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Results tagged with co.combinatorics
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user 101078
Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.
18
votes
Accepted
Can the equation $1+z+z^2=z^n$ for natural $n$ have multiple complex roots $z$?
Suppose that $z$ is our multiple root, $n\geq 4$. Since $z$ is a root of equation $(n-2)z^2+(n-1)z+n=0$, we have $|z|^2=\frac{n}{n-2}$. Indeed, the roots are non-real and conjugate, so the other root …
- 7,193
4
votes
Accepted
Limit associated with two Beatty sequences that are not a Beatty pair
I will give a complete answer when $1,1/r$ and $1/s$ are linearly independent over $\mathbb Q$ and a recipe to compute your $t$ otherwise.
First of all, notice that $n$ lies in a Beatty sequence $\lfl …
- 7,193
1
vote
Bounds for the sequence $a(n,A)=n*a(\lfloor (1-A)n \rfloor,A)$
Define the sequence $n_k$ via the recursive formula $n_0=n$ and $n_{k+1}=\lfloor(1-A)n_k \rfloor$. As $0<A<1$, the sequence is decreasing. Let $M$ be the largest solution of inequality $n_M\geq 1$. Fi …
- 7,193
5
votes
Accepted
Inequality for $3$-adic valuation
Notice that
$$
3^{-k-1}f(n,k+1)/3^{-k}f(n,k)=\frac{{2n+2k+2\choose n+k+1}}{{2n+2k\choose n+k}}\frac{{n+k+1\choose n-k-1}}{{n+k \choose n-k}}.
$$
The first factor here equals
$$
\frac{(2n+2k+2)!}{(2n+2 …
- 7,193