Identifier
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00314: Integer compositions —Foata bijection⟶ Integer compositions
Images
[1] => [[1]] => [1] => [1]
[2] => [[1,2]] => [2] => [2]
[1,1] => [[1],[2]] => [2] => [2]
[3] => [[1,2,3]] => [3] => [3]
[2,1] => [[1,3],[2]] => [3] => [3]
[1,1,1] => [[1],[2],[3]] => [3] => [3]
[4] => [[1,2,3,4]] => [4] => [4]
[3,1] => [[1,3,4],[2]] => [4] => [4]
[2,2] => [[1,2],[3,4]] => [2,2] => [2,2]
[2,1,1] => [[1,4],[2],[3]] => [4] => [4]
[1,1,1,1] => [[1],[2],[3],[4]] => [4] => [4]
[5] => [[1,2,3,4,5]] => [5] => [5]
[4,1] => [[1,3,4,5],[2]] => [5] => [5]
[3,2] => [[1,2,5],[3,4]] => [2,3] => [2,3]
[3,1,1] => [[1,4,5],[2],[3]] => [5] => [5]
[2,2,1] => [[1,3],[2,5],[4]] => [3,2] => [3,2]
[2,1,1,1] => [[1,5],[2],[3],[4]] => [5] => [5]
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [5] => [5]
[6] => [[1,2,3,4,5,6]] => [6] => [6]
[5,1] => [[1,3,4,5,6],[2]] => [6] => [6]
[4,2] => [[1,2,5,6],[3,4]] => [2,4] => [2,4]
[4,1,1] => [[1,4,5,6],[2],[3]] => [6] => [6]
[3,3] => [[1,2,3],[4,5,6]] => [3,3] => [3,3]
[3,2,1] => [[1,3,6],[2,5],[4]] => [3,3] => [3,3]
[3,1,1,1] => [[1,5,6],[2],[3],[4]] => [6] => [6]
[2,2,2] => [[1,2],[3,4],[5,6]] => [2,2,2] => [2,2,2]
[2,2,1,1] => [[1,4],[2,6],[3],[5]] => [4,2] => [4,2]
[2,1,1,1,1] => [[1,6],[2],[3],[4],[5]] => [6] => [6]
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [6] => [6]
[7] => [[1,2,3,4,5,6,7]] => [7] => [7]
[6,1] => [[1,3,4,5,6,7],[2]] => [7] => [7]
[5,2] => [[1,2,5,6,7],[3,4]] => [2,5] => [2,5]
[5,1,1] => [[1,4,5,6,7],[2],[3]] => [7] => [7]
[4,3] => [[1,2,3,7],[4,5,6]] => [3,4] => [3,4]
[4,2,1] => [[1,3,6,7],[2,5],[4]] => [3,4] => [3,4]
[4,1,1,1] => [[1,5,6,7],[2],[3],[4]] => [7] => [7]
[3,3,1] => [[1,3,4],[2,6,7],[5]] => [4,3] => [4,3]
[3,2,2] => [[1,2,7],[3,4],[5,6]] => [2,2,3] => [2,2,3]
[3,2,1,1] => [[1,4,7],[2,6],[3],[5]] => [4,3] => [4,3]
[3,1,1,1,1] => [[1,6,7],[2],[3],[4],[5]] => [7] => [7]
[2,2,2,1] => [[1,3],[2,5],[4,7],[6]] => [3,2,2] => [2,3,2]
[2,2,1,1,1] => [[1,5],[2,7],[3],[4],[6]] => [5,2] => [5,2]
[2,1,1,1,1,1] => [[1,7],[2],[3],[4],[5],[6]] => [7] => [7]
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [7] => [7]
[8] => [[1,2,3,4,5,6,7,8]] => [8] => [8]
[7,1] => [[1,3,4,5,6,7,8],[2]] => [8] => [8]
[6,2] => [[1,2,5,6,7,8],[3,4]] => [2,6] => [2,6]
[6,1,1] => [[1,4,5,6,7,8],[2],[3]] => [8] => [8]
[5,3] => [[1,2,3,7,8],[4,5,6]] => [3,5] => [3,5]
[5,2,1] => [[1,3,6,7,8],[2,5],[4]] => [3,5] => [3,5]
[5,1,1,1] => [[1,5,6,7,8],[2],[3],[4]] => [8] => [8]
[4,4] => [[1,2,3,4],[5,6,7,8]] => [4,4] => [4,4]
[4,3,1] => [[1,3,4,8],[2,6,7],[5]] => [4,4] => [4,4]
[4,2,2] => [[1,2,7,8],[3,4],[5,6]] => [2,2,4] => [2,2,4]
[4,2,1,1] => [[1,4,7,8],[2,6],[3],[5]] => [4,4] => [4,4]
[4,1,1,1,1] => [[1,6,7,8],[2],[3],[4],[5]] => [8] => [8]
[3,3,2] => [[1,2,5],[3,4,8],[6,7]] => [2,3,3] => [2,3,3]
[3,3,1,1] => [[1,4,5],[2,7,8],[3],[6]] => [5,3] => [5,3]
[3,2,2,1] => [[1,3,8],[2,5],[4,7],[6]] => [3,2,3] => [3,2,3]
[3,2,1,1,1] => [[1,5,8],[2,7],[3],[4],[6]] => [5,3] => [5,3]
[3,1,1,1,1,1] => [[1,7,8],[2],[3],[4],[5],[6]] => [8] => [8]
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [2,2,2,2] => [2,2,2,2]
[2,2,2,1,1] => [[1,4],[2,6],[3,8],[5],[7]] => [4,2,2] => [2,4,2]
[2,2,1,1,1,1] => [[1,6],[2,8],[3],[4],[5],[7]] => [6,2] => [6,2]
[2,1,1,1,1,1,1] => [[1,8],[2],[3],[4],[5],[6],[7]] => [8] => [8]
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [8] => [8]
[9] => [[1,2,3,4,5,6,7,8,9]] => [9] => [9]
[8,1] => [[1,3,4,5,6,7,8,9],[2]] => [9] => [9]
[7,2] => [[1,2,5,6,7,8,9],[3,4]] => [2,7] => [2,7]
[7,1,1] => [[1,4,5,6,7,8,9],[2],[3]] => [9] => [9]
[6,3] => [[1,2,3,7,8,9],[4,5,6]] => [3,6] => [3,6]
[6,2,1] => [[1,3,6,7,8,9],[2,5],[4]] => [3,6] => [3,6]
[6,1,1,1] => [[1,5,6,7,8,9],[2],[3],[4]] => [9] => [9]
[5,4] => [[1,2,3,4,9],[5,6,7,8]] => [4,5] => [4,5]
[5,3,1] => [[1,3,4,8,9],[2,6,7],[5]] => [4,5] => [4,5]
[5,2,2] => [[1,2,7,8,9],[3,4],[5,6]] => [2,2,5] => [2,2,5]
[5,2,1,1] => [[1,4,7,8,9],[2,6],[3],[5]] => [4,5] => [4,5]
[5,1,1,1,1] => [[1,6,7,8,9],[2],[3],[4],[5]] => [9] => [9]
[4,4,1] => [[1,3,4,5],[2,7,8,9],[6]] => [5,4] => [5,4]
[4,3,2] => [[1,2,5,9],[3,4,8],[6,7]] => [2,3,4] => [2,3,4]
[4,3,1,1] => [[1,4,5,9],[2,7,8],[3],[6]] => [5,4] => [5,4]
[4,2,2,1] => [[1,3,8,9],[2,5],[4,7],[6]] => [3,2,4] => [3,2,4]
[4,2,1,1,1] => [[1,5,8,9],[2,7],[3],[4],[6]] => [5,4] => [5,4]
[4,1,1,1,1,1] => [[1,7,8,9],[2],[3],[4],[5],[6]] => [9] => [9]
[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => [3,3,3] => [3,3,3]
[3,3,2,1] => [[1,3,6],[2,5,9],[4,8],[7]] => [3,3,3] => [3,3,3]
[3,3,1,1,1] => [[1,5,6],[2,8,9],[3],[4],[7]] => [6,3] => [6,3]
[3,2,2,2] => [[1,2,9],[3,4],[5,6],[7,8]] => [2,2,2,3] => [2,2,2,3]
[3,2,2,1,1] => [[1,4,9],[2,6],[3,8],[5],[7]] => [4,2,3] => [2,4,3]
[3,2,1,1,1,1] => [[1,6,9],[2,8],[3],[4],[5],[7]] => [6,3] => [6,3]
[3,1,1,1,1,1,1] => [[1,8,9],[2],[3],[4],[5],[6],[7]] => [9] => [9]
[2,2,2,2,1] => [[1,3],[2,5],[4,7],[6,9],[8]] => [3,2,2,2] => [2,2,3,2]
[2,2,2,1,1,1] => [[1,5],[2,7],[3,9],[4],[6],[8]] => [5,2,2] => [2,5,2]
[2,2,1,1,1,1,1] => [[1,7],[2,9],[3],[4],[5],[6],[8]] => [7,2] => [7,2]
[2,1,1,1,1,1,1,1] => [[1,9],[2],[3],[4],[5],[6],[7],[8]] => [9] => [9]
[1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => [9] => [9]
[10] => [[1,2,3,4,5,6,7,8,9,10]] => [10] => [10]
[9,1] => [[1,3,4,5,6,7,8,9,10],[2]] => [10] => [10]
[8,2] => [[1,2,5,6,7,8,9,10],[3,4]] => [2,8] => [2,8]
[8,1,1] => [[1,4,5,6,7,8,9,10],[2],[3]] => [10] => [10]
[7,3] => [[1,2,3,7,8,9,10],[4,5,6]] => [3,7] => [3,7]
>>> Load all 138 entries. <<<
[7,2,1] => [[1,3,6,7,8,9,10],[2,5],[4]] => [3,7] => [3,7]
[7,1,1,1] => [[1,5,6,7,8,9,10],[2],[3],[4]] => [10] => [10]
[6,4] => [[1,2,3,4,9,10],[5,6,7,8]] => [4,6] => [4,6]
[6,3,1] => [[1,3,4,8,9,10],[2,6,7],[5]] => [4,6] => [4,6]
[6,2,2] => [[1,2,7,8,9,10],[3,4],[5,6]] => [2,2,6] => [2,2,6]
[6,2,1,1] => [[1,4,7,8,9,10],[2,6],[3],[5]] => [4,6] => [4,6]
[6,1,1,1,1] => [[1,6,7,8,9,10],[2],[3],[4],[5]] => [10] => [10]
[5,5] => [[1,2,3,4,5],[6,7,8,9,10]] => [5,5] => [5,5]
[5,4,1] => [[1,3,4,5,10],[2,7,8,9],[6]] => [5,5] => [5,5]
[5,3,2] => [[1,2,5,9,10],[3,4,8],[6,7]] => [2,3,5] => [2,3,5]
[5,3,1,1] => [[1,4,5,9,10],[2,7,8],[3],[6]] => [5,5] => [5,5]
[5,2,2,1] => [[1,3,8,9,10],[2,5],[4,7],[6]] => [3,2,5] => [3,2,5]
[5,2,1,1,1] => [[1,5,8,9,10],[2,7],[3],[4],[6]] => [5,5] => [5,5]
[5,1,1,1,1,1] => [[1,7,8,9,10],[2],[3],[4],[5],[6]] => [10] => [10]
[4,4,2] => [[1,2,5,6],[3,4,9,10],[7,8]] => [2,4,4] => [2,4,4]
[4,4,1,1] => [[1,4,5,6],[2,8,9,10],[3],[7]] => [6,4] => [6,4]
[4,3,3] => [[1,2,3,10],[4,5,6],[7,8,9]] => [3,3,4] => [3,3,4]
[4,3,2,1] => [[1,3,6,10],[2,5,9],[4,8],[7]] => [3,3,4] => [3,3,4]
[4,3,1,1,1] => [[1,5,6,10],[2,8,9],[3],[4],[7]] => [6,4] => [6,4]
[4,2,2,2] => [[1,2,9,10],[3,4],[5,6],[7,8]] => [2,2,2,4] => [2,2,2,4]
[4,2,2,1,1] => [[1,4,9,10],[2,6],[3,8],[5],[7]] => [4,2,4] => [4,2,4]
[4,2,1,1,1,1] => [[1,6,9,10],[2,8],[3],[4],[5],[7]] => [6,4] => [6,4]
[4,1,1,1,1,1,1] => [[1,8,9,10],[2],[3],[4],[5],[6],[7]] => [10] => [10]
[3,3,3,1] => [[1,3,4],[2,6,7],[5,9,10],[8]] => [4,3,3] => [3,4,3]
[3,3,2,2] => [[1,2,7],[3,4,10],[5,6],[8,9]] => [2,2,3,3] => [2,2,3,3]
[3,3,2,1,1] => [[1,4,7],[2,6,10],[3,9],[5],[8]] => [4,3,3] => [3,4,3]
[3,3,1,1,1,1] => [[1,6,7],[2,9,10],[3],[4],[5],[8]] => [7,3] => [7,3]
[3,2,2,2,1] => [[1,3,10],[2,5],[4,7],[6,9],[8]] => [3,2,2,3] => [2,3,2,3]
[3,2,2,1,1,1] => [[1,5,10],[2,7],[3,9],[4],[6],[8]] => [5,2,3] => [2,5,3]
[3,2,1,1,1,1,1] => [[1,7,10],[2,9],[3],[4],[5],[6],[8]] => [7,3] => [7,3]
[3,1,1,1,1,1,1,1] => [[1,9,10],[2],[3],[4],[5],[6],[7],[8]] => [10] => [10]
[2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10]] => [2,2,2,2,2] => [2,2,2,2,2]
[2,2,2,2,1,1] => [[1,4],[2,6],[3,8],[5,10],[7],[9]] => [4,2,2,2] => [2,2,4,2]
[2,2,2,1,1,1,1] => [[1,6],[2,8],[3,10],[4],[5],[7],[9]] => [6,2,2] => [2,6,2]
[2,2,1,1,1,1,1,1] => [[1,8],[2,10],[3],[4],[5],[6],[7],[9]] => [8,2] => [8,2]
[2,1,1,1,1,1,1,1,1] => [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]] => [10] => [10]
[1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => [10] => [10]
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
peak composition
Description
The composition corresponding to the peak set of a standard tableau.
Let $T$ be a standard tableau of size $n$.
An entry $i$ of $T$ is a descent if $i+1$ is in a lower row (in English notation), otherwise $i$ is an ascent.
An entry $2 \leq i \leq n-1$ is a peak, if $i-1$ is an ascent and $i$ is a descent.
This map returns the composition $c_1,\dots,c_k$ of $n$ such that $\{c_1, c_1+c_2,\dots, c_1+\dots+c_k\}$ is the peak set of $T$.
Map
Foata bijection
Description
The Foata bijection for compositions.
The Foata bijection $\phi$ is a bijection on the set of words whose letters are positive integers. 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 $w_{i+1} < v_k$.
In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.
To compute $\phi([1,4,2,5,3])$, the sequence of words is
  • $1$
  • $|1|4 \to 14$
  • $|14|2 \to 412$
  • $|4|1|2|5 \to 4125$
  • $|4|125|3 \to 45123.$
In total, this gives $\phi([1,4,2,5,3]) = [4,5,1,2,3]$.
This bijection sends the major index St000769The major index of a composition regarded as a word. to the number of inversions St000766The number of inversions of an integer composition..