Let p¯k(n) be the number of partitions of n with largest part at most k which is  equivalent to partition into at most k parts. I do know an expression for that function. ( product of 1/1(1-n) through 1/(1-n)^k )   What I am searching for is an expression that gives that number of partitions under both restrictions, so all partitions of n that have at most k parts AND all parts &lt;= j.
How can I find the number of partitions of $n$ into exactly $k$ distinct parts, where each part is at most $M$?  The number of partitions $p_k(\leq M,n)$ of $n$ into at most $k$ parts, each of size at most $M$, is given by the generating function: $$  \binom{M+k}{k}_{x} = \prod_{j=1}^{k}\frac{1-x^{M+k-j+1}}{1-x^j}= \sum_{n=0}^{kM} p_{k}(\leq M,n) x^n $$  For the number of the partitions $p_k(\mathcal{D},n)$ of $n$ into at most $k$ parts there is the recurrence relationship: $$   p_{k}(\mathcal{D},n) = p_{k}(\mathcal{D},n-k) + p_{k-1}(\mathcal{D},n) $$  But what, if I want to count only the partitions with distinct parts and restricted number of parts and restricted part size?  Update: Now I know the generating function for the number of distinct restricted partitions $p_k(\leq M, \mathcal{D},n)$ of $n$ into exactly $k$ distinct parts, all at most $M$ is $$    \prod_{j=1}^{M} (1+xq^{j}) = \sum_{k,n=0}^{\infty}p_k(\leq M, \mathcal{D},n)x^{k}q^{n} $$ and there is also a recurrence relation $$    p_k(\leq M, \mathcal{D},n) =         p_{k-1}(\leq M-1, \mathcal{D},n-k) +         p_k(\leq M-1, \mathcal{D},n-k) $$ How can I prove this? Could you recommend a book, where I could read about this?
Finding $\lim_{n\to\infty}\binom{2n}{n}$ I found a limit problem which is $\lim_{n\to\infty}{2n \choose n}$ as we know that $\sum_{i=0}^{n}{n \choose i}^2={2n \choose n}$ now we could transform the given problem as $\lim_{n\to\infty} \sum_{i=0}^{n}{n \choose i}^2$ and as $\sum_{i=0}^{n}{n \choose i}^2$ is a polynomial in $n$, therefore, $\lim_{n\to\infty}\sum_{i=0}^{n}{n \choose i}^2=\infty$ Please tell me if my solution is correct or not,and if it is incorrect then please provide me a solution.  Thanks "...would have trouble imagining...": Is this present participle or gerund? Badge Name is Blue? Now some random badges on the Profile page for the mobile web is now blue! This is seemingly weird as the default and now permanent color for Badge name text is white. Here's the problem here:     Ahhhh! Now it has invaded the desktop/computer version too! Now seemingly, every version has badges with blue text. Android app has the same issues too! Can the badge color be fixed from blue to white on the mobile version please? How to calculate $$\int_{0}^{\pi/2}{\{\tan(x)\}dx}$$ where $\{x\}$ means the fractional part of $x$. I mentioned the zero points of $\{\tan(x)\}$. $\{\tan(x)\}$ equals to 0 iff $x=\arctan(k)$, $k$ is an integer. I tried to separate it into$$\sum_{k=0}^{\infty}{\int_{\arctan(k)}^{\arctan(k+1)}{(\tan(x)}-k)dx}$$ Then I found it hard to continue. I don't think I am on the right way. How do amortization schedules work and when are they used? Specifically, I notice that on a mortgage statement, my payment is "amortized" so that at the beginning of the loan, I am paying almost entirely interest and very little principal.  However, on a credit card balance, it seems interest is only charged on the balance.  Why do amortization schedules work the way they do, and are they typically used for most loans, and why? What is the formula for percent of frame filled at a specific distance and focal length? I am thinking about buying a new lens specifically to take photos of distant targets (birds, bears, etc). I currently own the Canon 70-300 IS USM, which is a nice lens, but I find that if I want to take a photo of anything further away than about 10 meters, the target does not fill a large amount of the frame (meaning I have to significantly crop in post production).  I am currently looking at the Canon prime 400mm, however I would like to determine at what distance an object will fill a reasonable proportion of the frame.  Is there a formula or rule of thumb I can apply that will help me in this situation? I realise the size of the target I am shooting will play a role here, so if we need to make an assumption about the size of the target please let me know. Is any particular punctuation necessary when a quotation is the subject of a sentence? For example, in the sentence below, should there be a comma after export? Or anything else?     "Laughter is America's most important export" is a well-known quote of Walt Disney's. Drag-and-drop functionality no longer working after upgrade to 19.04 After upgrading from 18.04 to 19.04 (with a necessary upgrade to 18.10 between) on a Lenovo Yoga 11e, I can no longer drag and drop anything (ie, files, windows, etc.) with the touchpad.  I can, however, move things around using the touchscreen.  Whenever I attempt to do so with the mouse, though, sometimes it turns into a hand.  Whether or not it turns into the hand or remains a pointer, the icon will not move anymore.  IOW, as soon as I left-click, the pointer freezes in place until I release the touchpad. Origins of possessive pronouns Number of ways in which $5$ boys and $5$ girls are ordered in such a way that exactly $4$ girls stand consecutively in the queue Let $n$ be the number of ways in which $5$ boys and $5$ girls can stand in a queue in such a way that all the girls stand consecutively in the queue. Let $m$ be the number of ways in which $5$ boys and $5$ girls in such a way that exactly $4$ girls stand consecutively in the queue. Then find $\frac{m}{n}$.    I could find $n$ correctly as $6!\times 5!$, but I am finding $m$ as $7!\times 4!\times \binom{5}{1}$, but the book says $m$ is $6!\times 4!\times 5\times \binom{5}{4}$. Where have I gone wrong? What is the proper logic to find $m$? Why do RAW images look worse than JPEGs in editing programs? I've found that when you load a RAW image into an editing program such as Lightroom/Aperture, the image is usually worse than if you just took the image as JPEG. Now I understand that the camera does some magic during the JPEG conversion. But I'm trying to understand what that "magic" is.   If I'm looking to implement that "magic" myself on the desktop, what kind of settings should I be trying? I find that the RAW files have more extreme contrast. The dark areas for example are way darker than JPEG. Why is this? And what is the best way to fix this? How to test API integration in marketing cloud? I have created a API package under Setup -> Installed Packages.   I have created API Integration component for the package .I have the below basic details for the component.   Client Id Client Secret Authentication Base URI REST Base URI   I want to test the created API package using any tool( Postman or other API testing tools).  I was able to find API integration testing links for SFDC but not for the Marketing Cloud.   What is the redirect URL to be used in the API package? I have created an API integration package of Integration type ( Web APP)   In web app properties, what is the redirect URL to be provided? Side effects of changing blackboard bold font with \setmathfont Moment generating function of the natural sufficient statistics of Gamma distribution I have $X_1,X_2 ... X_n$ Gamma-distributed r.v with density:    First of all I showed, that Gamma distribution belongs to exponential family and can be represented in form     I found, that for Gamma distribution $B(\theta) = log(\theta)$ where $\theta$ is $\alpha/\beta$  Then we know, that moment generating function of the random variable $T(X)$ is given by $$M(t)=\exp\{B(\theta+t)-B(\theta)\}$$  Natural sufficient statistics for Gamma distribution is:    But I didn't understand what should I do next? Because Natural sufficient statistics is two dimensional it confuses me:  $M(T) = exp[log(\theta + T) - log(\theta)]$  and $T = [s,t]^\intercal$ Say I have the following CSS and HTML code:    #header {   height: 150px; } &lt;div id="header"&gt;   &lt;h1&gt;Header title&lt;/h1&gt;   Header content (one or multiple lines) &lt;/div&gt;     The header section is fixed height, but the header content may change.  I would like the content of the header to be vertically aligned to the bottom of the header section, so the last line of text "sticks" to the bottom of the header section.  So if there is only one line of text, it would be like:   ----------------------------- | Header title | | | | header content (resulting in one line) -----------------------------   And if there were three lines:   ----------------------------- | Header title | | header content (which is so | much stuff that it perfectly | spans over three lines) -----------------------------   How can this be done in CSS? Indian passport threading coming off My Indian passport threading is coming off i.e. the cover of the passport is coming off, there are only 2 threads remaining.   Will it be a problem at the airport when going abroad ? Also anyone who has experience of getting a replacement passport fairly quick ? Because I need to get a passport in like 15 days. How can I output a list of all installed packages with installation date and additional information? How do I programmatically determine operating system in Java? I quite enjoy paying attention to how I design my documents and presentations. I usually spend hours thinking over and designing my slides for a workshop or presentation, so that they are aesthetically pleasing and as intuitive as possible. Likewise I recently started revising my CV I figured and I wanted to make it stand out a bit more. (Just to make it clear I don't mean making a clown of a document but just better use of colors, contrast and design elements.)  I have long wondered whether or not this is something that can backfire, since most documents in academic context are extremely plain, at least in my experience. It's very common to see the default Powerpoint slides (white bg, black arial text) or .   My question is as follows: is putting time and effort into design of academic documents something that can backfire? Will I risk being prejudged with first impressions such as "well he put much effort in the presentation his documents, perhaps because the content is sub-par"?     I realize that the question might be somewhat subjective from person to person but I encourage everyone to consider it in terms of this SE blog entry: . The ideal $I= \langle x,y \rangle\subset k[x,y]$ is not a principal ideal.   I don't know how to consider it. Any suggestions?