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Statistic identifier: St000949
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Collection: Dyck paths
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Description: Gives the number of generalised tilting modules of the corresponding LNakayama algebra.
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References: [1] [[https://en.wikipedia.org/wiki/Tilting_theory]]
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Code:
gap('LoadPackage("QPA");')
def NthRadical(M, n):
if n == 0:
f = gap.IdentityMapping(M)
else:
f = gap.RadicalOfModuleInclusion(M)
N = gap.Source(f)
for i in range(n-1):
h = gap.RadicalOfModuleInclusion(N);
N = gap.Source(h)
f = h * f
return f
def ARQuiverNak(A):
injA = gap.IndecInjectiveModules(A)
L = [gap.Source(NthRadical(inj, j))
for inj in injA
for j in range(gap.Dimension(inj).sage())]
return L
def kupisch(D):
"""
sage: [kupisch(D) for D in DyckWords(3)]
[[2, 2, 2, 1], [2, 3, 2, 1], [3, 2, 2, 1], [3, 3, 2, 1], [4, 3, 2, 1]]
sage: all(kupisch(D) == [a+2 for a in D.reverse().to_area_sequence()[::-1]] + [1] for D in DyckWords(5))
"""
H = D.heights()
return [1+H[i] for i, s in enumerate(D) if s == 0]+[1]
def statistic(D):
D = DyckWord(D)
K = kupisch(D)
A = gap.NakayamaAlgebra(K, gap.GF(3))
g = gap.GlobalDimensionOfAlgebra(A,30)
L = ARQuiverNak(A)
LL = [x for x in L
if not gap.IsProjectiveModule(x) or not gap.IsInjectiveModule(x)]
r = len(gap.SimpleModules(A)) - (len(L) - len(LL))
S = [[LL[i-1] for i in s] for s in Subsets(len(LL), r)]
return sum(1 for x in S
if gap.N_RigidModule(gap.DirectSumOfQPAModules(x) , g))
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Statistic values:
[1,0] => 2
[1,0,1,0] => 3
[1,1,0,0] => 5
[1,0,1,0,1,0] => 4
[1,0,1,1,0,0] => 7
[1,1,0,0,1,0] => 7
[1,1,0,1,0,0] => 9
[1,1,1,0,0,0] => 14
[1,0,1,0,1,0,1,0] => 5
[1,0,1,0,1,1,0,0] => 9
[1,0,1,1,0,0,1,0] => 10
[1,0,1,1,0,1,0,0] => 11
[1,0,1,1,1,0,0,0] => 19
[1,1,0,0,1,0,1,0] => 9
[1,1,0,0,1,1,0,0] => 16
[1,1,0,1,0,0,1,0] => 11
[1,1,0,1,0,1,0,0] => 15
[1,1,0,1,1,0,0,0] => 23
[1,1,1,0,0,0,1,0] => 19
[1,1,1,0,0,1,0,0] => 23
[1,1,1,0,1,0,0,0] => 28
[1,1,1,1,0,0,0,0] => 42
[1,0,1,0,1,0,1,0,1,0] => 6
[1,0,1,0,1,0,1,1,0,0] => 11
[1,0,1,0,1,1,0,0,1,0] => 13
[1,0,1,0,1,1,0,1,0,0] => 13
[1,0,1,0,1,1,1,0,0,0] => 24
[1,0,1,1,0,0,1,0,1,0] => 13
[1,0,1,1,0,0,1,1,0,0] => 23
[1,0,1,1,0,1,0,0,1,0] => 13
[1,0,1,1,0,1,0,1,0,0] => 18
[1,0,1,1,0,1,1,0,0,0] => 27
[1,0,1,1,1,0,0,0,1,0] => 26
[1,0,1,1,1,0,0,1,0,0] => 32
[1,0,1,1,1,0,1,0,0,0] => 33
[1,0,1,1,1,1,0,0,0,0] => 56
[1,1,0,0,1,0,1,0,1,0] => 11
[1,1,0,0,1,0,1,1,0,0] => 20
[1,1,0,0,1,1,0,0,1,0] => 23
[1,1,0,0,1,1,0,1,0,0] => 24
[1,1,0,0,1,1,1,0,0,0] => 43
[1,1,0,1,0,0,1,0,1,0] => 13
[1,1,0,1,0,0,1,1,0,0] => 24
[1,1,0,1,0,1,0,0,1,0] => 18
[1,1,0,1,0,1,0,1,0,0] => 22
[1,1,0,1,0,1,1,0,0,0] => 37
[1,1,0,1,1,0,0,0,1,0] => 32
[1,1,0,1,1,0,0,1,0,0] => 32
[1,1,0,1,1,0,1,0,0,0] => 43
[1,1,0,1,1,1,0,0,0,0] => 66
[1,1,1,0,0,0,1,0,1,0] => 24
[1,1,1,0,0,0,1,1,0,0] => 43
[1,1,1,0,0,1,0,0,1,0] => 27
[1,1,1,0,0,1,0,1,0,0] => 37
[1,1,1,0,0,1,1,0,0,0] => 57
[1,1,1,0,1,0,0,0,1,0] => 33
[1,1,1,0,1,0,0,1,0,0] => 43
[1,1,1,0,1,0,1,0,0,0] => 52
[1,1,1,0,1,1,0,0,0,0] => 76
[1,1,1,1,0,0,0,0,1,0] => 56
[1,1,1,1,0,0,0,1,0,0] => 66
[1,1,1,1,0,0,1,0,0,0] => 76
[1,1,1,1,0,1,0,0,0,0] => 90
[1,1,1,1,1,0,0,0,0,0] => 132
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Created: Aug 25, 2017 at 10:52 by Rene Marczinzik
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Last Updated: Aug 25, 2020 at 15:00 by Martin Rubey