Identifier
Values
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,1,2] => [3,1,2] => 2
[3,1,2] => [3,2,1] => [3,2,1] => 2
[3,2,1] => [2,3,1] => [1,3,2] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,4,2,3] => [1,4,2,3] => 2
[1,4,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[1,4,3,2] => [1,3,4,2] => [1,2,4,3] => 1
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1
[2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 2
[2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 3
[2,4,1,3] => [4,3,1,2] => [4,3,1,2] => 3
[2,4,3,1] => [3,4,1,2] => [2,4,1,3] => 2
[3,1,2,4] => [3,2,1,4] => [3,2,1,4] => 2
[3,1,4,2] => [4,2,1,3] => [4,2,1,3] => 3
[3,2,1,4] => [2,3,1,4] => [1,3,2,4] => 1
[3,2,4,1] => [2,4,1,3] => [2,4,1,3] => 2
[3,4,1,2] => [3,1,4,2] => [2,1,4,3] => 1
[3,4,2,1] => [4,1,3,2] => [4,1,3,2] => 3
[4,1,2,3] => [4,3,2,1] => [4,3,2,1] => 3
[4,1,3,2] => [3,4,2,1] => [1,4,3,2] => 2
[4,2,1,3] => [2,4,3,1] => [1,4,3,2] => 2
[4,2,3,1] => [2,3,4,1] => [1,2,4,3] => 1
[4,3,1,2] => [4,2,3,1] => [4,1,3,2] => 3
[4,3,2,1] => [3,2,4,1] => [2,1,4,3] => 1
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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 fireworks map
Description
Sends a permutation to an inverse fireworks permutation.
A permutation $\sigma$ is inverse fireworks if its inverse avoids the vincular pattern $3-12$. The inverse fireworks map sends any permutation $\sigma$ to an inverse fireworks permutation that is below $\sigma$ in left weak order and has the same Rajchgot index St001759The Rajchgot index of a permutation..
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.