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Identifier
Values
=>
Cc0020;cc-rep
([],1)=>0 ([],2)=>0 ([(0,1)],2)=>1 ([],3)=>0 ([(1,2)],3)=>1 ([(0,2),(1,2)],3)=>0 ([(0,1),(0,2),(1,2)],3)=>3 ([],4)=>0 ([(2,3)],4)=>2 ([(1,3),(2,3)],4)=>0 ([(0,3),(1,3),(2,3)],4)=>0 ([(0,3),(1,2)],4)=>4 ([(0,3),(1,2),(2,3)],4)=>1 ([(1,2),(1,3),(2,3)],4)=>3 ([(0,3),(1,2),(1,3),(2,3)],4)=>1 ([(0,2),(0,3),(1,2),(1,3)],4)=>0 ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>2 ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)=>12 ([],5)=>0 ([(3,4)],5)=>6 ([(2,4),(3,4)],5)=>0 ([(1,4),(2,4),(3,4)],5)=>0 ([(0,4),(1,4),(2,4),(3,4)],5)=>0 ([(1,4),(2,3)],5)=>4 ([(1,4),(2,3),(3,4)],5)=>1 ([(0,1),(2,4),(3,4)],5)=>2 ([(2,3),(2,4),(3,4)],5)=>6 ([(0,4),(1,4),(2,3),(3,4)],5)=>0 ([(1,4),(2,3),(2,4),(3,4)],5)=>1 ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)=>2 ([(1,3),(1,4),(2,3),(2,4)],5)=>0 ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)=>0 ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>2 ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)=>1 ([(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)=>0 ([(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)=>0 ([(0,1),(2,3),(2,4),(3,4)],5)=>6 ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)=>1 ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)=>4 ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)=>5 ([(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)=>1 ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)=>1 ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>12 ([(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),(3,4)],5)=>2 ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)=>2 ([(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)=>6 ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>60 ([],6)=>0 ([(4,5)],6)=>24 ([(3,5),(4,5)],6)=>0 ([(2,5),(3,5),(4,5)],6)=>0 ([(1,5),(2,5),(3,5),(4,5)],6)=>0 ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>0 ([(2,5),(3,4)],6)=>8 ([(2,5),(3,4),(4,5)],6)=>2 ([(1,2),(3,5),(4,5)],6)=>2 ([(3,4),(3,5),(4,5)],6)=>18 ([(1,5),(2,5),(3,4),(4,5)],6)=>0 ([(0,1),(2,5),(3,5),(4,5)],6)=>6 ([(2,5),(3,4),(3,5),(4,5)],6)=>2 ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>0 ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>2 ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>6 ([(2,4),(2,5),(3,4),(3,5)],6)=>0 ([(0,5),(1,5),(2,4),(3,4)],6)=>0 ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)=>0 ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)=>0 ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4 ([(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>1 ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)=>4 ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)=>0 ([(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)=>0 ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)=>0 ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>0 ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6 ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2 ([(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)=>0 ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>24 ([(0,5),(1,4),(2,3)],6)=>24 ([(1,5),(2,4),(3,4),(3,5)],6)=>0 ([(0,1),(2,5),(3,4),(4,5)],6)=>2 ([(1,2),(3,4),(3,5),(4,5)],6)=>6 ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)=>0 ([(1,4),(2,3),(2,5),(3,5),(4,5)],6)=>1 ([(0,1),(2,5),(3,4),(3,5),(4,5)],6)=>2 ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>1 ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>4 ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>4 ([(1,4),(1,5),(2,3),(2,5),(3,4)],6)=>5 ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)=>0 ([(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)=>1 ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>1 ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)=>3 ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>0 ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>0 ([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)=>1 ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)=>1 ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)=>1 ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)=>8 ([(0,5),(1,5),(2,3),(2,4),(3,4)],6)=>6 ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)=>0 ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)=>1 ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)=>0 ([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4 ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)=>2 ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)=>2 ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)=>0 ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>1 ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2 ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>24 ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>2 ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3 ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6 ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)=>2 ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)=>0 ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2 ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>0 ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>0 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)=>0 ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>1 ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2 ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2 ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>1 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>4 ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4 ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)=>0 ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>2 ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>0 ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>0 ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>0 ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)=>0 ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)=>2 ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)=>0 ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>1 ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>2 ([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)=>0 ([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>0 ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>0 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)=>0 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>0 ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)=>0 ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)=>2 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>0 ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>0 ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>0 ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6 ([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>1 ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2 ([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)=>0 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>3 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>1 ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)=>36 ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6 ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>0 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>0 ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>18 ([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)=>36 ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)=>1 ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)=>4 ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>24 ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>2 ([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3 ([(0,1),(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6 ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)=>0 ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)=>1 ([(0,3),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2 ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>1 ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2 ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>1 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>8 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>8 ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6)=>1 ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>2 ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>1 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>5 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)=>6 ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>0 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>0 ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>0 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2 ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>1 ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4 ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>1 ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3 ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>60 ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>12 ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6 ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>6 ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>2 ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>6 ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4 ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>2 ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>0 ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>8 ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>24 ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>360
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Description
The number of odd automorphisms of a graph.
Let $D$ be an arbitrary orientation of a graph $G$. Then an automorphism of $G$ is odd, if it reverses the orientation of an odd number of edges of $D$.
The graphs on $n$ vertices without any odd automorphisms are equinumerous with the number of non-isomorphic $n$-team tournaments, see [2].
The odd automorphisms of the complete graphs are precisely the even permutations.
References
[1] Number of outcomes of unlabeled n-team round-robin tournaments. OEIS:A000568
[2] Royle, G. F., Praeger, C. E., Glasby, S. P., Freedman, S. D., Devillers, A. Tournaments and Even Graphs are Equinumerous arXiv:2204.01947
Code
def is_odd_automorphism(D, g):
    count = 0
    for a, b in D.edges(labels=False):
        c, d = g(a), g(b)
        if D.has_edge(c, d) and not D.has_edge(d, c):
            pass
        elif D.has_edge(d, c) and not D.has_edge(c, d):
            count += 1
        else:
            raise ValueError("%s is mapped to %s" % ((a,b), (c,d)))
    return is_odd(count)

def statistic(G):
    D = next(G.orientations())
    count = 0
    for g in G.automorphism_group():
        if is_odd_automorphism(D, g):
            count += 1
    return count
Created
Apr 06, 2022 at 10:26 by Martin Rubey
Updated
Apr 27, 2022 at 10:37 by Martin Rubey