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Identifier
Values
[1,0] => [1,0] => [1,1,0,0] => [1,0,1,0] => 0
[1,0,1,0] => [1,1,0,0] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => 0
[1,1,0,0] => [1,0,1,0] => [1,1,0,1,0,0] => [1,1,0,0,1,0] => 1
[1,0,1,0,1,0] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 0
[1,0,1,1,0,0] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => 1
[1,1,0,0,1,0] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [1,1,1,0,0,0,1,0] => 1
[1,1,0,1,0,0] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => 1
[1,1,1,0,0,0] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0] => 2
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 0
[1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => 1
[1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => 1
[1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => 1
[1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => 2
[1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 1
[1,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => 1
[1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => 1
[1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => 1
[1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,1,0,0,1,0] => 1
[1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => 2
[1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 2
[1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => 2
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => 3
[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
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Description
Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
The modified algebra B is obtained from the stable Auslander algebra kQ/I by deleting all relations which contain walks of length at least three (conjectural this step of deletion is not necessary as the stable higher Auslander algebras might be quadratic) and taking as B then the algebra kQ^(op)/J when J is the quadratic perp of the ideal I.
See www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Loewy length and Nakayama algebras associated to Dyck paths.
Map
decomposition reverse
Description
This map is recursively defined as follows.
The unique empty path of semilength 0 is sent to itself.
Let D be a Dyck path of semilength n>0 and decompose it into 1D10D2 with Dyck paths D1,D2 of respective semilengths n1 and n2 such that n1 is minimal. One then has n1+n2=n1.
Now let ˜D1 and ˜D2 be the recursively defined respective images of D1 and D2 under this map. The image of D is then defined as 1˜D20˜D1.
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.
Map
swap returns and last descent
Description
Return a Dyck path with number of returns and length of the last descent interchanged.
This is the specialisation of the map Φ in [1] to Dyck paths. It is characterised by the fact that the number of up steps before a down step that is neither a return nor part of the last descent is preserved.