All Questions
7 questions
2
votes
0
answers
189
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Component-wise sums of permutations
Given a set $S$ containing all possible permutations of a vector $v = (1, 2, 3, ..., n-1, n)$, find the size of the set $P$, where $P$ is defined as the set of possible component-wise sums obtained by ...
- 21
0
votes
0
answers
266
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Is there a permutation $\tau\in S_n$ with $\tau(1)^{\tau(2)}+\cdots+\tau(n-1)^{\tau(n)}+\tau(n)^{\tau(1)}$ a square?
Let $n>1$ be an integer, and let $S_n$ be the symmetric group of all the permutatins of $\{1,\ldots,n\}$.
I'm curious whether there is a permutation $\tau\in S_n$ such that
$$\tau(1)^{\tau(2)}+\...
- 15.6k
7
votes
0
answers
116
views
A conjecture on circular permutations of n elements in an abelian group of odd order
In 2013 I formulated the following conjecture in additive combinatorics.
Conjecture. Let $G$ be an additive abelian group of odd order, and let $A$ be a subset of $G$ with $|A|=n>2$. Then, there is ...
- 15.6k
3
votes
0
answers
282
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A new combinatorial problem for finite groups
In a recent preprint arXiv:1811.10503, I proved that if $a_1,\ldots,a_n$ are distinct elements of a torsion-free additive abelian group $G$, then there is a permutation $\pi\in S_n$ such that all ...
- 15.6k
3
votes
0
answers
193
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A conjectural lower bound for $|\{\sum_{k=1}^nka_k:\ a_1,\ldots,a_n\ \text{are distinct elements of }\ A\}|$
Motivated by Question 315568 of mine, I'm interested in the set
$$S(n):=\bigg\{\sum_{k=1}^n k\pi(k):\ \pi\in S_n\bigg\}.$$
It is easy to see that
$$S(1)=\{1\},\ S(2)=\{4,5\}\ \text{and}\ S(3)=\{10,...
- 15.6k
7
votes
1
answer
509
views
A permutation problem
Here I ask a question on permutations of $n$ distinct real numbers.
QUESTION: Let $a_1,a_2,\ldots,a_n\ (n>1)$ be (pairwise) distinct real numbers. Is there a permutation $b_1,\ldots,b_n$ of $a_1,\...
- 15.6k
5
votes
1
answer
331
views
Sets $A$ stable under $(x,f(x))\mapsto x+f(x)$
Let $A$ be a finite set of real numbers or integers. We know how to characterize, broadly speaking, sets $A$ such that $A+A$ is not much larger than $A$ (Freiman's theorem). I have a question that ...
- 20.2k