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Matching statistic: St001498
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-,+] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[-,+,+] => [2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[+,-,+] => [1,3,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-,-,+] => [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[-,+,-] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[-,3,2] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[2,1,+] => [1,3,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[3,+,1] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[3,-,1] => [1,3,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[-,+,+,+] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[+,-,+,+] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[+,+,-,+] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[-,-,+,+] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[-,+,-,+] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[-,+,+,-] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[+,-,-,+] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[+,-,+,-] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[-,-,-,+] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[-,-,+,-] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[-,+,-,-] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[-,+,4,3] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[+,-,4,3] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[-,-,4,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[+,3,2,+] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[-,3,2,+] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[-,3,2,-] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[-,3,4,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[-,4,2,3] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[+,4,+,2] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[-,4,+,2] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[+,4,-,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[-,4,-,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,1,+,+] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,1,-,+] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,1,+,-] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[2,1,4,3] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[2,3,1,+] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,4,1,3] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[2,4,+,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,4,-,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,2,+] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,+,1,+] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[3,-,1,+] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,+,1,-] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[3,-,1,-] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[3,+,4,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[3,-,4,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[3,4,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[4,1,+,2] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[4,1,-,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
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