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Identifier

St001632: Posets ⟶ ℤ

Values

([(0,1)],2) => 1

([(0,1),(0,2)],3) => 2

([(0,2),(2,1)],3) => 1

([(0,2),(1,2)],3) => 1

([(0,1),(0,2),(0,3)],4) => 0

([(0,2),(0,3),(3,1)],4) => 2

([(0,1),(0,2),(1,3),(2,3)],4) => 2

([(0,3),(3,1),(3,2)],4) => 1

([(0,3),(1,3),(3,2)],4) => 1

([(0,3),(1,3),(2,3)],4) => 0

([(0,3),(1,2),(1,3)],4) => 0

([(0,2),(0,3),(1,2),(1,3)],4) => 0

([(0,3),(2,1),(3,2)],4) => 1

([(0,3),(1,2),(2,3)],4) => 1

([(0,1),(0,2),(0,3),(0,4)],5) => 0

([(0,2),(0,3),(0,4),(4,1)],5) => 0

([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => 1

([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 0

([(0,3),(0,4),(4,1),(4,2)],5) => 1

([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2

([(0,3),(0,4),(3,2),(4,1)],5) => 2

([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => 2

([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => 2

([(0,4),(4,1),(4,2),(4,3)],5) => 1

([(0,4),(1,4),(4,2),(4,3)],5) => 0

([(0,4),(1,4),(2,4),(4,3)],5) => 0

([(0,4),(1,4),(2,4),(3,4)],5) => 0

([(0,4),(1,3),(2,3),(2,4)],5) => 0

([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 0

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 0

([(0,4),(1,4),(2,3),(4,2)],5) => 1

([(0,4),(1,3),(2,3),(3,4)],5) => 0

([(0,4),(1,4),(2,3),(2,4)],5) => 0

([(0,4),(1,4),(2,3),(3,4)],5) => 1

([(0,4),(1,2),(1,4),(2,3)],5) => 0

([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => 1

([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => 0

([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => 2

([(0,4),(1,2),(1,4),(4,3)],5) => 0

([(0,4),(1,2),(1,3),(1,4)],5) => 0

([(0,2),(0,4),(3,1),(4,3)],5) => 2

([(0,4),(1,2),(1,3),(3,4)],5) => 1

([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 2

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 2

([(0,3),(0,4),(1,2),(1,4)],5) => 0

([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 0

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 0

([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => 0

([(0,3),(1,2),(1,4),(3,4)],5) => 1

([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => 1

([(0,3),(3,4),(4,1),(4,2)],5) => 1

([(0,4),(1,2),(2,4),(4,3)],5) => 1

([(0,4),(3,2),(4,1),(4,3)],5) => 1

([(0,4),(1,2),(2,3),(2,4)],5) => 1

([(0,4),(2,3),(3,1),(4,2)],5) => 1

([(0,3),(1,2),(2,4),(3,4)],5) => 2

([(0,4),(1,2),(2,3),(3,4)],5) => 1

([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 2

([(0,1),(0,2),(0,3),(0,4),(0,5)],6) => 0

([(0,2),(0,3),(0,4),(0,5),(5,1)],6) => 0

([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6) => 0

([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6) => 1

([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6) => 0

([(0,3),(0,4),(0,5),(5,1),(5,2)],6) => 0

([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6) => 0

([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6) => 1

([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6) => 2

([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 1

([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 0

([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6) => 0

([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 1

([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6) => 1

([(0,3),(0,4),(0,5),(4,2),(5,1)],6) => 0

([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6) => 0

([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6) => 0

([(0,4),(0,5),(5,1),(5,2),(5,3)],6) => 1

([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6) => 2

([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6) => 2

([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 2

([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6) => 2

([(0,4),(0,5),(4,3),(5,1),(5,2)],6) => 1

([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6) => 1

([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6) => 2

([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6) => 2

([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) => 2

([(0,5),(5,1),(5,2),(5,3),(5,4)],6) => 1

([(0,4),(4,5),(5,1),(5,2),(5,3)],6) => 1

([(0,5),(1,5),(5,2),(5,3),(5,4)],6) => 0

([(0,5),(1,5),(2,5),(5,3),(5,4)],6) => 0

([(0,5),(1,5),(2,5),(3,5),(5,4)],6) => 0

([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 0

([(0,5),(1,5),(2,5),(3,4),(5,3)],6) => 0

([(0,5),(1,5),(2,5),(3,4),(5,4)],6) => 0

([(0,5),(1,5),(2,5),(3,4),(3,5)],6) => 0

([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 1

([(0,5),(1,4),(2,4),(2,5),(5,3)],6) => 0

([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6) => 0

([(0,5),(1,4),(1,5),(2,4),(2,5),(4,3)],6) => 0

([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6) => 2

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6) => 0

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6) => 0

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Description

The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.

Code

gap('LoadPackage("QPA");')

def incidence_algebra_gap(P):
    P = P.relabel()
    return gap("PosetAlgebra(GF(3), Poset(%s, %s))" % (list(P), P.cover_relations()))

def statistic(P):
    A = incidence_algebra_gap(P)
    RegA = A.IndecProjectiveModules().DirectSumOfQPAModules()
    injectives = A.IndecInjectiveModules()
    return sum(1 for M in injectives if M.ExtOverAlgebra(RegA)[2].Size().sage() == 1)

Created

Oct 03, 2020 at 21:04 by Rene Marczinzik

Updated

Nov 13, 2022 at 12:12 by Martin Rubey