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Well, if $\mathcal O$ is the set of all subsets of $U$ then there are $2^{2^u}$ subsets of $\mathcal O$, where $u$ is the cardinality of $U$, and most of these will have union equal to $U$. In fact, a …

Let $\lambda$ denote the uniform measure on the powerset of $\mathbb N$ (also known as the Lebesgue measure and the fair-coin measure). Let $\tau$ be the product topology of the discrete topology on $ …

It seems that Kleitman and Rothschild (Proc. AMS, 1970 and Trans. AMS, 1975) defined $O_n(=\text{top}(n))$ to be the number of preorders and $P_n(=\text{top}_0(n))$ to be the number of partial orders …

Of course all subfamilies of $2^{[n/2-1]}$ are sparse. For any $k$, the number of union-closed families $\mathcal F\subseteq 2^{[k]}$ is $u(k)=2^{\binom{k}{\lfloor k/2\rfloor}(1+o(1))}$ (link), where …

In computability theory, the function $r\mapsto j_r$ is called the principal function of the set $J=\{j_1<\dots<j_m\}$ and denoted $p_J$. The relation with your function being that $p_r=j_r-r$, i.e., …

Let $B$ be the minimum number of bins required. Since you asked about limits with respect to any of the variables, let's observe that the obvious inequality $$1\le B\le \min(k,N)$$ is completely sharp …

See The Longest Run of Heads by Mark F. Schilling which gives a recursive formula for $\Pr (R_N\le M)$, whereas you are looking for $\Pr (R_N\le M) -\Pr (R_N\le M-1)$.

This is $\mathbb E (X_1^{n-1} X_2) $ where $ X_i $ is the $ i $ th order statistic of a sample from the uniform distribution on $[0, t] $. To evaluate it can try using the joint pdf of these order sta …

Also not an answer, but may be useful. A somewhat similar kind of word is mentioned at the end of section 1.5 of Salomaa: Jewels of Formal Language Theory: There is an infinite word over a 3-letter …

Regarding the 3rd question, I will show this: Theorem. For a random binary word of length $n$, the expected number of $h$th powers is $$ \sim \frac{n}{2^{h-1}-1}. $$ Proof. A basic event about occurr …

Here is the case $k=1$, $S=2$, and $L\le 2$. So we have two players who each choose $L$ many numbers among $N$ possibilities, aiming to choose something the other does not choose. So let $p_{N,L}$ b …

Let $s$ consist of $2^{2^k}$ zeros, followed by the same number of ones, for increasing $k$: $$0^21^2\, 0^{16}1^{16}\, 0^{256}1^{256}\,0^{65536}1^{65536}\dots$$ Observe that all but finitely many bloc …

Let us choose $\alpha$ many sets of size $n/3$ at random. The cardinality $X=|U\cap V|$ of the intersection of two of them is a hypergeometric random variable (well, given $U$, but the value of $U$ do …

Euler in 1769 conjectured that for all integers n and k greater than 1, if the sum of k nth powers of positive integers is itself a nth power, then k is greater than or equal to n: $$a^n_1 + a^n_2 …

This is studied in Reverse Mathematics as the Chain Antichain Principle (CAC) and it is observed that it follows from Ramsey's theorem.