a3nm
Reputation
245
Top tag
Next privilege 250 Rep.
 Mar 28 comment Width of a product of chains Which sum of math.stackexchange.com/a/299958 are you referring to? Is it the last one? Then could you explain a bit why it should only go to $r-1$? (From my understanding, you should sum across all the $r$ chains.) Mar 26 comment Width of a product of chains Thanks! As far as I can see, this paper points out that a product of chains has the Sperner property, and then is concerned about which products have the strict Sperner property, i.e., all their maximal antichains correspond to a rank. I don't think it does any estimation of the width itself, which @Brian M. Scott did in his answer. Sep 28 comment Extending a partial order to antichains Thanks for this remark! In my answer I was implicitly thinking about finite posets, even though the question was more general. I rephrased accordingly. Jul 8 comment Undistinguishable elements in posets Thanks again for pointing this out! Jul 8 comment Undistinguishable elements in posets @YannPequignot: You are entirely right, my "indistinguishable sets" are exactly this notion of interval, thanks a lot for pointing this out. Please post your comment as an answer and I will accept it. Thanks! :) May 13 comment Small posets with prescribed number of linear extensions @talegari: No I mean that their sizes (number of elements) should be $O(\log n)$. May 15 comment Prove (without quoting any theorems) that polynomials on [0,1] are continous I don't understand the point of the first question. Shouldn't you replace $P([0, 1])$ by $C^0[0, 1]$ in this question? You can then combine both questions to show the desired result. (What your version of the first question asks you to prove is true but I fail to see where you will need to use it.) Feb 12 comment Width of a product of chains Thanks a lot for this detailed answer! Is this new, or are those results from an existing source? Feb 10 comment Product-Decomposition of distributive lattices Feb 6 comment Number of upper sets of size $n$ in a finite tree Yes, you can indeed show by induction on the tree $T$ that $f_T$ is log-concave. Thanks a lot for pointing me to this, I believe this was the right notion needed for the induction hypothesis. If you edit your answer to mention this, I'll accept it. Feb 4 comment Number of upper sets of size $n$ in a finite tree If it is indeed true that the convolution of two "mountains" is indeed a "mountains", then the result is proved and I'm happy, but in fact I'm not sure this is always true. Feb 4 comment Number of upper sets of size $n$ in a finite tree Thanks for this remark. In fact, I'm already aware of the following result: for any binary tree $T$ with root subtrees $L$ and $R$ (not just the divisibility example you gave), we have $f_T(k) = \sum_i f_L(i) f_R(k-i-1)$. (Any upper set of size $i$ in $T$ is obtained by taking one upper set in $L$ and one in $R$, with all possible repartitions of sizes.) This generalizes to trees of arbitrary arity, and, indeed, it's a convolution. However, I can't manage to do an induction from here. I'm not sure if we don't need a stronger induction hypothesis that the result I conjectured. Jan 29 comment How to understand the duality between Dilworth's theorem and Mirsky's theorem? I'm interested in finite partial orders, I edited the question to reflect this. Thanks for this remark! Jan 1 comment Extending a partial order to antichains WimC: A more detailed paper about the same thing by the same authors: isg.rhul.ac.uk/~jason/Pubs/imj.pdf Aug 15 comment Understanding what $\sqrt{p}$ means for an event of probability $p$ Ross Millikan: I thought of this, but I couldn't understand exactly what it should mean. Aug 15 comment Understanding what $\sqrt{p}$ means for an event of probability $p$ did: I'm not well aware that the problem is deep, I just thought about it and did not find useful references. Your Bernoulli factory reference is interesting, it would be frustrating if there is indeed no simpler interpretation for the simple case of the square root. PhiNotPi: Yes, this is a good argument against the existence of a simple interpretation...