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
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.
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.