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Aug
15
comment On pentagonal tilings
Thanks, although the answers posted now prompt the question: what exactly is the difference between, e.g., patterns 4 (dark blue) and 8 (medium gray)?
Aug
15
accepted On pentagonal tilings
Aug
14
asked On pentagonal tilings
Aug
14
comment On counting and generating all $k$-permutations of a multiset
Thanks! In the meantime I realized that the generation problem can be "factored" into two generation problems, arranged as an outer loop and an inner loop; the outer loop is the generation of all $k$-combinations (i.e. $k$-submultisets) from the multiset; the inner one is the generation of all the permutations of the current $k$-submultiset. There are many non-recursive algorithms for the inner loop generation. I have not found a non-recursive algorithm for the outer loop one, but I guess it would not be difficult to convert one of these to a relatively simple iteration, as you did here.
Aug
14
accepted On counting and generating all $k$-permutations of a multiset
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
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Jul
14
revised On the importance of natural transformations
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Jul
14
revised On the importance of natural transformations
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Jul
14
revised On the importance of natural transformations
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Jul
14
revised On the importance of natural transformations
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Jul
14
asked On the importance of natural transformations
Jul
11
awarded  Disciplined