I want to count the number of necklaces, with $n$ beads in total, where the alphabet of beads is $\{1,\ldots,k\}$, and where the number of beads with color $i$ is $n_i$. For example, if $n=4$, and $n_1 = n_2 = 2$, the following necklaces are the only possible $$1122$$ $$1212$$  I have been able to find formulas for different variations of the problem, but not this one. If there is no nice formula, a generating function is always a nice compromise.
Given n distinct objects, there are $n!$ permutations of the objects and $n!/n$ "circular permutations" of the objects (orientation of the circle matters, but there is no starting point, so $1234$ and $2341$ are the same, but $4321$ is different).  Given $n$ objects of $k$ types (where the objects within each type are indistinguishable), $r_i$ of the $i^{th}$ type, there are   \begin{equation*} \frac{n!}{r_1!r_2!\cdots r_k!} \end{equation*}  permutations.  How many circular permutations are there of such a set?
I need to convert one into 1, two into 2 and so on.  Is there a way to do this with a library or a class or anything? "It was recommended that" + verb in present tense It was recommended that the international development partners develop guidelines or a roadmap to help countries take the necessary steps to select the right tool for their purposes.   Why is it using present tense "develop" in the sentence which is actually a past tense "It was recommended"? I tried to compile a (flawlessly working) LaTeX document on a different computer, and suddenly was faced with the above error message printed by the xparse package (after that, many errors follow).  I'm working on both computers with miktex 2.9 (and Texnic-Center) on Windows 7 (64 bit).  I tried EVERYTHING I found on stackexchange or anywhere else on the web, including:   Updating all packages Removing the l3 packages and reinstalling them Doing the above for both the current user and in admin mode Removing the complete miktex distribution and re-installing it   for only the current user for "anyone who uses this computer" with all packages only the basic packages  Downloading the newest l3 packages manually from CTAN and copying them into the miktex folder Refreshing the FNDB and updating formats afterwards   Nothing worked. Right now, I'm just out of ideas. ANY suggestions??  EDIT: After performing UlrikeFischer's suggestion, I get different errors. After a long compilation process, this is the log file I get: Automorphisms of the complex plane How can it be shown that $$Aut(\mathbb{C})=\{f|f(z)=az+b,a\neq 0\},$$ where an automorphism of $\mathbb{C}$ is defined as a bijective entire function with entire inverse?  If $f$ is of the form $f(z)=az+b$, with $a\neq 0$, then obviously $f$ is in $Aut(\mathbb{C})$. How can I prove the converse? I see stuff like command 1&gt; out or with 2&gt;&amp;1 to redirect stderr, but sometimes I also see &amp;&gt; by itself, etc.  What is the best way to understand &amp; and what it means exactly? On every loading of a lib, I get the error:     no version information available   This lib has been compiled on another PC (ubuntu 10.04) than the one running it (mandriva 2010.2).  $ ldd /usr/local/gnu-eabi-4.5.2/i686-pc-linux-gnu/arm-eabi/lib/libbfd-2.21.so /usr/local/gnu-eabi-4.5.2/i686-pc-linux-gnu/arm-eabi/lib/libbfd-2.21.so: /lib/libz.so.1: no version information available (required by /usr/local/gnu-eabi-4.5.2/i686-pc-linux-gnu/arm-eabi/lib/libbfd-2.21.so)  linux-gate.so.1 =&gt;  (0xffffe000)  libz.so.1 =&gt; /lib/libz.so.1 (0xb77a7000)  libc.so.6 =&gt; /lib/i686/libc.so.6 (0xb7655000)  /lib/ld-linux.so.2 (0xb787a000)   Edit: the workaround didn't work. Why and how does C# allow accessing private variables outside the class itself when it's within the same containing class? How do I get the last column text to be at the bottom of the row?  That is I want the XXX to be aligned with the line containing the arrow. I tried playing around with \multirow{3}{-3ex}{XXX}, but obviously I don't understand \multirow. Code: \documentclass{article} \usepackage{tabularx} \usepackage{booktabs} \usepackage{multirow}  \begin{document}     \begin{tabularx}{\linewidth}[t]{l X p{1.3cm}}         &amp; Header &amp; Title \\\cmidrule(lr){2-2}\cmidrule(lr){3-3}         a. &amp; Some text text text text text text text text text text text text text text              text text text text text text text text text text              Want XXX on this line $\rightarrow$%          &amp; XXX\\     \end{tabularx} \end{document} Drupal 7. Custom module. I have a content type 'RFI' with a handful of fields I created with Drupal UI. I want a user to: fill out a form &amp; click submit while drupal: saves fields to database, creates a new node of 'RFI' and populates the fields of the node with the content provided by the form's corresponding fields.  From what I understand:  Build a form with fields (taking from the 'DBTNG examples' module just for clarity):  function rfi_build_form($form, &amp;$form_state) { $form = array();  $form['add'] = array( '#type'  =&gt; 'fieldset', '#title' =&gt; t('Add a person entry'), ); $form['add']['name'] = array( '#type'  =&gt; 'textfield', '#title' =&gt; t('Name'), '#size'  =&gt; 15, ); $form['add']['surname'] = array( '#type'  =&gt; 'textfield', '#title' =&gt; t('Surname'), '#size'  =&gt; 15, ); $form['add']['age'] = array( '#type'  =&gt; 'textfield', '#title' =&gt; t('Age'), '#size'  =&gt; 5, '#description' =&gt; t("Values greater than 127 will cause an exception. Try it - it's a    great example why exception handling is needed with DTBNG."), ); $form['add']['submit'] = array( '#type'  =&gt; 'submit', '#value' =&gt; t('Add'), ); return $form; }   Here the form fields are saved to database with database API.  function rfi_build_submit($form, &amp;$form_state) { global $user;  // Save the submitted entry. $entry = array( 'name'    =&gt; $form_state['values']['name'], 'surname' =&gt; $form_state['values']['surname'], 'age'     =&gt; $form_state['values']['age'], 'uid'     =&gt; $user-&gt;uid, ); $return = dbtng_example_entry_insert($entry); if ($return) { drupal_set_message(t("Created entry @entry", array('@entry' =&gt; print_r($entry, TRUE)))); } }   Now here is how I know how to create a node of RFI type. How do I take the values of the fields above [name], [surname], [age] and have them related to corresponding fields in my content type?  function rfi_build_type($form, &amp;$form_state){ $node = new stdClass(); $node-&gt;type = 'rfi'; $node-&gt;name = ''; $node-&gt;surname = ''; $node-&gt;age = ''; $node-&gt;language = 'en'; $node-&gt;uid = 1; node_save($node); }   I was thinking that since the fields have been saved already, that I could use a database query to get these values, but intuition tells me I'm overlooking something.  Thank you! At the end of the game, how are medals awarded?   As a secondary, what is the relation between bronze silver and gold medals, and how much experience do they reward? What really allows airplanes to fly? Is it bad to marinate meat for too long? From my experience it is always best to marinate meat for a decent amount of time, so the meat can "soak" up the marinade and be more flavourful.  However, I came across this recipe here:   In the directions, it says: Add pork and marinade for a minimum of 6 hours or overnight, but no longer than 12 hours.  I was wondering, why does it say "no longer than 12 hours"? Isn't it good to let meat marinate for as long as possible?  Thanks. Is the question mark misused in affirmative sentences? Word caption style -- different style for label and number? This should be a common problem for anyone involved in technical writing.  When you insert an image (or table etc.) caption in Word, you get  dialog. You enter your caption,  "A happy bicycle" (in yellow).  This might be fine for a happy bicycle, but I want to have captions that have one style for the "enumeration" (label and number) and one style for the description (caption text, as called by Word), as such:    I have tried searching for this but I haven't found anything yet... App crash when editing comment or uploading image So, when I point a laser at a piece of wire (pointed at a specific angle), a circle of light appears on a wall behind it (see image below). I am trying to see why this happens and if there are any readings on this. Could anyone give some tips on what I should search for or any references?   ()  So this is a problem for a physics tournament, but I haven't tried it myself yet. However I have found a video depicting what I mean. How do I enumerate the properties of a JavaScript object? How do I enumerate the properties of a JavaScript object?  I actually want to list all the defined variables and their values, but I've learned that defining a variable actually creates a property of the window object. Why does my item name not show up in 1.13? How to fix task manager memory column if it shows lies I have a problem in showing memory in task manager with memory and paging columns. Why it shows so small size usage? And maybe anyone knows how to fix it?  Here is a screenshot(working set, memory, page pool) sorted by most consuming: Actually I have installed the new Ubuntu 15.04 64bit, and when I'm trying to install Steam on it, there is a problem. In fact I have downloaded the deb package from steam website, installed it and when I start it nothing happens. I tried to start it from terminal and what I've got  $ steam Running Steam on ubuntu 15.04 64-bit STEAM_RUNTIME is enabled automatically Installing breakpad exception handler for appid(steam)/version(0_client) libGL error: unable to load driver: r600_dri.so libGL error: driver pointer missing libGL error: failed to load driver: r600 libGL error: unable to load driver: swrast_dri.so libGL error: failed to load driver: swrast