Identifier
Values
{{1,2}} => [2,1] => [2,1] => [2,1] => 1
{{1},{2}} => [1,2] => [1,2] => [1,2] => 0
{{1,2,3}} => [2,3,1] => [3,1,2] => [3,2,1] => 2
{{1,2},{3}} => [2,1,3] => [2,1,3] => [2,1,3] => 1
{{1,3},{2}} => [3,2,1] => [3,2,1] => [2,3,1] => 2
{{1},{2,3}} => [1,3,2] => [1,3,2] => [1,3,2] => 1
{{1},{2},{3}} => [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}} => [2,3,4,1] => [4,1,2,3] => [4,3,2,1] => 3
{{1,2,3},{4}} => [2,3,1,4] => [3,1,2,4] => [3,2,1,4] => 2
{{1,2,4},{3}} => [2,4,3,1] => [4,1,3,2] => [3,4,2,1] => 3
{{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1
{{1,2},{3},{4}} => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
{{1,3,4},{2}} => [3,2,4,1] => [4,2,1,3] => [2,4,3,1] => 3
{{1,3},{2,4}} => [3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 2
{{1,3},{2},{4}} => [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 2
{{1,4},{2,3}} => [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 3
{{1},{2,3,4}} => [1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 2
{{1},{2,3},{4}} => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}} => [4,2,3,1] => [4,2,3,1] => [2,3,4,1] => 3
{{1},{2,4},{3}} => [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 2
{{1},{2},{3,4}} => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
{{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
inverse first fundamental transformation
Description
Let $\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
Map
inverse
Description
Sends a permutation to its inverse.