Stupid notation question: is there a standard way of expressing "the multisets of size k whose elements are in a set S" or to put it another way the unordered k-tuples, I guess. I want something like Sᵏ

@ccppurcell With a quotient, maybe? Like \(S^k/\sim\) where \(\sim\) is the permutation relation of tuples?

@MartinShadok I do like that approach! But I want to keep things feeling combinatorial and that feels more algebraic to me. If there is no confusion, I would probably just define Sₖ to be what I needed in the preamble. The case I have in mind is actually ℕₖ so that is probably fine; in general Sₖ could be all manner of objects.

@ccppurcell Depending on the context and how frequently you're going use it in equations and things, I'd advocate for something more verbose like Multi(k,S) or Multi_k(S). Not standard, but if you define it, it'll be easier for a reader to keep straight than yet another meaning of exponentiation / subscripting.

@jsiehler I agree that that's an issue, but in the case of ℕ I don't think there's much else it could mean... integers greater than k perhaps? In the context of programming I would definitely do as you suggest, so perhaps I should do it here too.

@ccppurcell @jsiehler the usual notation for k-subsets of S is \(\binomial{S}{k}\).
For multisets, add extra pair of (), see

@dimpase I like the binomial notation for subsets, but I don't like the double brackets for multisets - doing that suggests that you could generalise to n brackets but I don't see an obvious generalisation. On the other hand, that does solve all the problems that have previously been addressed!

@ccppurcell well, double vs single pair of brackets is nothing unsual, one e.g. does \(k[[x]]\) for the ring of formal power series in \(x\) over \(k\)

@dimpase well bra-ket notation in quantum physics is no longer unusual (in the frequency sense) but I don't have to like it!

@ccppurcell bra-ket was always there in quantum physics, I just wish physicists made an effort to avoid it to be understood by mathematicians

@dimpase I know! I would have hoped that TCS would rebel and invent sensible notation for it once quantum computing became a subfield in its own right. I guess we are stuck with it now. It's not as bad as sin² vs sin⁻¹ in my opinion.


@ccppurcell bra-ket is more or less a Hermitean scalar product, IIRC. Nothing needs to be invented there.

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