2
votes
0answers
170 views
A set-theoretic combinatoric problem
Let $A=\{a_i\in\mathbb{Q}:a_i> 1\land 1\le i \le b-2\}$, (cardinality of $A$ is $b-2$) and $\displaystyle B=\bigcup_{c=0}^{\infty} \left[\dfrac{bc+1}{d},\dfrac{b(c+1)-1}{d}\righ …
1
vote
0answers
78 views
Periodic tilings of the plane with fundamental domain given by $k$ squares of prescribed size
Given $k$ strictly positive real numbers $l_1,\dots,l_k$, can one decide the existence of a periodic tiling of the plane whose fundamental domain is the union of $k$ squares
of le …
7
votes
1answer
216 views
Non-enumerative proof that there are many simple permutations?
Terence Tao asked for a non-enumerative proof that a positive proportion of permutations are derangements and got a great answer. Inspired by this, I'd like to ask about another fa …
3
votes
2answers
205 views
The proportion between permutations and derangements.
Denote the number of derangements by $D_N$. It's known that $D_N/N! \rightarrow 1/e$. Therefore $N!/e$ is an approximation for $D_N$.
I'm trying to bound the difference between th …
6
votes
1answer
259 views
A sum-product estimate in Z/p^2Z
We are interested in a sum-product type estimate. Let $p$ be an odd prime, and let $A$ be the order $p-1$ subgroup of $(\mathbb{Z}/p^2\mathbb{Z})^\times$. That is, let $A = \langle …
6
votes
2answers
355 views
Why are negative sets multisets? (Reference request)
It is easy to establish that
$$
\left({n\choose k}\right)=(-1)^k{-n \choose k},
$$
where the symbol on the left-hand-side counts the number of multisets of $k$ elements from $n$.
…
0
votes
0answers
79 views
Probability distribution : a number as a randomly built up partition [closed]
A row of $\mathbb{L}$ empty cells is available to represent a number as follows
Number
1 : -------- Choose any
cell at random and add 1.
2 …
12
votes
0answers
422 views
How to explain the picturesque patterns in François Brunault’s matrix?
How to explain the patterns in the matrix defined in François Brunault's
answer to the question Freeness of a Z[x] module depicted below? --
Choosing colors according to the highe …
1
vote
2answers
206 views
Upper Limit on the Central Binomial Coefficient
What is the tightest upper bound we can establish on the central binomial coefficients $ 2n \choose n$ ?
I just tried to proceed a bit, like this:
$ n! > n^{\frac{n}{2}} $
for a …
16
votes
0answers
194 views
congruence on words: having the same (scattered) subwords of length at most n
For a fixed finite alphabet $A=\{a,b,...\}$, write $x \sim_n y$ if the two words $x$ and $y$ have the same (scattered) subwords of length at most $n$. The relation $\sim_n$ is a co …
5
votes
0answers
144 views
Can Sudoku be solved using matroid theory?
Advanced sudoku-solving seems rather streamlined:
for each square write down the acceptable values
Using a standard set of techniques - many of which I did not know by name - ded …
33
votes
7answers
2k views
Conway’s game of life for random initial position
What is the behavior of Conway's game of life when the initial position is random? -- We can ask this question on an infinite grid or on an $n$ by $n$ table (planar or on a torus). …
2
votes
1answer
130 views
Independence of an interval and a product set in $\mathbb Z/L\mathbb Z$.
I have $k$ distinct prime numbers $\ell_1 < \dots <\ell_k$, and for each $i=1,\dots,k$, a subset $A_i$ of $\mathbb Z / \ell_i \mathbb Z$. Let $L=\ell_1 \dots \ell_k$. Now usi …
24
votes
3answers
1k views
the following inequality is true,but I can’t prove it
The inequality is
\begin{equation*}
\sum_{k=1}^{2d}\left(1-\frac{1}{2d+2-k}\right)\frac{d^k}{k!}>e^d\left(1-\frac{1}{d}\right)
\end{equation*}
for all integer $d\geq 1$. I use com …
1
vote
2answers
112 views
Discriminants of indefinite integral binary quadratic forms admitting 3 or 6 torsion.
Are there any results known about the discriminants of indefinite integral binary quadratic forms admitting automorphisms of order 3 or 6? It seems reasonable to expect that any p …

