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Results tagged with co.combinatorics
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user 106551
Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.
1
vote
1
answer
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Limit of quotients of polynomials at fixed value
Let $0<t<1$ be a parameter. Let $n\in\mathbb{Z}$, $n>0$. For $i\in\{0,1,\dots,2^n-1\}$, we always consider $i$ as having $n$ binary digits (positions $1$ to $n$), putting $0$s if necessary. Let $f(i)$ …
7
votes
What are the odds of a tie in a random election with k candidates?
In general, the probability is
$$
\frac{1}{k^N}\sum_{j=2}^k{k \choose j}\sum_{m=0}^{\lfloor N/j\rfloor}{N \choose mj}\frac{(mj)!}{(m!)^j}\sum_{\substack{(a_1,\dots,a_{k-j})\\a_1+\dots+a_{k-j}=N-mj\\0\ …
1
vote
Asymptotic growth of antichains in divisibility posets
Here is an expression for $c$ based on Mikhail Tikhomirov's nice answer to the question (divisibility independence). Let us also use his notation from that answer. With $1\leq x \leq n$, $x\equiv 1 \b …
1
vote
Permutations of squares and finite fields
Let $A_q=\{x^2:x\in\mathbb{F}_q^{*}\}$.
Let $\pi^{\prime}$ be the permutation on $A_q$ defined by $$\pi^{\prime}(a_k)=a_{\pi(k)}.$$
Then
$$
\sum_{k=1}^{(q-1)/2}a_ka_{\pi(k)}=\sum_{a\in A_q}(a\pi^{\pri …