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Every partition counted in $p(n,k)$ is one of the following two forms. One of the subsets in the partition is $\{n\}$. There are $p(n-1,k-1)$ of these, since all you need to count is the number of ways to partition the remaining elements $1,\ldots, n-1$ into $k-1$ subsets. $\{n\}$ is not one of the subsets. Then, you need to count the number of ways of ...


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The question if this integral is zero, is one of the classical $NP$-complete Problems given in the book of Garey and Johnson, Computers and Intractability, 1979. p. 252. There called AN14 and with the upper limit of the integral $2\pi$ instead of $\pi$.


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This problem has a solution using ordinary generating functions. First question. Observe that $$\sum_{q\ge 0} q z^q = \frac{z}{(1-z)^2}.$$ Therefore the generating function of the contribution from partitions with $k$ terms is given by $$\left(\frac{z}{(1-z)^2}\right)^{k-1} \frac{z}{1-z}$$ and the contribution from all partitions is $$p(z) = \sum_{k\ge ...


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Let $\omega(n)$ be the amount of ways you can express $n$ as $a_0!a_1!\cdots c_k!$. Then obviously $\omega(n!) = \Omega(n)$. Since $n\geq \max \lbrace a_i\rbrace_{1\leq i\leq n} \geq P_\max$. Then $\max\lbrace a_i\rbrace$ can be any number, $k$, between $P_\max$ and $n$. For each of those numbers, we have $n! = k!\frac{n!}{k!}$ so we need to express ...


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The partitioning follows directly from the definition of a total order (specifically, this is a strict total order, like < on the integers). Strict total orders exhibit trichotomy: any pair of objects is either in order one way, or the other, or else it is the same object. These three categories are exclusive, thus it partitions the set. For the second ...


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This paper provides an example of a distribution on $\Bbb N$ which has infinite entropy. Thus, in your case $P$ is any partition satisfying $$ m(I_i) = \frac{1}{\lg(i+1)} - \frac{1}{\lg(i+2)}, \qquad i\in \Bbb N, $$ where $\lg i$ is a logarithm base $2$. For example, $I_1 = [1,\frac1{\lg3})$, $I_2 = [\frac1{\lg 3},\frac12)$, $I_3 = [\frac12,\frac1{\lg5})$ ...


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Your description of reflexivity is correct. For symmetry it means that the subset $R$ is "symmetric" around the line $y = x$, this means that for any point $(a, b)\in R$ its mirror point $(b, a)\in R$ (it's the point you get by doing reflection in the line $y=x$), i.e. either none of the two points $(a, b)$ and $(b, a)$ is included in $R$, or both of them ...


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First, you have to define clearly how you are representing a relation between two real numbers graphically. For example, if $a \sim b$, are you plotting the point $(a, b)$ on the graph? Then it would make sense to plot each point $(a, a)$ for every $a \in S$ (but it will not make up the entire line $y = x$ unless $S = \mathbb R$). If this is your ...



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