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Identifier

St001799: Graphs ⟶ ℤ

Values

([],1) => 0

([],2) => 1

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

([],3) => 6

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

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

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

([],4) => 25

([(2,3)],4) => 16

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

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

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

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

([(1,2),(1,3),(2,3)],4) => 7

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

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

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

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

([],5) => 90

([(3,4)],5) => 63

([(2,4),(3,4)],5) => 45

([(1,4),(2,4),(3,4)],5) => 33

([(0,4),(1,4),(2,4),(3,4)],5) => 25

([(1,4),(2,3)],5) => 42

([(1,4),(2,3),(3,4)],5) => 30

([(0,1),(2,4),(3,4)],5) => 28

([(2,3),(2,4),(3,4)],5) => 36

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

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

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

([(1,3),(1,4),(2,3),(2,4)],5) => 21

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

([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 18

([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 14

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

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 7

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

([(0,4),(1,3),(2,3),(2,4)],5) => 18

([(0,1),(2,3),(2,4),(3,4)],5) => 21

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

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

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

([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 7

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

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

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

([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 7

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

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

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

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

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

([],6) => 301

([(4,5)],6) => 220

([(3,5),(4,5)],6) => 166

([(2,5),(3,5),(4,5)],6) => 130

([(1,5),(2,5),(3,5),(4,5)],6) => 106

([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 90

([(2,5),(3,4)],6) => 157

([(2,5),(3,4),(4,5)],6) => 121

([(1,2),(3,5),(4,5)],6) => 115

([(3,4),(3,5),(4,5)],6) => 139

([(1,5),(2,5),(3,4),(4,5)],6) => 91

([(0,1),(2,5),(3,5),(4,5)],6) => 87

([(2,5),(3,4),(3,5),(4,5)],6) => 103

([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 71

([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 79

([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 63

([(2,4),(2,5),(3,4),(3,5)],6) => 94

([(0,5),(1,5),(2,4),(3,4)],6) => 81

([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 67

([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 61

([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 85

([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 73

([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 65

([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 49

([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 61

([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 53

([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 45

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 52

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 45

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 35

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 49

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 41

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 33

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 26

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 25

([(0,5),(1,4),(2,3)],6) => 108

([(1,5),(2,4),(3,4),(3,5)],6) => 85

([(0,1),(2,5),(3,4),(4,5)],6) => 80

([(1,2),(3,4),(3,5),(4,5)],6) => 94

([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 60

([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 70

([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => 66

([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 50

([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 58

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 42

([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 61

([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 44

([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 52

([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 40

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Description

The number of proper separations of a graph.
A separation of a graph with vertex set $V$ is a set $\{A, B\}$ of subsets of $V$ such that $A\cup B = V$ and there is no edge between $A\setminus B$ and $B\setminus A$. A separation $\{A, B\}$ is proper $A\setminus B$ and $B\setminus A$ are non-empty.
For example, the number of proper separations of the empty graph on $n$ vertices $\{1,\dots,n\}$ is the Stirling number of the second kind $S(n+1, 3)$, i.e., the number of set partitions of $\{0,1,\dots,n\}$ into three subsets, where the subset containing $0$ corresponds to $A \cap B$.

References

[1] Carmesin, J., Kurkofka, J. Entanglements arXiv:2205.11488

Code

def statistic(G):
    G = G.relabel(inplace=False)
    n = G.num_verts()
    count = 0
    if n < 2:
        return 0
    for S in SetPartitions(range(n+1), 3):
        S0, S1, S2 = S
        if n in S0:
            A, B, D = S1, S2, S0
        elif n in S1:
            A, B, D = S0, S2, S1
        else:
            A, B, D = S0, S1, S2
            
        if not any((u in A and w in B) or (w in A and u in B)
                   for u, w in G.edges(labels=False)):
            count += 1
    return count

Created

May 24, 2022 at 10:08 by Martin Rubey

Updated

May 24, 2022 at 10:08 by Martin Rubey