# Tagged Questions

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### Partitions tending to a constant

$P_{k}(n)$ = the number of partitions of n into k parts. Now, if we fix some $t\ge 0$ , then $\lim_{n\to\infty}P_{n-t}(n)\to$ c, c being some constant. Please help me with this! I believe ...
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### asymptotic notation rearrangment

I'm having a look at this paper http://arxiv-web3.library.cornell.edu/pdf/0903.3048v1.pdf namely Theorem 5 and why it implies Theorem 2 immediately. Basically, I'm hoping somebody could explain to ...
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### OEIS sequence A086449

OEIS sequence A086449 http://oeis.org/A086449 is defined by: $a(0)=1$, $a(2n+1)=a(n)$, $a(2n) = a(n)+a(n-1)+\ldots+a(n-2^m)+\ldots$ $= a(n)+\sum_{i=0}^{\lfloor\lg n\rfloor}a(n-2^i)$ One can show ...
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### Integer Partition into Powers

Is there any way to count the number of integer partitions of a number N into powers of two such that each size is repeated a power of two times? Ok so the recurrence can be expressed by: $a(0)=1$, ...
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### Big-Oh for size of a Sperner family

I'm developing an algorithm that will generate a collection of subsets of a ground set having the property that no subset in the collection is a subset of any other, and I'd like to give a Big-Oh ...
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### Rearranging asymptotic notation

If $a \le b^{\frac{1+\log_{2}b}{2}}(1+o(1))$, then what is $b$ in terms of $a$? Whenever I try to rearrange this, I get in a huge mess... Any help would be appreciated. Thanks.
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### Decreasing integers on the blackboard

There are $n\geq 2$ copies of an integer $k>0$ written on the blackboard. A move consists of choosing an integer $m>0$ on the blackboard, and replacing it as well as one other integer on the ...
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### Euler numbers grow $2\left(\frac{2}{ \pi }\right)^{2 n+1}$-times slower than the factorial?

Stirling's approximation of the factorial for even numbers is given by $$(2n)! \sim \left(\frac{2n}{e}\right)^{2n}\sqrt{4 \pi n}. \tag{1}$$ Further, the Euler numbers grow quite rapidly for large ...
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### Help understanding solution to growth of partition function

I'm currently a Combinatorics student trying to parse through this solution. I do not understand the proof currently. Any help understanding it is greatly appreciated. Question Let the number of ...
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### representing integers as linear combination of integers

Let $a,b,a',b'$ be $r-\epsilon_1$ bit positive integers. Let $c,d$ be $s+\epsilon_2$ bit positive integers. Fix a pair $c,d$ and vary $a,b$ over all $r-\epsilon_1$ bit numbers. Do we have almost ...
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### Asymptotics of coefficients in the expansion of $\log\cos x$

Let $c_n$ be the coefficient of $x^{2n}$ in the Maclauren expansion of $\log\cos x$. What can be said about the asymptotics of $c_n$ as $n\to\infty$? I expect that this question is routine, but I ...
I'm interested in some general information on the following question: Consider the collection of partitions of an $n$-set into $m$ blocks as a uniform probability space. Let $X$ be the random ...