Identifier
Values
[1,0,1,0] => [1,2] => [1,2] => [2,1] => 1
[1,1,0,0] => [2,1] => [2,1] => [1,2] => 0
[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => [3,2,1] => 2
[1,0,1,1,0,0] => [1,3,2] => [1,3,2] => [3,1,2] => 2
[1,1,0,0,1,0] => [2,1,3] => [2,1,3] => [2,3,1] => 2
[1,1,0,1,0,0] => [2,3,1] => [1,3,2] => [3,1,2] => 2
[1,1,1,0,0,0] => [3,2,1] => [3,2,1] => [1,2,3] => 0
[1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 3
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 3
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 3
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,2,4,3] => [4,3,1,2] => 3
[1,0,1,1,1,0,0,0] => [1,4,3,2] => [1,4,3,2] => [4,1,2,3] => 3
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 3
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 3
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [1,3,2,4] => [4,2,3,1] => 3
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [1,2,4,3] => [4,3,1,2] => 3
[1,1,0,1,1,0,0,0] => [2,4,3,1] => [1,4,3,2] => [4,1,2,3] => 3
[1,1,1,0,0,0,1,0] => [3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 3
[1,1,1,0,0,1,0,0] => [3,2,4,1] => [2,1,4,3] => [3,4,1,2] => 3
[1,1,1,0,1,0,0,0] => [4,2,3,1] => [4,1,3,2] => [1,4,2,3] => 2
[1,1,1,1,0,0,0,0] => [4,3,2,1] => [4,3,2,1] => [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
complement
Description
Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$
Map
to non-crossing permutation
Description
Sends a Dyck path $D$ with valley at positions $\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation $\pi$ having descents $\{i_1,\ldots,i_k\}$ and whose inverse has descents $\{j_1,\ldots,j_k\}$.
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to $n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..
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..