1\ldots l_n \in \mathbf{N}^n$ with $l_i \leq i$.
* The major code $M(\sigma)$ of a permutation $\sigma \in \mathfrak{S}_n$ is a way to encode a permutation as a sequence $m_1 m_2 \ldots m_n$ with $m_i \geq i$. To define $m_i$, let $\operatorname{del}_i(\sigma)$ be the normalized permutation obtained by removing all $\sigma_j < i$ from the one-line notation of $\sigma$. The $i$-th index is then given by
$$m_i = \operatorname{maj}(\operatorname{del}_i(\sigma)) - \operatorname{maj}(\operatorname{del}_{i-1}(\sigma)).$$
For example, the permutation $[9,3,5,7,2,1,4,6,8]$ has major code $[5, 0, 1, 0, 1, 2, 0, 1, 0]$ since
$$\operatorname{maj}([8,2,4,6,1,3,5,7]) = 5, \quad \operatorname{maj}([7,1,3,5,2,4,6]) = 5, \quad \operatorname{maj}([6,2,4,1,3,5]) = 4,$$
$$\operatorname{maj}([5,1,3,2,4]) = 4, \quad \operatorname{maj}([4,2,1,3]) = 3, \quad \operatorname{maj}([3,1,2]) = 1, \quad \operatorname{maj}([2,1]) = 1.$$
Observe that the sum of the major code of $\sigma$ equals the major index of $\sigma$.
The Lehmer code encodes the inversions of a permutation and the major code encodes its major index. In particular, the number of inversions of a permutation equals the major index of its image under this map.
* The Lehmer code of a permutation $\sigma$ is given by $L(\sigma) = l_1 \ldots l_n$ with $l_i = \# \{ j > i : \sigma_j < \sigma_i \}$. In particular, $l_i$ is the number of boxes in the $i$-th column of the Rothe diagram. For example, the Lehmer code of $\sigma = [4,3,1,5,2]$ is $32010$. The Lehmer code $L : \mathfrak{S}_n\ \tilde\longrightarrow\ S_n$ is a bijection between permutations of size $n$ and sequences $l_1\ldots l_n \in \mathbf{N}^n$ with $l_i \leq i$.
* The major code $M(\sigma)$ of a permutation $\sigma \in \mathfrak{S}_n$ is a way to encode a permutation as a sequence $m_1 m_2 \ldots m_n$ with $m_i \geq i$. To define $m_i$, let $\operatorname{del}_i(\sigma)$ be the normalized permutation obtained by removing all $\sigma_j < i$ from the one-line notation of $\sigma$. The $i$-th index is then given by
$$m_i = \operatorname{maj}(\operatorname{del}_i(\sigma)) - \operatorname{maj}(\operatorname{del}_{i-1}(\sigma)).$$
For example, the permutation $[9,3,5,7,2,1,4,6,8]$ has major code $[5, 0, 1, 0, 1, 2, 0, 1, 0]$ since
$$\operatorname{maj}([8,2,4,6,1,3,5,7]) = 5, \quad \operatorname{maj}([7,1,3,5,2,4,6]) = 5, \quad \operatorname{maj}([6,2,4,1,3,5]) = 4,$$
$$\operatorname{maj}([5,1,3,2,4]) = 4, \quad \operatorname{maj}([4,2,1,3]) = 3, \quad \operatorname{maj}([3,1,2]) = 1, \quad \operatorname{maj}([2,1]) = 1.$$
Observe that the sum of the major code of $\sigma$ equals the major index of $\sigma$.
Map
Foata bijection
Description
Sends a permutation to its image under the Foata bijection.
The Foata bijection $\phi$ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$.
At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).
The Foata bijection $\phi$ is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word:
Given a word $w_1 w_2 ... w_n$, compute the image inductively by starting with $\phi(w_1) = w_1$.
At the $i$-th step, if $\phi(w_1 w_2 ... w_i) = v_1 v_2 ... v_i$, define $\phi(w_1 w_2 ... w_i w_{i+1})$ by placing $w_{i+1}$ on the end of the word $v_1 v_2 ... v_i$ and breaking the word up into blocks as follows.
- If $w_{i+1} \geq v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} \geq v_k$.
- If $w_{i+1} < v_i$, place a vertical line to the right of each $v_k$ for which $w_{i+1} < v_k$.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
- $1$
- $|1|4 \to 14$
- $|14|2 \to 412$
- $|4|1|2|5 \to 4125$
- $|4|125|3 \to 45123.$
This bijection sends the major index (St000004The major index of a permutation.) to the number of inversions (St000018The number of inversions of a permutation.).
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