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As $k\to\infty$, $\binom{2k-2}{k-1}\sim 2^{2k-2}/\sqrt{\pi k}$. As $N-k\to\infty$, $\binom{2N-2k}{N-1}\sim 2^{2N-2k}/\sqrt{\pi (N-k)}$. Now approximate the sum by an integral: $$\int_0^N \frac{\ln k} …

The bound of Ruciński and Vince is for strongly balanced, which is a more strict condition. If only balanced is required, the example of connected regular graphs provides a bound much greater than $n^ …

THIS ANSWER IS INCORRECT, SEE THE COMMENTS. According to Maple, $$ f(x) = -\frac{\gamma+\log x}{x} +\tfrac14 e^2 x - \tfrac19 e^3x^2 + \tfrac{1}{32}e^4x^3 - \tfrac{1}{150}e^5x^4 + O(x^5).$$ A converge …

Let $\bar x_0$ be your approximation of $x_0$. Find a small $\delta$ such that $f(\bar x_0)$ and $f(\bar x_0+\delta)$ have opposite sign. Then you know $x_0$ to an accuracy of $|\delta|$ (assuming $f$ …

Taking $f(a,b)=\frac{a+b}{a}$ as in the other question, I conjecture that the optimal $C$ is $e$. One bit of evidence is that the limit of the ratio for $a=cb$ and $b\to\infty$ is $(c+1)^{1/c}$ which …

This is closely related to Lucia's answer. For parameter $t$, let $X_t$ be the random variable with probability generating function $$ p_t(x) = \sum_{i=0}^c \mathrm{P}(X_t=i)\, x^i = \sum_{i=0}^c \fr …

This function has an interesting asymptotic expansion. $$\prod_{k\ge1}(1-n^{-k}) = 1 - n^{-1} - n^{-2} + n^{-5} + n^{-7} - n^{-12} - n^{-15} + \cdots\,\,. $$ The coefficients are all $\pm 1$ and the e …

Many unsolved asymptotics problems can be written as inclusion-exclusion sums. There is no general method for solving them. …

The limit is at least $2/\sqrt 3\approx 1.1547$. Write $\alpha=1/\sqrt 3$. If any strip length $\alpha n$ or more is used more than once, then obviously area $2\alpha$ can be chosen. On the other han …

The expected number of values hit is asymptotic to $\sqrt{n\log n}$. Start with Stirling's formula: $$ P(n,k) := 2^{-n}\binom{n}{n/2+k} = \sqrt{\frac{2}{\pi n}} e^{-2k^2/n} (1 + O(1/n)), $$ provided …

The following question is a simple case of a type of problem that occurs in combinatorial enumeration problems. Define $$F(x_1,\ldots,x_n) = \frac{1}{(2\pi)^{n/2}}\exp\biggl( -\frac12\sum_{j=1}^n x_ …

(Edited after Christian's comments.) For $0\le i\le n^{4/7}$, $$n!/(n-i)! = n^i \exp(-i(i-1)/(2n)+O(i^3/n^2)).$$ Approximate the sum for that range by the corresponding integral (a gaussian with the …

Since the vast majority graphs has trivial groups, and this is even more true (to exponential precision) for graphs with likely numbers of edges, the answer will be the same if you consider labelled g …

OP didn't say where the problem came from, so maybe I am just reverse-engineering it here. If $P$ is a partition of a finite set $X$, define $\kappa(P)$ to be the number of pairs $x,y\in X$ such that …

For two random permutations of $n$ letters, let $p_1(n)$ be the probability they are conjugate and $p_2(n)$ be the probability they have the same order. I computed these exactly up to $n=70$. In the …