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Keywords here are "block design", "Steiner triple system" and "resolvable". We want to find $8$ resolution classes: each partitioning $\{1,2,\ldots,30\}$ into $10$ sets of size $3$. This is going to be an ugly case, since $30 \not\equiv 1,3 \pmod 6$ (a condition for the existence of a Steiner triple system). [The question would be more mathematically ...


2

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 ...


1

The number of partitions of $n$ into $k_2$ positive parts, no part exceeding $k_1$, is the coefficient of $x^n$ in $(x+x^2+x^3+\cdots+x^r)^s$, where I have written $r$ for $k_1$, and $s$ for $k_2$. This is the coefficient of $x^t$ in $(1+x+x^2+\cdots+x^{r-1})^s$, where I have written $t$ for $n-s$. Now ...


1

I'm assuming from your list of examples that what you are looking for is the number of partitions of $n$ into $k$ distinct parts with largest part equal to $m$ (this is different from the question in the title, but seems to fit your examples best). The number of partitions of $n$ with at most $k$ parts, each of length at most $m$, is the coefficient of ...


1

The partition function is asymptotically $\frac{1}{4n\sqrt{3}}e^{\pi\sqrt{2n/3}}$ and the Fibonacci numbers are asymptotically $\frac{1}{\sqrt{5}}\varphi^n$, where $\varphi$ is the golden ratio. The quotient of these two is less than $1$ if $n$ is large enough: $$\frac{\sqrt{5}e^{\pi\sqrt{2n/3}}}{4n\sqrt{3}\varphi^n} = ...


1

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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