I would like to enumerate All the $1$-factors, or perfect matchings, of the complete directed graph $K_{2n}$. Is there an algorithm or method to enumerate all the perfect matchings?
In wikipedia FKT algorithm is given for planar graphs. Not anything for complete graphs. I need to find the number of perfect matchings in complete graph of six vertices.
Counting homomorphisms between $\Bbb Z$ and $\Bbb Z_n$ and also between $\Bbb Z_n$ and $\Bbb Q$ How do you prevent web pages from appearing in the Google search index? Count and limit the number of files uploaded (HTML file input) Games like Super Metroid and Aquaria present the terrain with the other facing parts having rocks and stuff while deeper behind them (i.e. underground) there's different detail or just black.      I would like to do something similar using polygons. Terrain is created in my current level as a set of overlapping square boxes. I'm not sure if this rendering method will work such a system for creating terrain but if anyone has ideas I'd love to hear them. Otherwise I'd like to know how I should re-write the terrain rendering system so it actually works to draw terrain in this manner... What does the 'static' keyword do in a class? Solving $a \sin\theta + b \cos\theta = c$ Could someone help me with the steps for solving the below equation $$a \sin\theta + b \cos\theta = c$$ I know that the solution is $$\theta = \tan^{-1} \frac{c}{^+_-\sqrt{a^2 + b^2 - c^2}} - \tan^{-1} \frac{a}{b} $$ I just can't figure out the right steps to arrive at this solution. Navigate folder structure in Open/Save File Dialog with keyboard? After hitting ⌘-O, how can I navigate my folders and select a file to open using only my keyboard?   In particular, how can I get the focus into the folder list? (From there I can use arrows or start typing the first letters of the desired file/folder.)  In some programs I can rely on hitting TAB 5 times, but this isn't reliable and I figure there must be a better way. Compute device list not showing card name when using OpenCL I have an AMD Radeon R7 M265, and when I force OPENCL with the batch file it only shows Iceland, Intel HD and the CPU.  I followed this instructions:   R, type CMD, press &#9166; Enter, go to your blender folder. ex:  cd C:\Program Files\Blender Foundation\Blender  Press &#9166; Enter and then type: set CYCLES_OPENCL_TEST=all Start Blender with blender.exe Inside blender go to File -> User preferences -> System Under compute device select OpenCL and then your card Close the preferences and go to the render panel and select device: GPU Compute.  I have the latest beta driver from AMD, but when I try to change the compute device it only shows this list:     Am I doing something wrong? Is my GPU is not supported? Or is it because it's in development and not every card is supported yet?  I've tried blender v2.74.5, v2.62, and v2.63. Every one gives the same list. I observed some naive examples. Spheres, for example, when we cut out one point, can be embedded into $\mathbb{R}^n$. And if we cut out a measure zero set of a projective space, it can be embedded into the Euclidean space of the same dimension. So I wonder if all manifolds can be embedded into a same dimensional Euclidean space when we cut out a measure zero set? Can anyone prove it or disprove it by giving me some counterexamples? Examples of topologies in which all open sets are regular? An open subset U of a space X is regular if it equals the interior of its closure, as we learn from the Wikipedia . Furthermore, the regular open subsets of a space (any space) form a complete Boolean algebra.  I'm coming to this from logic and algebra, with not much background in topology. I can't figure out which topologies have "interesting" collections of regular open sets. For example, in the trivial topology and the discrete topology, every open set is regular, if I'm not mistaken. Those are not "interesting" topologies. I assume there are other topologies in which the space X and the null set are the only regular open sets. If so, those aren't "interesting" either, at least not with regard to their regular open ets.  I believe what I'm looking for is topologies in which every open set is regular, other than the ones I've just described.  Thanks for any help. C++ cout hex values? Prove that, for all positive real numbers $x$, $y$ and $z$, $$4(x + y + z)^3 &gt; 27(x^2y + y^2z + z^2x)$$  I've tried expressing it as a sum of squares, but haven't got anywhere.   Hints are also welcome. Basically the &quot;updates page&quot; loads in a non rich html format, and if theres an update it's missing the tabs on the update page as well.  I can't figure out if this is just because apple may have stopped supporting El Capitan or if I have something else going on.  Any help would be appreciated.  I've trashed the caches in library and the user profile. But this page is the only one that still loads like this (see pic below) on my app store. RGB and CMYK: Why both? Any of you who have at least little experience with graphic design software, photography or professional printing, must have -almost undoubtedly- have gone through this: 'the struggle' of RGB and CMYK colour spaces.  Now, there are many questions regarding matters of 'which should I choose', 'what is what' and converting problems, but that's not really what I'm after here.  As a graphic designer with only a couple of years of experience, I still have this struggle from time, and find the whole CMYK/RGB choosing/converting/saving situation quite confusing, to say the least. I do know that RGB is meant for digital end purposes, and CMYK is suitable for printing.  I have always been wondering about the neccesity of this whole thing. So, my question is: why are there these two colour profiles, instead of just having one?   Wouldn't it be a lot easier if printers or software could convert any RGB file to a CMYK printing document instead? I'm pretty sure it does already exist, by the way, because printing an RGB file sometimes simply prints the right colours for me. Alas, not always (doesn't seem to be related to the printer of software I print from).    I have gone trough some RGB vs. CMYK-related questions and found this, a.o., (), an answer of user DKuntz2 (thanks DKuntz2):     RGB is a light-based theory. All colors begin with black "darkness",   to which different color "lights" are added to produce visible colors.   RGB "maxes" at white, which is the equivalent of having all "lights"   on at full brightness (red, green, blue).      CMYK is a color-based theory. All colors start with white "paper", to   which different color "inks" are added to produce output colors. CMYK   "maxes" at black, at which point all "inks" are applied at 100% (cyan,   magenta, yellow, black).   Not sure if this is entirely right, though, because some people disagreed with DKuntz2:     I disagree. With RGB you start with black - the absence of light; with   CMYK you start with white paper.   – e100  Although this does clarify the differences fairly well, I still don't see the point of the two colour profiles(/spaces)    Side question: What happens when you work in an RGB document in photoshop, then press ctrl+y? Photoshop says the document is RGB/8/CMYK, but it can't be both. Right?  EDIT:  Big thanks for the answers, the (current) three below are very informative and interesting. I understand the whole thing a bit better, although it's all very complicated and in-depth to me. Especially when I read about terms like colour models, colour profiles, colour spaces, colour spectra and gamut all one after another.  As much as I would like to accept an answer as the desired one, I feel that my question in the core isn't fully answered yet. What I would really like to emphasize is what I asked about earlier: Wouldn't it be a lot easier if printers or software could convert any RGB file to a CMYK printing document instead?. As Alan Gilbertson explained, the reason is - if I understand correctly - basically a 'lost-in-translation' thing that makes the conversion not 100% accurate, and the colours to possibly be a bit distorted.  However, I still don't see how the solution cannot be an automated conversion done by a printer. Should be possible with modern technology; if you can convert RGB to CMYK in Photoshop, why can't a printer do the exact same for you? As was also said in the answers, the slightest differences in colours and colour spaces, can not be seen by the human eye. And that's what it all comes down to: what we want to see. Be it on a screen or on a sheet of paper.  The fact that this question has three very long answers, or even moreso the fact that there are so many questions about these two colour models (I now know the right term, yay) kind of proves that this is way more complicated than it should be. INT(5) vs SMALLINT(5): numbers in parenthesis after the numeric type In MySQL table definitions is there a different between INT(5) and SMALLINT(5)?  Or do they both represent the same size? HTML and CSS is not working? Opposite of caregiver I want to write a book about the unfair expectations of a caregiver. A caregiver refers to somebody to looks after somebody else.  What is the term for that 'somebody else'.  'Patient' doesn't seem all encompassing. Correct position of "only" Which is grammatically correct?  I can only do so much in this time.  or  I can do only so much in this time. Background  Recently  was implemented. With this feature the amount of questions with new activity is shown in parenthesis at the beginning of the page title, e.g.     (2) Meta Stack Overflow   means that MSO has two questions with new activity since the user last viewed it.   this feature is unrelated to inbox updates. Here is an example to illustrate this:    Even though there are two inbox messages in this screenshot this is not reflected by the page title. The page title is currently only sensitive to questions with new activity.  Suggestion  I suggest that the page title also reflects inbox updates. For the screenshot above this would mean that the page title shows that there are two inbox updates and two questions with new activity.  I am not sure what format is best for showing this. For chat * in the page title is used to show that there are messages directed at you which is similar in meaning to inbox updates. However, * if is placed after the number in parenthesis to denote inbox updates there might be confusion as there is no connection between the number in parenthesis and * like in chat. Thus, one might prefer another format such as one of the following (all are meant to reflect the status of the screenshot above):      * (2) Meta Stack Overflow     2* (2) Meta Stack Overflow     [2] (2) Meta Stack Overflow     2i 2q  Meta Stack Overflow     (2/2)  Meta Stack Overflow     (2|2)  Meta Stack Overflow    For the alternatives without an explicit hint of which number refers to what it might be best to place the number for questions with new activity first as it will in most cases be greater. This will to a greater degree avoid combinations such as "(0/2)" which may be confused zero of two.  An advantage of the suggested feature is that if the user has a browser tab with a Stack Exchange site then they will be instantly notified of inbox updates. They do not need to focus a tab with a Stack Exchange site to know if there are inbox updates, they merely have to check the relevant page titles among their tabs.  A disadvantage of the suggested feature is that it might be experienced as verbose. I suggest that it is implemented such that it is as unobtrusive as possible while still being informative. Think of a situation where a user has   one Meta Stack Overflow tab with two inbox updates relating to Meta Stack Overflow and two questions with new activity, and one Super User tab with one inbox updates relating to Super User and three questions with new activity.   Then the sum of the inbox notification in the page titles should be local to sites and not global. The page titles for the tabs above should be something like the following pair:      2i 2q  Meta Stack Overflow     1i 3q  Super User Mail vs. mail what is the difference