Identifier
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00226: Standard tableaux —row-to-column-descents⟶ Standard tableaux
Images
[1] => [[1]] => [[1]]
[2] => [[1,2]] => [[1,2]]
[1,1] => [[1],[2]] => [[1],[2]]
[3] => [[1,2,3]] => [[1,2,3]]
[2,1] => [[1,2],[3]] => [[1,3],[2]]
[1,1,1] => [[1],[2],[3]] => [[1],[2],[3]]
[4] => [[1,2,3,4]] => [[1,2,3,4]]
[3,1] => [[1,2,3],[4]] => [[1,3,4],[2]]
[2,2] => [[1,2],[3,4]] => [[1,3],[2,4]]
[2,1,1] => [[1,2],[3],[4]] => [[1,4],[2],[3]]
[1,1,1,1] => [[1],[2],[3],[4]] => [[1],[2],[3],[4]]
[5] => [[1,2,3,4,5]] => [[1,2,3,4,5]]
[4,1] => [[1,2,3,4],[5]] => [[1,3,4,5],[2]]
[3,2] => [[1,2,3],[4,5]] => [[1,3,5],[2,4]]
[3,1,1] => [[1,2,3],[4],[5]] => [[1,4,5],[2],[3]]
[2,2,1] => [[1,2],[3,4],[5]] => [[1,4],[2,5],[3]]
[2,1,1,1] => [[1,2],[3],[4],[5]] => [[1,5],[2],[3],[4]]
[1,1,1,1,1] => [[1],[2],[3],[4],[5]] => [[1],[2],[3],[4],[5]]
[6] => [[1,2,3,4,5,6]] => [[1,2,3,4,5,6]]
[5,1] => [[1,2,3,4,5],[6]] => [[1,3,4,5,6],[2]]
[4,2] => [[1,2,3,4],[5,6]] => [[1,3,5,6],[2,4]]
[4,1,1] => [[1,2,3,4],[5],[6]] => [[1,4,5,6],[2],[3]]
[3,3] => [[1,2,3],[4,5,6]] => [[1,3,5],[2,4,6]]
[3,2,1] => [[1,2,3],[4,5],[6]] => [[1,4,6],[2,5],[3]]
[3,1,1,1] => [[1,2,3],[4],[5],[6]] => [[1,5,6],[2],[3],[4]]
[2,2,2] => [[1,2],[3,4],[5,6]] => [[1,4],[2,5],[3,6]]
[2,2,1,1] => [[1,2],[3,4],[5],[6]] => [[1,5],[2,6],[3],[4]]
[2,1,1,1,1] => [[1,2],[3],[4],[5],[6]] => [[1,6],[2],[3],[4],[5]]
[1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6]] => [[1],[2],[3],[4],[5],[6]]
[7] => [[1,2,3,4,5,6,7]] => [[1,2,3,4,5,6,7]]
[6,1] => [[1,2,3,4,5,6],[7]] => [[1,3,4,5,6,7],[2]]
[5,2] => [[1,2,3,4,5],[6,7]] => [[1,3,5,6,7],[2,4]]
[5,1,1] => [[1,2,3,4,5],[6],[7]] => [[1,4,5,6,7],[2],[3]]
[4,3] => [[1,2,3,4],[5,6,7]] => [[1,3,5,7],[2,4,6]]
[4,2,1] => [[1,2,3,4],[5,6],[7]] => [[1,4,6,7],[2,5],[3]]
[4,1,1,1] => [[1,2,3,4],[5],[6],[7]] => [[1,5,6,7],[2],[3],[4]]
[3,3,1] => [[1,2,3],[4,5,6],[7]] => [[1,4,6],[2,5,7],[3]]
[3,2,2] => [[1,2,3],[4,5],[6,7]] => [[1,4,7],[2,5],[3,6]]
[3,2,1,1] => [[1,2,3],[4,5],[6],[7]] => [[1,5,7],[2,6],[3],[4]]
[3,1,1,1,1] => [[1,2,3],[4],[5],[6],[7]] => [[1,6,7],[2],[3],[4],[5]]
[2,2,2,1] => [[1,2],[3,4],[5,6],[7]] => [[1,5],[2,6],[3,7],[4]]
[2,2,1,1,1] => [[1,2],[3,4],[5],[6],[7]] => [[1,6],[2,7],[3],[4],[5]]
[2,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7]] => [[1,7],[2],[3],[4],[5],[6]]
[1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7]] => [[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] => [[1,2,3,4,5,6,7],[8]] => [[1,3,4,5,6,7,8],[2]]
[6,2] => [[1,2,3,4,5,6],[7,8]] => [[1,3,5,6,7,8],[2,4]]
[6,1,1] => [[1,2,3,4,5,6],[7],[8]] => [[1,4,5,6,7,8],[2],[3]]
[5,3] => [[1,2,3,4,5],[6,7,8]] => [[1,3,5,7,8],[2,4,6]]
[5,2,1] => [[1,2,3,4,5],[6,7],[8]] => [[1,4,6,7,8],[2,5],[3]]
[5,1,1,1] => [[1,2,3,4,5],[6],[7],[8]] => [[1,5,6,7,8],[2],[3],[4]]
[4,4] => [[1,2,3,4],[5,6,7,8]] => [[1,3,5,7],[2,4,6,8]]
[4,3,1] => [[1,2,3,4],[5,6,7],[8]] => [[1,4,6,8],[2,5,7],[3]]
[4,2,2] => [[1,2,3,4],[5,6],[7,8]] => [[1,4,7,8],[2,5],[3,6]]
[4,2,1,1] => [[1,2,3,4],[5,6],[7],[8]] => [[1,5,7,8],[2,6],[3],[4]]
[4,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8]] => [[1,6,7,8],[2],[3],[4],[5]]
[3,3,2] => [[1,2,3],[4,5,6],[7,8]] => [[1,4,7],[2,5,8],[3,6]]
[3,3,1,1] => [[1,2,3],[4,5,6],[7],[8]] => [[1,5,7],[2,6,8],[3],[4]]
[3,2,2,1] => [[1,2,3],[4,5],[6,7],[8]] => [[1,5,8],[2,6],[3,7],[4]]
[3,2,1,1,1] => [[1,2,3],[4,5],[6],[7],[8]] => [[1,6,8],[2,7],[3],[4],[5]]
[3,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8]] => [[1,7,8],[2],[3],[4],[5],[6]]
[2,2,2,2] => [[1,2],[3,4],[5,6],[7,8]] => [[1,5],[2,6],[3,7],[4,8]]
[2,2,2,1,1] => [[1,2],[3,4],[5,6],[7],[8]] => [[1,6],[2,7],[3,8],[4],[5]]
[2,2,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8]] => [[1,7],[2,8],[3],[4],[5],[6]]
[2,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8]] => [[1,8],[2],[3],[4],[5],[6],[7]]
[1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8]] => [[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] => [[1,2,3,4,5,6,7,8],[9]] => [[1,3,4,5,6,7,8,9],[2]]
[7,2] => [[1,2,3,4,5,6,7],[8,9]] => [[1,3,5,6,7,8,9],[2,4]]
[7,1,1] => [[1,2,3,4,5,6,7],[8],[9]] => [[1,4,5,6,7,8,9],[2],[3]]
[6,3] => [[1,2,3,4,5,6],[7,8,9]] => [[1,3,5,7,8,9],[2,4,6]]
[6,2,1] => [[1,2,3,4,5,6],[7,8],[9]] => [[1,4,6,7,8,9],[2,5],[3]]
[6,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9]] => [[1,5,6,7,8,9],[2],[3],[4]]
[5,4] => [[1,2,3,4,5],[6,7,8,9]] => [[1,3,5,7,9],[2,4,6,8]]
[5,3,1] => [[1,2,3,4,5],[6,7,8],[9]] => [[1,4,6,8,9],[2,5,7],[3]]
[5,2,2] => [[1,2,3,4,5],[6,7],[8,9]] => [[1,4,7,8,9],[2,5],[3,6]]
[5,2,1,1] => [[1,2,3,4,5],[6,7],[8],[9]] => [[1,5,7,8,9],[2,6],[3],[4]]
[5,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9]] => [[1,6,7,8,9],[2],[3],[4],[5]]
[4,4,1] => [[1,2,3,4],[5,6,7,8],[9]] => [[1,4,6,8],[2,5,7,9],[3]]
[4,3,2] => [[1,2,3,4],[5,6,7],[8,9]] => [[1,4,7,9],[2,5,8],[3,6]]
[4,3,1,1] => [[1,2,3,4],[5,6,7],[8],[9]] => [[1,5,7,9],[2,6,8],[3],[4]]
[4,2,2,1] => [[1,2,3,4],[5,6],[7,8],[9]] => [[1,5,8,9],[2,6],[3,7],[4]]
[4,2,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9]] => [[1,6,8,9],[2,7],[3],[4],[5]]
[4,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9]] => [[1,7,8,9],[2],[3],[4],[5],[6]]
[3,3,3] => [[1,2,3],[4,5,6],[7,8,9]] => [[1,4,7],[2,5,8],[3,6,9]]
[3,3,2,1] => [[1,2,3],[4,5,6],[7,8],[9]] => [[1,5,8],[2,6,9],[3,7],[4]]
[3,3,1,1,1] => [[1,2,3],[4,5,6],[7],[8],[9]] => [[1,6,8],[2,7,9],[3],[4],[5]]
[3,2,2,2] => [[1,2,3],[4,5],[6,7],[8,9]] => [[1,5,9],[2,6],[3,7],[4,8]]
[3,2,2,1,1] => [[1,2,3],[4,5],[6,7],[8],[9]] => [[1,6,9],[2,7],[3,8],[4],[5]]
[3,2,1,1,1,1] => [[1,2,3],[4,5],[6],[7],[8],[9]] => [[1,7,9],[2,8],[3],[4],[5],[6]]
[3,1,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8],[9]] => [[1,8,9],[2],[3],[4],[5],[6],[7]]
[2,2,2,2,1] => [[1,2],[3,4],[5,6],[7,8],[9]] => [[1,6],[2,7],[3,8],[4,9],[5]]
[2,2,2,1,1,1] => [[1,2],[3,4],[5,6],[7],[8],[9]] => [[1,7],[2,8],[3,9],[4],[5],[6]]
[2,2,1,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8],[9]] => [[1,8],[2,9],[3],[4],[5],[6],[7]]
[2,1,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9]] => [[1,9],[2],[3],[4],[5],[6],[7],[8]]
[1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9]] => [[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] => [[1,2,3,4,5,6,7,8,9],[10]] => [[1,3,4,5,6,7,8,9,10],[2]]
[8,2] => [[1,2,3,4,5,6,7,8],[9,10]] => [[1,3,5,6,7,8,9,10],[2,4]]
[8,1,1] => [[1,2,3,4,5,6,7,8],[9],[10]] => [[1,4,5,6,7,8,9,10],[2],[3]]
[7,3] => [[1,2,3,4,5,6,7],[8,9,10]] => [[1,3,5,7,8,9,10],[2,4,6]]
>>> Load all 148 entries. <<<
[7,2,1] => [[1,2,3,4,5,6,7],[8,9],[10]] => [[1,4,6,7,8,9,10],[2,5],[3]]
[7,1,1,1] => [[1,2,3,4,5,6,7],[8],[9],[10]] => [[1,5,6,7,8,9,10],[2],[3],[4]]
[6,4] => [[1,2,3,4,5,6],[7,8,9,10]] => [[1,3,5,7,9,10],[2,4,6,8]]
[6,3,1] => [[1,2,3,4,5,6],[7,8,9],[10]] => [[1,4,6,8,9,10],[2,5,7],[3]]
[6,2,2] => [[1,2,3,4,5,6],[7,8],[9,10]] => [[1,4,7,8,9,10],[2,5],[3,6]]
[6,2,1,1] => [[1,2,3,4,5,6],[7,8],[9],[10]] => [[1,5,7,8,9,10],[2,6],[3],[4]]
[6,1,1,1,1] => [[1,2,3,4,5,6],[7],[8],[9],[10]] => [[1,6,7,8,9,10],[2],[3],[4],[5]]
[5,5] => [[1,2,3,4,5],[6,7,8,9,10]] => [[1,3,5,7,9],[2,4,6,8,10]]
[5,4,1] => [[1,2,3,4,5],[6,7,8,9],[10]] => [[1,4,6,8,10],[2,5,7,9],[3]]
[5,3,2] => [[1,2,3,4,5],[6,7,8],[9,10]] => [[1,4,7,9,10],[2,5,8],[3,6]]
[5,3,1,1] => [[1,2,3,4,5],[6,7,8],[9],[10]] => [[1,5,7,9,10],[2,6,8],[3],[4]]
[5,2,2,1] => [[1,2,3,4,5],[6,7],[8,9],[10]] => [[1,5,8,9,10],[2,6],[3,7],[4]]
[5,2,1,1,1] => [[1,2,3,4,5],[6,7],[8],[9],[10]] => [[1,6,8,9,10],[2,7],[3],[4],[5]]
[5,1,1,1,1,1] => [[1,2,3,4,5],[6],[7],[8],[9],[10]] => [[1,7,8,9,10],[2],[3],[4],[5],[6]]
[4,4,2] => [[1,2,3,4],[5,6,7,8],[9,10]] => [[1,4,7,9],[2,5,8,10],[3,6]]
[4,4,1,1] => [[1,2,3,4],[5,6,7,8],[9],[10]] => [[1,5,7,9],[2,6,8,10],[3],[4]]
[4,3,3] => [[1,2,3,4],[5,6,7],[8,9,10]] => [[1,4,7,10],[2,5,8],[3,6,9]]
[4,3,2,1] => [[1,2,3,4],[5,6,7],[8,9],[10]] => [[1,5,8,10],[2,6,9],[3,7],[4]]
[4,3,1,1,1] => [[1,2,3,4],[5,6,7],[8],[9],[10]] => [[1,6,8,10],[2,7,9],[3],[4],[5]]
[4,2,2,2] => [[1,2,3,4],[5,6],[7,8],[9,10]] => [[1,5,9,10],[2,6],[3,7],[4,8]]
[4,2,2,1,1] => [[1,2,3,4],[5,6],[7,8],[9],[10]] => [[1,6,9,10],[2,7],[3,8],[4],[5]]
[4,2,1,1,1,1] => [[1,2,3,4],[5,6],[7],[8],[9],[10]] => [[1,7,9,10],[2,8],[3],[4],[5],[6]]
[4,1,1,1,1,1,1] => [[1,2,3,4],[5],[6],[7],[8],[9],[10]] => [[1,8,9,10],[2],[3],[4],[5],[6],[7]]
[3,3,3,1] => [[1,2,3],[4,5,6],[7,8,9],[10]] => [[1,5,8],[2,6,9],[3,7,10],[4]]
[3,3,2,2] => [[1,2,3],[4,5,6],[7,8],[9,10]] => [[1,5,9],[2,6,10],[3,7],[4,8]]
[3,3,2,1,1] => [[1,2,3],[4,5,6],[7,8],[9],[10]] => [[1,6,9],[2,7,10],[3,8],[4],[5]]
[3,3,1,1,1,1] => [[1,2,3],[4,5,6],[7],[8],[9],[10]] => [[1,7,9],[2,8,10],[3],[4],[5],[6]]
[3,2,2,2,1] => [[1,2,3],[4,5],[6,7],[8,9],[10]] => [[1,6,10],[2,7],[3,8],[4,9],[5]]
[3,2,2,1,1,1] => [[1,2,3],[4,5],[6,7],[8],[9],[10]] => [[1,7,10],[2,8],[3,9],[4],[5],[6]]
[3,2,1,1,1,1,1] => [[1,2,3],[4,5],[6],[7],[8],[9],[10]] => [[1,8,10],[2,9],[3],[4],[5],[6],[7]]
[3,1,1,1,1,1,1,1] => [[1,2,3],[4],[5],[6],[7],[8],[9],[10]] => [[1,9,10],[2],[3],[4],[5],[6],[7],[8]]
[2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10]] => [[1,6],[2,7],[3,8],[4,9],[5,10]]
[2,2,2,2,1,1] => [[1,2],[3,4],[5,6],[7,8],[9],[10]] => [[1,7],[2,8],[3,9],[4,10],[5],[6]]
[2,2,2,1,1,1,1] => [[1,2],[3,4],[5,6],[7],[8],[9],[10]] => [[1,8],[2,9],[3,10],[4],[5],[6],[7]]
[2,2,1,1,1,1,1,1] => [[1,2],[3,4],[5],[6],[7],[8],[9],[10]] => [[1,9],[2,10],[3],[4],[5],[6],[7],[8]]
[2,1,1,1,1,1,1,1,1] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]] => [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
[1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
[12] => [[1,2,3,4,5,6,7,8,9,10,11,12]] => [[1,2,3,4,5,6,7,8,9,10,11,12]]
[6,6] => [[1,2,3,4,5,6],[7,8,9,10,11,12]] => [[1,3,5,7,9,11],[2,4,6,8,10,12]]
[4,2,2,2,2] => [[1,2,3,4],[5,6],[7,8],[9,10],[11,12]] => [[1,6,11,12],[2,7],[3,8],[4,9],[5,10]]
[2,2,2,2,2,2] => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]] => [[1,7],[2,8],[3,9],[4,10],[5,11],[6,12]]
[1,1,1,1,1,1,1,1,1,1,1,1] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]] => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
[7,7] => [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]] => [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
[8,8] => [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14,15,16]] => [[1,3,5,7,9,11,13,15],[2,4,6,8,10,12,14,16]]
[] => [] => []
[10,10] => [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16,17,18,19,20]] => [[1,3,5,7,9,11,13,15,17,19],[2,4,6,8,10,12,14,16,18,20]]
[9,9] => [[1,2,3,4,5,6,7,8,9],[10,11,12,13,14,15,16,17,18]] => [[1,3,5,7,9,11,13,15,17],[2,4,6,8,10,12,14,16,18]]
Map
initial tableau
Description
Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.
Map
row-to-column-descents
Description
Return a standard tableau whose column descent set equals the row descent set of the original tableau.
A column descent in a standard tableau is an entry $i$ such that $i+1$ appears in a column to the left of the cell containing $i$, in English notation.
A row descent is an entry $i$ such that $i+1$ appears in a row above of the cell containing $i$.