Suppose $Sub(G)$ is the set of all subgroups of a group $G$. What is the asymptotic of $|Sub(S_n)|$ with $n \to \infty$?  One can get the following bounds on $|Sub(S_n)|$:  $$2^{2^{\langle \frac{n}{2} \rangle}} - 1 \leq |Sub(S_n)| \leq 2^{n!}$$  Proof:  On one hand:  $|Sub(S_n)| \leq |P(S_n)| = 2^{n!}$  On the other hand:  Suppose $k = \langle \frac{n}{2} \rangle$. Then $S_n$ has a subgroup isomorphic to $Sym(\{1, … , k\} \times \{0, 1\})$. For all $A \subset \{1, … , k\} \times \{0, 1\})$ define $\sigma_A: (p, q) \mapsto (p, q + I_A(p) \mod 2)$. It is not hard to see, that $\{\sigma_A| A \subset \{1, … , k\}\} \cong C_2^{2^k}$. And it is known, that $C_2^m$ always has $2^m - 1$ subgroups isomorphic to $C_2$.  The only information on $|Sub(S_n)|$ I was able to find online was the list of exact values of $|Sym(S_n)|$ for $n \leq 18$, calculated by Derek Holt in .
Is there an efficient way to enumerate the unique subgroups of the symmetric group? Naïvely, for the symmetric group $S_n$ of order $\left | S_n \right | = n!$, there are $2^{n!}$ subsets of the group members that could potentially form a subgroup. In addition, many of these subgroups are going to be isomorphic to each other. I feel that the question has an easy answer in terms of the conjugacy classes, but I don't see how. If the answer generalizes to all finite groups, please elaborate!
How do I include a custom class file in a module? In Drupal 7 module development, I could include a custom inc, api, or php class file using the  directive in the module.info file or the drupal_load_included() function.   With D8 using PSR, what is the proper way to package up a "plain ole PHP object" and reference it in DefaultController.php? Think something like: $a = new POPO();.  Is just giving the PHP file containing the class a namespace sufficient? How does the controller (or module in general) know to reference or scan for that class? We are in the early stages of development of the official Stack Exchange Android app and we need your help with testing. To set a few expectations: there will be bugs, crashes, and missing features, all of which will be fixed in .    We are initially targeting Android devices running  (Android 4.1 or higher) starting with phones (up to ~5 inch) and eventually moving our way to tablets.    To register for alpha testing .  After you register we will send you an invite to a private Google community where you can find a link to download our app directly from the  store. The invitations will be sent out in waves, so if you don’t get an invite immediately, be patient.  Crash reports will be automatically reported back to us using . Bugs or feature requests should be reported here on meta and tagged . Please do not use the Google community to report bugs.  As always, we welcome your feedback and are committed to building the best mobile app possible for browsing the Stack Exchange sites. This is the error I get when I open the software center. I've tried rebooting and running sudo apt-get update but I see no improvement. It worked earlier but I don't know what went wrong since then. I'm using Ubuntu 16.04 final release on a Dell Inspiron 15R. Any advice will be appreciated. Thanks in advance.  EDIT: The problem magically solved itself without me doing anything special (or maybe inadvertently doing something, I don't know). So I'd still like to get to the root of this. This is almost surely an Arzela-Ascoli question, since it comes from an old exam of which such problems are quite common. Unfortunately, I can't seem to get it though.  $\{ f_n \}$ is a sequence of functions $[0,1] \to \mathbb{R}$ satisfying $|f_n'(x)| \leq \frac{1 + |\ln (x)|}{\sqrt{x}}$ and $-10 \leq \int_0^1 f_n(x) dx \leq 10$ for every $n$. The question is to show the existence of a uniformly convergent subsequence.  In these kinds of situations, the bounds given usually turn into something nice  to force uniform boundedness. Maybe I could hope to establish equicontinuity using the integral somehow, but I'm not told that the $f_n$ are always positive, so the absolute values would probably screw things up anyway.  Is it an easy problem I'm missing or is this totally the wrong track? If so, what's the right track? Arbitrary products of quasi-coherent sheaves? Position an exampleblock in front of content I need to position an example block:  \begin{exampleblock}{}    Proposed and implemented \end{exampleblock}   Without disturbing other frame content. Something like:  -------------------- ------------- -------------------- -------------------------- -----------    --/============\--------- -----|            |-- -----| Proposed   |-------- -----| and        |-------- -----| implemented|------ -----\============/ -------------------- --------------------------    -----------------   Let it overlay the text behind it. And not to be taken into account during layouting.  Update The Claudio Fiandrino's way is not apropriate to me, because, i need an Itemize environment working inside, and it would be great, if block style is the same as theme-style. How to open a serial port by friendly name? In Salesforce Marketing Cloud (SFMC), I want to unsubscribe a contact from a single publication list by using a form on a landing page.  I also want to create an Unsub event in SFMC when they submit the form.  There is some sample code from SFMC Help documents  that seems to do that for external sites. Will it work in a SFMC hosted landing page? Do I need to edit the code much? Error mounting NTFS partition after hybrid shutdown with Windows 8 I've got this error and I can't access my files located in a different partition when I'm using Ubuntu, but when I'm using Windows 8 my files are accessible. I'm using Ubuntu as my main OS. I tried shutting down Windows 8 properly but still nothing happened. Is there any way where this can be fixed?  Error mounting /dev/sda4 at /media/kris/BackUp Files:  Command-line `mount -t "ntfs" -o "uhelper=udisks2,nodev,nosuid,uid=1000,gid=1000,dmask=0077,fmask=0177" "/dev/sda4" "/media/kris/BackUp Files"' exited with non-zero exit status 14:  The disk contains an unclean file system (0, 0). Metadata kept in Windows cache, refused to mount. Failed to mount '/dev/sda4': Operation not permitted The NTFS partition is in an unsafe state. Please resume and shutdown Windows fully (no hibernation or fast restarting), or mount the volume read-only with the 'ro' mount option. How to align checkboxes and their labels consistently cross-browsers This is one of the minor CSS problems that plague me constantly. How do folks around Stack Overflow vertically align checkboxes and their labels consistently cross-browser? Whenever I align them correctly in Safari (usually using vertical-align: baseline on the input), they're completely off in Firefox and IE. Fix it in Firefox, and Safari and IE are inevitably messed up. I waste time on this every time I code a form. Here's the standard code that I work with:   &lt;form&gt;     &lt;div&gt;         &lt;label&gt;&lt;input type="checkbox" /&gt; Label text&lt;/label&gt;     &lt;/div&gt; &lt;/form&gt;    I usually use Eric Meyer's reset, so form elements are relatively clean of overrides. Looking forward to any tips or tricks that you have to offer! Find $cot\space x_n$ if $sin(x_{n+1}-x_n)+2^{-(n+1)}sin\space x_nsin \space x_{n+1}=0$ for $n\ge1$ Also Given     Suppose $x_1=tan^{-1}2&gt;x_2&gt;x_3&gt;......$ are positive real numbers   I cannot understand as to how I should approach but I have a hunch which is a bit hand-wavy way to prove it but tried to show that  $lim_{n \to \infty}x_n =\dfrac{\pi}{4}$ (For the proof see my edits in @hypernova's answer)  But can anybody prove it and then use it to find $cot \space x_n$ please? I'd like to extract the text from an HTML file using Python.  I want essentially the same output I would get if I copied the text from a browser and pasted it into notepad.    I'd like something more robust than using regular expressions that may fail on poorly formed HTML.  I've seen many people recommend Beautiful Soup, but I've had a few problems using it.  For one, it picked up unwanted text, such as JavaScript source.  Also, it did not interpret HTML entities.  For example, I would expect &amp;#39; in HTML source to be converted to an apostrophe in text, just as if I'd pasted the browser content into notepad.  Update html2text looks promising. It handles HTML entities correctly and ignores JavaScript.  However, it does not exactly produce plain text; it produces markdown that would then have to be turned into plain text. It comes with no examples or documentation, but the code looks clean.    Related questions: For which values of $\alpha$ and $\beta$ does the integral $\int\limits_2^{\infty}\frac{dx}{x^{\alpha}\ln^{\beta}x}$ converge? Show that product of primes, $\prod_{k=1}^{\pi(n)} p_k < 4^n$ This an interesting problem my friend has been working on for a while now (I just saw it an hour ago, but could not come up with anything substantial besides some PMI attempts).  Here's the full problem:  Let $x_{1}, x_{2}, x_{3}, \cdots x_{y}$ be all the primes that are less than a given $n&gt;1,n \in \mathbb{N}$.   Prove that $$x_{1}x_{2}x_{3}\cdots x_{y} &lt; 4^n$$  Any ideas very much appreciated! Professionals: Would you choose GIMP over Photoshop and/or Inkscape over Illustrator? Multiple domains for different products? I have a website with software applications. Is it good for SEO to choose one keyword rich domain name for each of our software products or should we stick to a single domain?  From a user's perspective I think it would be easier to remember a domain that is keyword rich as the user will instantly know what this product is for.  But I have read articles that the latest trend in SEO is to stick to one domain for all of your products and invest on this single domain website.  Is that true? What do you advise? Should I register a separate domain for each of our products or should I use only one single domain?  Should I do a 301 redirect with a .htaccess to a single domain? And what about the sitemaps? Should I register all sites in Google Webmaster Tools and post a separate sitemap for each one of them? should my main site sitemap include all pages or should separate domains have their own sitemaps? The merits of a partitionless filesystem I ran into something a couple of weeks ago that I'd never seen before: A filesystem (ext3 I believe) installed to a storage device without a partition. In essence /dev/sdb was the entire filesystem. I know many filesystems can be extended into empty space, so doing this allows extending without dealing with LVM or some other kind of volume manager, but are there any other advantages for setting up storage this way?  The specific case I saw was as the ephemeral data volume for a number crunching server, the boot and root volumes were traditional partitions on a different storage device entirely. - How can I initialize normal array member variable in initializer list? How can I send the stdout of one process to multiple processes using (preferably unnamed) pipes in Unix (or Windows)? I'd like to redirect the stdout of process proc1 to two processes proc2 and proc3:           proc2 -&gt; stdout        /  proc1        \           proc3 -&gt; stdout   I tried   proc1 | (proc2 &amp; proc3)   but it doesn't seem to work, i.e.   echo 123 | (tr 1 a &amp; tr 1 b)   writes   b23   to stdout instead of    a23  b23 notify-send not working under ssh