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 Aug 13 comment On counting and generating all $k$-permutations of a multiset @hardmath: I need to think about it... Naively, I would have expected that such an algorithm would, in general, produce some configurations more than once. On the other hand, if the generation happens in lexicographic order, I suppose that it would be relatively inexpensive to weed these duplicates out. I need to work out the details. Thanks for your suggestion. Aug 13 asked On counting and generating all $k$-permutations of a multiset Aug 12 awarded Popular Question Aug 5 awarded Stellar Question Jul 26 accepted On the importance of natural transformations Jul 23 awarded Yearling Jul 14 comment On the importance of natural transformations Also, as I think more about your answer, I'd say that, in a way, it just reiterates what I wrote in my question: the definition of natural transformation adds nothing to what I already knew. (It's like learning that one has been "speaking prose all one's life without knowing it". What does one do with this fact?) I don't think I ever saw the proof of the fact that the complex eigenvalues of a matrix in $\mathrm{GL}_n\mathbb{R}$ are also eigenvalues of the same matrix when viewed as belonging to $\mathrm{GL}_n\mathbb{C}$, but this fact strikes me as more or less obvious. Jul 14 comment On the importance of natural transformations Thanks. I think I get the gist of your answer, but I'm confused by your use of the expression $\mathbb{R}[t] \hookrightarrow \mathbb{C}[t]$, and more specifically, the trailing $[t]$'s. If I were to write out how I understand your answer, I would have written something beginning with: "let $f$ be the insertion of $\mathbb{R} \hookrightarrow \mathbb{C}$", etc. Am I right? Jul 14 revised On the importance of natural transformations added 2 characters in body Jul 14 revised On the importance of natural transformations added 10 characters in body Jul 14 revised On the importance of natural transformations added 113 characters in body Jul 14 revised On the importance of natural transformations added 113 characters in body Jul 14 revised On the importance of natural transformations deleted 72 characters in body Jul 14 asked On the importance of natural transformations Jul 11 awarded Disciplined Jul 6 comment Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$” The suitability of this approach, then, hinges on the details of the proof of $A = \frac{1}{2} rC$... Since you don't give them, I assume that they are very straightforward, but I can't find a particularly simple proof of this assertion. In any case, thanks for posting this strategy. Jul 6 revised What's the name given to the ratio $P^2/A$ for a closed figure in the Euclidean plane? added 19 characters in body Jul 6 accepted What's the name given to the ratio $P^2/A$ for a closed figure in the Euclidean plane? Jul 5 asked What's the name given to the ratio $P^2/A$ for a closed figure in the Euclidean plane? Jul 5 revised Looking for elementary proof of “for a circle, $C^2/A = 4 \pi$” fixed typo in (3)