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Identifier

St001957: Graphs ⟶ ℤ

Values

([],1) => 1

([],2) => 1

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

([],3) => 1

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

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

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

([],4) => 1

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

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

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

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

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

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

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

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

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

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

([],5) => 1

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

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

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

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

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

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

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

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

([(0,4),(1,4),(2,3),(3,4)],5) => 12

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

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

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

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 10

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

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

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

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

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

([(0,4),(1,3),(2,3),(2,4)],5) => 10

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([],6) => 1

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

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

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

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

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

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

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

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

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

([(1,5),(2,5),(3,4),(4,5)],6) => 12

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

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

([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 16

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

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

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

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

([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 10

([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 24

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

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

([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 9

([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 15

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

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

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

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

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 13

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 12

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

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

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

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

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

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

([(1,5),(2,4),(3,4),(3,5)],6) => 10

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

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

([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 20

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

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

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

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

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

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

([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 12

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

([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 20

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$F_{1} = q$

$F_{2} = 2\ q$

$F_{3} = 1 + 2\ q + q^{3}$

$F_{4} = 4 + 3\ q + q^{2} + q^{3} + 2\ q^{4}$

$F_{5} = 20 + 3\ q + 2\ q^{2} + 2\ q^{3} + 2\ q^{4} + 2\ q^{5} + 2\ q^{10} + q^{12}$

$F_{6} = 118 + 4\ q + 3\ q^{2} + 3\ q^{3} + 4\ q^{4} + 2\ q^{5} + 5\ q^{6} + q^{7} + q^{9} + 2\ q^{10} + 3\ q^{12} + 2\ q^{13} + q^{14} + q^{15} + 2\ q^{16} + 3\ q^{20} + q^{24}$

Description

The number of Hasse diagrams with a given underlying undirected graph.
In particular, this statistic vanishes if the graph contains a triangle.
This is the size of the preimage of Mp00074to graph.

Code

def mapping(elt):
    return Graph(elt.hasse_diagram(), immutable=True)

@cached_function
def preimages(level):
    print("computing preimages for level", level)
    result = dict()
    for el in posets(level):
        image = mapping(el).canonical_label().copy(immutable=True)
        result[image] = result.get(image, 0) + 1
    return result

def statistic(x):
    return preimages(x.num_verts()).get(x.canonical_label().copy(immutable=True), 0)

Created

Aug 06, 2024 at 13:28 by Martin Rubey

Updated

Aug 06, 2024 at 13:28 by Martin Rubey