Questions tagged [combinatorial-number-theory]

Filter by
Sorted by
Tagged with
5 votes
3 answers

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 ...
qifeng618's user avatar
  • 706
5 votes
1 answer

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&...
Sebastien Palcoux's user avatar
6 votes
2 answers

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 ...
Sebastien Palcoux's user avatar
5 votes
0 answers

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 ...
Theodore Slaman's user avatar
7 votes
1 answer

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 ...
Jakub Konieczny's user avatar
17 votes
5 answers

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 ...
Esfan Haghverdi's user avatar
6 votes
2 answers

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:...
Joseph O'Rourke's user avatar
11 votes
3 answers

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 $...
darij grinberg's user avatar