NecklaceCount on IntegerPartition
The colorings of a cycle with these color multiplicities, up to rotation: the size's divisors d of the greatest common divisor of the parts, each contributing totient(d) multinomials, over the size.
FindStat's St001605 (the cyclic group); its title says colourings of a cycle.
Called as CombinatorialStat(x, "NecklaceCount"). The head's page has its examples.
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Distribution
Over IntegerPartitions at each size, tallied here in your browser; sizes past 30 are not offered.
FindStat St001605
The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
| Object | Value |
|---|---|
[3] | 1 |
[4] | 1 |
[5] | 1 |
[6] | 1 |
[7] | 1 |
[8] | 1 |
[9] | 1 |
[10] | 1 |
Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St001605 has the rest.