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.