Identifier
Values
{{1,2}} => {{1,2}} => [2,1] => [1,2] => 0
{{1},{2}} => {{1},{2}} => [1,2] => [1,2] => 0
{{1,2,3}} => {{1,2,3}} => [2,3,1] => [1,2,3] => 0
{{1,2},{3}} => {{1},{2,3}} => [1,3,2] => [1,2,3] => 0
{{1,3},{2}} => {{1,3},{2}} => [3,2,1] => [1,3,2] => 1
{{1},{2,3}} => {{1,2},{3}} => [2,1,3] => [1,2,3] => 0
{{1},{2},{3}} => {{1},{2},{3}} => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}} => {{1,2,3,4}} => [2,3,4,1] => [1,2,3,4] => 0
{{1,2,3},{4}} => {{1},{2,3,4}} => [1,3,4,2] => [1,2,3,4] => 0
{{1,2,4},{3}} => {{1,3},{2,4}} => [3,4,1,2] => [1,3,2,4] => 1
{{1,2},{3,4}} => {{1,2},{3,4}} => [2,1,4,3] => [1,2,3,4] => 0
{{1,2},{3},{4}} => {{1},{2},{3,4}} => [1,2,4,3] => [1,2,3,4] => 0
{{1,3,4},{2}} => {{1,2,4},{3}} => [2,4,3,1] => [1,2,4,3] => 1
{{1,3},{2,4}} => {{1,3,4},{2}} => [3,2,4,1] => [1,3,4,2] => 2
{{1,3},{2},{4}} => {{1},{2,4},{3}} => [1,4,3,2] => [1,2,4,3] => 1
{{1,4},{2,3}} => {{1,4},{2,3}} => [4,3,2,1] => [1,4,2,3] => 2
{{1},{2,3,4}} => {{1,2,3},{4}} => [2,3,1,4] => [1,2,3,4] => 0
{{1},{2,3},{4}} => {{1},{2,3},{4}} => [1,3,2,4] => [1,2,3,4] => 0
{{1,4},{2},{3}} => {{1,4},{2},{3}} => [4,2,3,1] => [1,4,2,3] => 2
{{1},{2,4},{3}} => {{1,3},{2},{4}} => [3,2,1,4] => [1,3,2,4] => 1
{{1},{2},{3,4}} => {{1,2},{3},{4}} => [2,1,3,4] => [1,2,3,4] => 0
{{1},{2},{3},{4}} => {{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => 0
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Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
Wachs-White
Description
A transformation of set partitions due to Wachs and White.
Return the set partition of $\{1,...,n\}$ corresponding to the set of arcs, interpreted as a rook placement, applying Wachs and White's bijection $\gamma$.
Note that our index convention differs from the convention in [1]: regarding the rook board as a lower-right triangular grid, we refer with $(i,j)$ to the cell in the $i$-th column from the right and the $j$-th row from the top.
Map
cycle-as-one-line notation
Description
Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.