Identifier
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00189: Skew partitions —rotate⟶ Skew partitions
Images
=>
Cc0028;cc-rep-1Cc0028;cc-rep-2
[1]=>[[1],[]]=>[[1],[]] [1,1]=>[[1,1],[]]=>[[1,1],[]] [2]=>[[2],[]]=>[[2],[]] [1,1,1]=>[[1,1,1],[]]=>[[1,1,1],[]] [1,2]=>[[2,1],[]]=>[[2,2],[1]] [2,1]=>[[2,2],[1]]=>[[2,1],[]] [3]=>[[3],[]]=>[[3],[]] [1,1,1,1]=>[[1,1,1,1],[]]=>[[1,1,1,1],[]] [1,1,2]=>[[2,1,1],[]]=>[[2,2,2],[1,1]] [1,2,1]=>[[2,2,1],[1]]=>[[2,2,1],[1]] [1,3]=>[[3,1],[]]=>[[3,3],[2]] [2,1,1]=>[[2,2,2],[1,1]]=>[[2,1,1],[]] [2,2]=>[[3,2],[1]]=>[[3,2],[1]] [3,1]=>[[3,3],[2]]=>[[3,1],[]] [4]=>[[4],[]]=>[[4],[]] [1,1,1,1,1]=>[[1,1,1,1,1],[]]=>[[1,1,1,1,1],[]] [1,1,1,2]=>[[2,1,1,1],[]]=>[[2,2,2,2],[1,1,1]] [1,1,2,1]=>[[2,2,1,1],[1]]=>[[2,2,2,1],[1,1]] [1,1,3]=>[[3,1,1],[]]=>[[3,3,3],[2,2]] [1,2,1,1]=>[[2,2,2,1],[1,1]]=>[[2,2,1,1],[1]] [1,2,2]=>[[3,2,1],[1]]=>[[3,3,2],[2,1]] [1,3,1]=>[[3,3,1],[2]]=>[[3,3,1],[2]] [1,4]=>[[4,1],[]]=>[[4,4],[3]] [2,1,1,1]=>[[2,2,2,2],[1,1,1]]=>[[2,1,1,1],[]] [2,1,2]=>[[3,2,2],[1,1]]=>[[3,2,2],[1,1]] [2,2,1]=>[[3,3,2],[2,1]]=>[[3,2,1],[1]] [2,3]=>[[4,2],[1]]=>[[4,3],[2]] [3,1,1]=>[[3,3,3],[2,2]]=>[[3,1,1],[]] [3,2]=>[[4,3],[2]]=>[[4,2],[1]] [4,1]=>[[4,4],[3]]=>[[4,1],[]] [5]=>[[5],[]]=>[[5],[]] [1,1,1,1,1,1]=>[[1,1,1,1,1,1],[]]=>[[1,1,1,1,1,1],[]] [1,1,1,1,2]=>[[2,1,1,1,1],[]]=>[[2,2,2,2,2],[1,1,1,1]] [1,1,1,2,1]=>[[2,2,1,1,1],[1]]=>[[2,2,2,2,1],[1,1,1]] [1,1,1,3]=>[[3,1,1,1],[]]=>[[3,3,3,3],[2,2,2]] [1,1,2,1,1]=>[[2,2,2,1,1],[1,1]]=>[[2,2,2,1,1],[1,1]] [1,1,2,2]=>[[3,2,1,1],[1]]=>[[3,3,3,2],[2,2,1]] [1,1,3,1]=>[[3,3,1,1],[2]]=>[[3,3,3,1],[2,2]] [1,1,4]=>[[4,1,1],[]]=>[[4,4,4],[3,3]] [1,2,1,1,1]=>[[2,2,2,2,1],[1,1,1]]=>[[2,2,1,1,1],[1]] [1,2,1,2]=>[[3,2,2,1],[1,1]]=>[[3,3,2,2],[2,1,1]] [1,2,2,1]=>[[3,3,2,1],[2,1]]=>[[3,3,2,1],[2,1]] [1,2,3]=>[[4,2,1],[1]]=>[[4,4,3],[3,2]] [1,3,1,1]=>[[3,3,3,1],[2,2]]=>[[3,3,1,1],[2]] [1,3,2]=>[[4,3,1],[2]]=>[[4,4,2],[3,1]] [1,4,1]=>[[4,4,1],[3]]=>[[4,4,1],[3]] [1,5]=>[[5,1],[]]=>[[5,5],[4]] [2,1,1,1,1]=>[[2,2,2,2,2],[1,1,1,1]]=>[[2,1,1,1,1],[]] [2,1,1,2]=>[[3,2,2,2],[1,1,1]]=>[[3,2,2,2],[1,1,1]] [2,1,2,1]=>[[3,3,2,2],[2,1,1]]=>[[3,2,2,1],[1,1]] [2,1,3]=>[[4,2,2],[1,1]]=>[[4,3,3],[2,2]] [2,2,1,1]=>[[3,3,3,2],[2,2,1]]=>[[3,2,1,1],[1]] [2,2,2]=>[[4,3,2],[2,1]]=>[[4,3,2],[2,1]] [2,3,1]=>[[4,4,2],[3,1]]=>[[4,3,1],[2]] [2,4]=>[[5,2],[1]]=>[[5,4],[3]] [3,1,1,1]=>[[3,3,3,3],[2,2,2]]=>[[3,1,1,1],[]] [3,1,2]=>[[4,3,3],[2,2]]=>[[4,2,2],[1,1]] [3,2,1]=>[[4,4,3],[3,2]]=>[[4,2,1],[1]] [3,3]=>[[5,3],[2]]=>[[5,3],[2]] [4,1,1]=>[[4,4,4],[3,3]]=>[[4,1,1],[]] [4,2]=>[[5,4],[3]]=>[[5,2],[1]] [5,1]=>[[5,5],[4]]=>[[5,1],[]] [6]=>[[6],[]]=>[[6],[]] [1,1,1,1,1,1,1]=>[[1,1,1,1,1,1,1],[]]=>[[1,1,1,1,1,1,1],[]] [1,1,1,1,1,2]=>[[2,1,1,1,1,1],[]]=>[[2,2,2,2,2,2],[1,1,1,1,1]] [1,1,1,1,2,1]=>[[2,2,1,1,1,1],[1]]=>[[2,2,2,2,2,1],[1,1,1,1]] [1,1,1,1,3]=>[[3,1,1,1,1],[]]=>[[3,3,3,3,3],[2,2,2,2]] [1,1,1,2,1,1]=>[[2,2,2,1,1,1],[1,1]]=>[[2,2,2,2,1,1],[1,1,1]] [1,1,1,2,2]=>[[3,2,1,1,1],[1]]=>[[3,3,3,3,2],[2,2,2,1]] [1,1,1,3,1]=>[[3,3,1,1,1],[2]]=>[[3,3,3,3,1],[2,2,2]] [1,1,1,4]=>[[4,1,1,1],[]]=>[[4,4,4,4],[3,3,3]] [1,1,2,1,1,1]=>[[2,2,2,2,1,1],[1,1,1]]=>[[2,2,2,1,1,1],[1,1]] [1,1,2,1,2]=>[[3,2,2,1,1],[1,1]]=>[[3,3,3,2,2],[2,2,1,1]] [1,1,2,2,1]=>[[3,3,2,1,1],[2,1]]=>[[3,3,3,2,1],[2,2,1]] [1,1,2,3]=>[[4,2,1,1],[1]]=>[[4,4,4,3],[3,3,2]] [1,1,3,1,1]=>[[3,3,3,1,1],[2,2]]=>[[3,3,3,1,1],[2,2]] [1,1,3,2]=>[[4,3,1,1],[2]]=>[[4,4,4,2],[3,3,1]] [1,1,4,1]=>[[4,4,1,1],[3]]=>[[4,4,4,1],[3,3]] [1,1,5]=>[[5,1,1],[]]=>[[5,5,5],[4,4]] [1,2,1,1,1,1]=>[[2,2,2,2,2,1],[1,1,1,1]]=>[[2,2,1,1,1,1],[1]] [1,2,1,1,2]=>[[3,2,2,2,1],[1,1,1]]=>[[3,3,2,2,2],[2,1,1,1]] [1,2,1,2,1]=>[[3,3,2,2,1],[2,1,1]]=>[[3,3,2,2,1],[2,1,1]] [1,2,1,3]=>[[4,2,2,1],[1,1]]=>[[4,4,3,3],[3,2,2]] [1,2,2,1,1]=>[[3,3,3,2,1],[2,2,1]]=>[[3,3,2,1,1],[2,1]] [1,2,2,2]=>[[4,3,2,1],[2,1]]=>[[4,4,3,2],[3,2,1]] [1,2,3,1]=>[[4,4,2,1],[3,1]]=>[[4,4,3,1],[3,2]] [1,2,4]=>[[5,2,1],[1]]=>[[5,5,4],[4,3]] [1,3,1,1,1]=>[[3,3,3,3,1],[2,2,2]]=>[[3,3,1,1,1],[2]] [1,3,1,2]=>[[4,3,3,1],[2,2]]=>[[4,4,2,2],[3,1,1]] [1,3,2,1]=>[[4,4,3,1],[3,2]]=>[[4,4,2,1],[3,1]] [1,3,3]=>[[5,3,1],[2]]=>[[5,5,3],[4,2]] [1,4,1,1]=>[[4,4,4,1],[3,3]]=>[[4,4,1,1],[3]] [1,4,2]=>[[5,4,1],[3]]=>[[5,5,2],[4,1]] [1,5,1]=>[[5,5,1],[4]]=>[[5,5,1],[4]] [1,6]=>[[6,1],[]]=>[[6,6],[5]] [2,1,1,1,1,1]=>[[2,2,2,2,2,2],[1,1,1,1,1]]=>[[2,1,1,1,1,1],[]] [2,1,1,1,2]=>[[3,2,2,2,2],[1,1,1,1]]=>[[3,2,2,2,2],[1,1,1,1]] [2,1,1,2,1]=>[[3,3,2,2,2],[2,1,1,1]]=>[[3,2,2,2,1],[1,1,1]] [2,1,1,3]=>[[4,2,2,2],[1,1,1]]=>[[4,3,3,3],[2,2,2]] [2,1,2,1,1]=>[[3,3,3,2,2],[2,2,1,1]]=>[[3,2,2,1,1],[1,1]] [2,1,2,2]=>[[4,3,2,2],[2,1,1]]=>[[4,3,3,2],[2,2,1]] [2,1,3,1]=>[[4,4,2,2],[3,1,1]]=>[[4,3,3,1],[2,2]] [2,1,4]=>[[5,2,2],[1,1]]=>[[5,4,4],[3,3]] [2,2,1,1,1]=>[[3,3,3,3,2],[2,2,2,1]]=>[[3,2,1,1,1],[1]] [2,2,1,2]=>[[4,3,3,2],[2,2,1]]=>[[4,3,2,2],[2,1,1]] [2,2,2,1]=>[[4,4,3,2],[3,2,1]]=>[[4,3,2,1],[2,1]] [2,2,3]=>[[5,3,2],[2,1]]=>[[5,4,3],[3,2]] [2,3,1,1]=>[[4,4,4,2],[3,3,1]]=>[[4,3,1,1],[2]] [2,3,2]=>[[5,4,2],[3,1]]=>[[5,4,2],[3,1]] [2,4,1]=>[[5,5,2],[4,1]]=>[[5,4,1],[3]] [2,5]=>[[6,2],[1]]=>[[6,5],[4]] [3,1,1,1,1]=>[[3,3,3,3,3],[2,2,2,2]]=>[[3,1,1,1,1],[]] [3,1,1,2]=>[[4,3,3,3],[2,2,2]]=>[[4,2,2,2],[1,1,1]] [3,1,2,1]=>[[4,4,3,3],[3,2,2]]=>[[4,2,2,1],[1,1]] [3,1,3]=>[[5,3,3],[2,2]]=>[[5,3,3],[2,2]] [3,2,1,1]=>[[4,4,4,3],[3,3,2]]=>[[4,2,1,1],[1]] [3,2,2]=>[[5,4,3],[3,2]]=>[[5,3,2],[2,1]] [3,3,1]=>[[5,5,3],[4,2]]=>[[5,3,1],[2]] [3,4]=>[[6,3],[2]]=>[[6,4],[3]] [4,1,1,1]=>[[4,4,4,4],[3,3,3]]=>[[4,1,1,1],[]] [4,1,2]=>[[5,4,4],[3,3]]=>[[5,2,2],[1,1]] [4,2,1]=>[[5,5,4],[4,3]]=>[[5,2,1],[1]] [4,3]=>[[6,4],[3]]=>[[6,3],[2]] [5,1,1]=>[[5,5,5],[4,4]]=>[[5,1,1],[]] [5,2]=>[[6,5],[4]]=>[[6,2],[1]] [6,1]=>[[6,6],[5]]=>[[6,1],[]] [7]=>[[7],[]]=>[[7],[]] [1,1,1,1,1,1,1,1]=>[[1,1,1,1,1,1,1,1],[]]=>[[1,1,1,1,1,1,1,1],[]] [1,1,1,1,1,1,2]=>[[2,1,1,1,1,1,1],[]]=>[[2,2,2,2,2,2,2],[1,1,1,1,1,1]] [1,1,1,1,1,3]=>[[3,1,1,1,1,1],[]]=>[[3,3,3,3,3,3],[2,2,2,2,2]] [1,1,1,1,4]=>[[4,1,1,1,1],[]]=>[[4,4,4,4,4],[3,3,3,3]] [1,1,1,5]=>[[5,1,1,1],[]]=>[[5,5,5,5],[4,4,4]] [1,1,6]=>[[6,1,1],[]]=>[[6,6,6],[5,5]] [1,7]=>[[7,1],[]]=>[[7,7],[6]] [8]=>[[8],[]]=>[[8],[]] [1,1,1,1,1,1,1,1,1]=>[[1,1,1,1,1,1,1,1,1],[]]=>[[1,1,1,1,1,1,1,1,1],[]] [1,1,1,1,1,1,1,2]=>[[2,1,1,1,1,1,1,1],[]]=>[[2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1]] [1,1,1,1,1,1,3]=>[[3,1,1,1,1,1,1],[]]=>[[3,3,3,3,3,3,3],[2,2,2,2,2,2]] [1,1,1,1,1,4]=>[[4,1,1,1,1,1],[]]=>[[4,4,4,4,4,4],[3,3,3,3,3]] [1,1,1,1,5]=>[[5,1,1,1,1],[]]=>[[5,5,5,5,5],[4,4,4,4]] [1,1,1,6]=>[[6,1,1,1],[]]=>[[6,6,6,6],[5,5,5]] [1,1,7]=>[[7,1,1],[]]=>[[7,7,7],[6,6]] [1,8]=>[[8,1],[]]=>[[8,8],[7]] [9]=>[[9],[]]=>[[9],[]] [1,1,1,1,1,1,1,1,1,1]=>[[1,1,1,1,1,1,1,1,1,1],[]]=>[[1,1,1,1,1,1,1,1,1,1],[]] [1,1,1,1,1,1,1,1,2]=>[[2,1,1,1,1,1,1,1,1],[]]=>[[2,2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1,1]] [1,1,1,1,1,1,1,3]=>[[3,1,1,1,1,1,1,1],[]]=>[[3,3,3,3,3,3,3,3],[2,2,2,2,2,2,2]] [1,1,1,1,1,1,4]=>[[4,1,1,1,1,1,1],[]]=>[[4,4,4,4,4,4,4],[3,3,3,3,3,3]] [1,1,1,1,1,5]=>[[5,1,1,1,1,1],[]]=>[[5,5,5,5,5,5],[4,4,4,4,4]] [1,1,1,1,6]=>[[6,1,1,1,1],[]]=>[[6,6,6,6,6],[5,5,5,5]] [1,1,1,7]=>[[7,1,1,1],[]]=>[[7,7,7,7],[6,6,6]] [1,1,8]=>[[8,1,1],[]]=>[[8,8,8],[7,7]] [1,9]=>[[9,1],[]]=>[[9,9],[8]] [10]=>[[10],[]]=>[[10],[]] [1,10]=>[[10,1],[]]=>[[10,10],[9]] [1,1,1,1,1,1,1,1,1,1,1,1]=>[[1,1,1,1,1,1,1,1,1,1,1,1],[]]=>[[1,1,1,1,1,1,1,1,1,1,1,1],[]] [1,1,1,1,1,1,1,1,1,2]=>[[2,1,1,1,1,1,1,1,1,1],[]]=>[[2,2,2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1,1,1]] [12]=>[[12],[]]=>[[12],[]] [11]=>[[11],[]]=>[[11],[]] [1,1,1,1,1,1,1,1,1,1,1]=>[[1,1,1,1,1,1,1,1,1,1,1],[]]=>[[1,1,1,1,1,1,1,1,1,1,1],[]] [1,1,1,1,1,1,1,1,1,1,1,1,1]=>[[1,1,1,1,1,1,1,1,1,1,1,1,1],[]]=>[[1,1,1,1,1,1,1,1,1,1,1,1,1],[]]
Map
to ribbon
Description
The ribbon shape corresponding to an integer composition.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
Map
rotate
Description
The rotation of a skew partition.
This is the skew partition obtained by rotating the diagram by 180 degrees. Equivalently, given a skew partition $\lambda/\mu$, its rotation $(\lambda/\mu)^\natural$ is the skew partition with cells $\{(a-i, b-j)| (i, j) \in \lambda/\mu\}$, where $b$ and $a$ are the first part and the number of parts of $\lambda$ respectively.