Number of colorings of a complete graph

Question

This surely is well known, but googling gave too much false positives. Let $K_n$ be the complete graph with $n$ vertices. (Which has $e=n(n-1)/2$ edges.) Now "edge coloring" was defined as "no touching colors on any vertex", but I'm interested in the much simpler "ignore that restriction". Obviously, there are $m^e$ unrestricted edge colorings of $K_n$ with $m$ colors (if you distinguish vertices), but many of them are topologically indistinguishable under vertex swaps. Example: $K_4, e=6$ has $6$ unrestricted bicolorings ($6/0: 1,5/1: 1,4/2: 2,3/3: 2$).

At the moment I'm only interested in the $m=2$ formula, but surely the general one is known anyway.

Maybe you mean $m^e$? And in your example shouldn't the bicolorings be $7$ (you have 7 choices for the number of edges of the first colour)? — Dark Magician, Sep 04 '22 at 10:47
Silly me, corrected. Also one bicoloring is invariant under color swap, thus I miscounted. — Hauke Reddmann, Sep 04 '22 at 15:00
The post How many $2$-edge-colorings of $K_n$ are there? addresses $m=2$ but the Polya enumeration theorem gives the generalization for all $m$ almost immediately. Note that the top answer counts colorings where vertices are indistinguishable but the colors are distinguishable; see Chris Godsil's answer for the correction to apply if you only want to count colorings up to color swaps, which it looks like you do. — Misha Lavrov, Sep 04 '22 at 18:36
Ah, I knew this must have been adressed before even here, alas, the "suggested questions" gizmo failed again...Sorry! — Hauke Reddmann, Sep 05 '22 at 07:17

Number of colorings of a complete graph

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