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 Jan 25 comment The lattice of partial partitions @BrianM.Scott- thanks for your comment. I'd been using the order for $\Pi_{\le n}$ (a la Hanlon, Hersh, and Shareshian) instead of $Q_n$ and didn't notice these were different. Jan 25 comment The lattice of partial partitions It didn't occur to me that the partial order they used might be different than the one I had in mind, so obviously this is what I missed. Thanks! Jan 25 comment The lattice of partial partitions @hardmath - your mapping is why the two lattices clearly have the same size. But when I draw $Q_2$ I get two maximal chains, viz. $() \le (1) \le (1)(2) \le (12)$ and $() \le (2) \le (1)(2) \le (12)$, while for $\Pi_3$ the three maximal chains are $(1)(2)(3) \le (12)(3) \le (123)$, $(1)(2)(3) \le (13)(2) \le (123)$, and $(1)(2)(3) \le (1)(23) \le (123)$. Jun 9 comment Markov chain: join states in Transition Matrix en.wikipedia.org/wiki/Lumpability Jul 30 comment Why is the volume of a sphere $\frac{4}{3}\pi r^3$? The negative solution to Hilbert's third problem (en.wikipedia.org/wiki/Hilbert%27s_third_problem) strongly suggests (if not shows outright) that this and related questions require calculus in an essential way, and that calculus-free "derivations" are just hiding something. Apr 29 comment Meaning of pullback Consider $f = g \circ \alpha$. The universal property implies that the corresponding pullback satisfies (using Wikipedia notation current at the time of writing) $P = \text{dom } \alpha$, $p_1 = id$, $p_2 = \alpha$, so that $\alpha$ is the pullback of $f = g \circ \alpha$ along $\alpha$. Jan 22 comment How can I explicitly construct a *nice* conformal mapping from a triangle to a square in MATLAB? Jan 22 comment How can I explicitly construct a *nice* conformal mapping from a triangle to a square in MATLAB? @Arkamis--Look at the polygon p1. The stuff you're preoccupied with doesn't have anything to do with the actual polygon, just an easy way to generate points inside it. Jan 22 comment How can I explicitly construct a *nice* conformal mapping from a triangle to a square in MATLAB? I should also note that I've seen mathfaculty.fullerton.edu/mathews/c2003/… and this (or similar maps) doesn't really solve my problem AFAIK. Jan 22 comment How can I explicitly construct a *nice* conformal mapping from a triangle to a square in MATLAB? @Arkamis--I don't think so. That's just the y-limits of the equilateral triangle, which is centered at the origin. Jul 26 comment Number of item distributions in buckets of different sizes @VladimirDotsenko: I wish I could have read your comment back in 2001! Jul 26 comment Number of item distributions in buckets of different sizes I would hope that this is not closed: if there is a simple answer, I am not aware of it and would like to know it. I previously wondered about this very question many years ago and based on that experience I do not think it should be migrated. For instance, I could not figure out how to make the straightforward appeal to inclusion-exclusion actually work in practice. Jul 26 comment Number of item distributions in buckets of different sizes The case where $c_b \ge I$ gives $\binom{I+B-1}{I}$, by "stars and bars": en.wikipedia.org/wiki/Stars_and_bars_(combinatorics) Dec 6 comment What discrete memoryless channels have zero capacity? Edited to remove the question's ambiguity. Nov 1 comment Bizarre appearance of Cauchy-like density estimate Ah, you caught my mistake, thanks. I should have said Sibuya: this is implicit in my MO question. Oct 31 comment Bizarre appearance of Cauchy-like density estimate Thanks, this is interesting. However, it's not just the tail that is a good fit (and I have just posted to MO about this at mathoverflow.net/questions/79623), it's the whole domain of the data (incl. near 0). Your answer does however remind me that discrete stable distributions can be obtained by combining (IIRC) a Poissonian number of Poisson RVs. Oct 25 comment Bizarre appearance of Cauchy-like density estimate At root, this is a question about whether or not the generalized CLT has anything to say about what I'm seeing. To me, this is a mathematical question. If I were primarily concerned about bandwidth selection or was sure I'd made a mistake, I would agree with the votes for "off-topic", but that isn't the motivation at all. I expect a decent answer will be mathematical in character. Oct 21 comment Are there any (pairs of) simple distributions that give rise to a power law ratio? I have what appears to be such a data set (corroborated by MLE but I haven't done anything exhaustive). The fits appear excellent. However, it's a small enough data set that I can't easily ID plausible distributions for the ratio (these seem noisier when considered separately). Mar 30 comment What is $\prod_{k=1}^n (1-x^k)$? I see that equation 1.30 in Stanley gives the connection with partitions. In fact the reciprocal of the product that is in the title is what actually interested me, so this is quite convenient. Thanks again! Mar 30 comment What is $\prod_{k=1}^n (1-x^k)$? Great, thanks. I don't have Andrews but I will see if it's in Stanley's books, which I have at home.