All questions with tag [math: arithmetic-combinatorics]


Multiset Combination in Combinatorics

I want to buy a $k$-combination of doughnuts, where $k$ is any amount less than or equal to the total doughnuts available. At the bakery there are $n$ different types of doughnuts but there are restricted amount left for each type of doughnut. For example, In this case, the total doughnuts available is 4 + 2 + 3 + 2 + 7 + 2 + 8 = 28 and n = 7 ...
2022-07-25 19:42:12

Given a list of $2^n$ nonzero vectors in $GF(2^n)$, do some $2^{n-1}$ of them sum to 0?

Let $G=(\mathbb{Z/2Z})^n$ written additively, $n>1$. (you can think of it as $\mathbb{F}_{2^n}$ but I didn't find that useful... yet) Let $v_i$ be nonzero elements of $G$ for $i \in \{1 \dots 2^n \}$ so that $\sum_{i=1}^{2^n} v_i=0$. Is it possible to find a subset $J \subset \{1 \dots 2^n \}$, $|J|=2^{n-1}$ so that $$\sum_{j \in J} v_j=0...
2022-07-08 20:32:28

Expectation of a Random Subset of the Roots of Unity.

Let $p$ be a prime. If $1_A(x)$ represents the indication function of the set $A\subset\mathbb{Z}/p\mathbb{Z}$ and also $$\hat{1}_A(t):=\frac{1}{p}\sum_{n=1}^p 1_A(n)e^{2\pi i \frac{nt}{p}}$$ represents the Fourier change of $1_A$, after that what can be claimed concerning $$\mathbb{E}\left( \sup_{t\neq 0} |\hat{1}_A(t)|\right)?$$ Do we have an ...
2022-07-04 20:52:15

Uses of Chevalley-Warning

In the current IMC 2011, the last trouble of the 1st day (no. 5, the hardest of that day) was as adheres to: We have $4n-1$ vectors in $F_2^{2n-1}$: $\{v_i\}_{i=1}^{4n-1}$. The trouble asks: Confirm the presence of a part $A \subseteq [4n-1], |A| = 2n$ such that $\sum_{i \in A} v_i = \vec{0}$. Exists a remedy that makes use of the Chevalley-Wa...
2022-06-09 01:05:34