- FindStat

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Identifier

St001343: Posets ⟶ ℤ

Values

([],1) => 1

([],2) => 1

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

([],3) => 1

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

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

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

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

([],4) => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([],5) => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([],6) => 1

([(4,5)],6) => 2

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The dimension of the reduced incidence algebra of a poset.
The reduced incidence algebra of a poset is the subalgebra of the incidence algebra consisting of the elements which assign the same value to any two intervals that are isomorphic to each other as posets.
Thus, this statistic returns the number of non-isomorphic intervals of the poset.

References

[1] wikipedia:Incidence_algebra#Reduced_incidence_algebras

Code

def statistic(P):
    return P.incidence_algebra(QQ).reduced_subalgebra().dimension()

Created

Jan 05, 2019 at 17:28 by Martin Rubey

Updated

Jun 22, 2019 at 10:57 by Martin Rubey