- FindStat

Identifier

St000514: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [2]=>2
[1,1]=>2
[3]=>2
[2,1]=>4
[1,1,1]=>8
[4]=>4
[3,1]=>4
[2,2]=>16
[2,1,1]=>16
[1,1,1,1]=>64
[5]=>4
[4,1]=>8
[3,2]=>8
[3,1,1]=>16
[2,2,1]=>64
[2,1,1,1]=>128
[1,1,1,1,1]=>1024
[6]=>8
[5,1]=>8
[4,2]=>32
[4,1,1]=>32
[3,3]=>32
[3,2,1]=>32
[3,1,1,1]=>128
[2,2,2]=>512
[2,2,1,1]=>512
[2,1,1,1,1]=>2048
[1,1,1,1,1,1]=>32768
[7]=>8
[6,1]=>16
[5,2]=>16
[5,1,1]=>32
[4,3]=>16
[4,2,1]=>128
[4,1,1,1]=>256
[3,3,1]=>128
[3,2,2]=>128
[3,2,1,1]=>256
[3,1,1,1,1]=>2048
[2,2,2,1]=>4096
[2,2,1,1,1]=>8192
[2,1,1,1,1,1]=>65536
[1,1,1,1,1,1,1]=>2097152
[8]=>16
[7,1]=>16
[6,2]=>64
[6,1,1]=>64
[5,3]=>16
[5,2,1]=>64
[5,1,1,1]=>256
[4,4]=>256
[4,3,1]=>64
[4,2,2]=>1024
[4,2,1,1]=>1024
[4,1,1,1,1]=>4096
[3,3,2]=>256
[3,3,1,1]=>1024
[3,2,2,1]=>1024
[3,2,1,1,1]=>4096
[3,1,1,1,1,1]=>65536
[2,2,2,2]=>65536
[2,2,2,1,1]=>65536
[2,2,1,1,1,1]=>262144
[2,1,1,1,1,1,1]=>4194304
[1,1,1,1,1,1,1,1]=>268435456
                    

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Description

The number of invariant simple graphs when acting with a permutation of given cycle type.
Put differently, let $o$ be the number of orbits of the action of a permutation of given cycle type on the set of edges of the complete graph. Then this statistic is $2^o$.

References

[1] Bergeron, F., Labelle, G., Leroux, P. Combinatorial species and tree-like structures MathSciNet:1629341

Code

def statistic(la):
    G = LazyCombinatorialSpecies(QQ, "X").Graphs().cycle_index_series()
    return la.aut() * G[la.size()].coefficient(la)

Created

May 26, 2016 at 21:28 by Martin Rubey

Updated

Sep 27, 2025 at 01:40 by Martin Rubey