Questions tagged [combinatorial-number-theory]
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10 questions
3
votes
0
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Does a Motzkin-prefix greedy construction for distinct subset sums recover the Conway–Guy sequence?
A finite set $P=\{p_1>\dots>p_n\}\subset\mathbb{N}$ has distinct subset sums (DSS) if all $2^n$ subset sums are different. A problem of Erdős asks for $f(n)$, the minimal possible value of ...
- 131
1
vote
0
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78
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Lower bounds on the degree of polynomials vanishing on lattice points in a ball
Is there a known lower bound on the degree $D$ of a Real polynomial $P$ of $d$ variables which is non-zero yet vanishes on all the lattice points in the closed ball of radius $r$ about the origin? How ...
- 111
5
votes
3
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How to prove the combinatorial identity $\sum_{k=\ell}^{n}\binom{2n-k-1}{n-1}k2^k=2^\ell n\binom{2n-\ell}{n}$ for $n\ge\ell\ge0$?
With the aid of the simple identity
\begin{equation*}
\sum_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^{k}}=2^n
\end{equation*}
in Item (1.79) on page 35 of the monograph
R. Sprugnoli, Riordan Array Proofs of ...
- 1,209
5
votes
1
answer
693
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Eulerian ordering of the integers modulo n
Let $n>1$ be an integer. Consider the set $C_n := \{0,1, \dots , n-1\}$.
An Eulerian ordering of $C_n$ is an ordering $r_1, \dots, r_n$ of its elements such that:
$$\forall i \le n \ \forall j&...
- 27.6k
6
votes
2
answers
561
views
Divisibility labeling on a boolean lattice and positive Euler totient
Let $B_n$ be the rank $n$ boolean lattice (i.e. the subset lattice of $\{1,2, \dots , n \}$). Let $\hat{0}$ and $\hat{1}$ be the minimum and the maximum of $B_n$. Let $f: B_n \to \mathbb{N}$ be a ...
- 27.6k
5
votes
0
answers
287
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Partitions of numbers with restrictions on repetitons
I have a question about restricted partitions of numbers:
For $n$ and $k$ positive integers let $M$ be the multiset in which each positive integer less than n appears exactly $k$ times. I want to ...
- 1,955
7
votes
1
answer
253
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Are irrational multiples of central sets again central?
Let me begin by giving the relevant definitions. A set $A \subset \mathbb{N}$ is said to be central if and only if there exists a topological system $(X,T)$ (with $X$ a compact metric space, $T$ a ...
- 1,944
17
votes
5
answers
3k
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Category Theory and Ergodic Theory
I am very much interested in finding out about any category theoretical work on dynamical systems and on ergodic theory. On the face of it, it seems that a categorical language can go a long way, at ...
- 179
6
votes
2
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347
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Repeatedly indexing into an $\infty$-sequence of integers
Suppose one has in hand an infinite sequence $s$ of distinct natural numbers,
for example,
$$s=s_1=(1, 3, 5, 7, 9, 11, 13, 15, 17, 19,\ldots) \;.$$
So this sequence can be considered an injection
$f:...
- 153k
13
votes
3
answers
3k
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Gauss sum (with sign) through algebra
Let $p$ be an odd prime, and $\zeta$ a primitive $p$-th root of unity over a field of characteristic $0$.
Let $G = \sum\limits_{j=0}^{p-1} \zeta^{j\left(j-1\right)/2}$ be the standard Gauss sum for $...
- 35.5k