All Questions
7 questions
6
votes
0
answers
171
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An inequality involving integer partitions
For integers $n\ge k\ge0$, let $p(n,k)$ denote the number of ways to write $n$ as a sum of $k$ positive integers (repetition allowed). For example, $p(6,3)=3$ since
$$6=1+1+4=1+2+3=2+2+2.$$
QUESTION. ...
- 15.6k
3
votes
0
answers
214
views
A family of polynomials related to integer partitions
For a positive integer $n$, let $p(n)$ be the number of partitions of $n$.
For $1\le k\le n$, let $p(n,k)$ denote the number of partitions of $n$ having exactly $k$ terms; in other words, $p(n,k)$ is ...
- 15.6k
3
votes
2
answers
459
views
Short sequence beats long sequence
I have encountered some comparison between two binomial sums. It was amusing how the one with "fewer" summands exceeds (in value) than the other which consists of many more terms. In fact, ...
- 43.2k
1
vote
0
answers
150
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How should the first n natural numbers be arranged in a circle to minimize the sum of the products of adjacent pairs? [closed]
I was able to find (and prove) arrangements that would result in the sum of the products of adjacent pairs attain the maximum.
I am able to conjecture that the arrangement that would result in the ...
- 11
2
votes
3
answers
365
views
Is this number theoretic quantity bounded above?
I am considering a combinatorial argument which involves the following quantity. We use the prime counting function $\pi(n)$ and to save on exponents we set $h=\pi(n/2)$. The quantity as a function ...
- 13k
9
votes
0
answers
365
views
How to count integer lattice points close to a subspace of $\mathbb R^n$?
Consider $m$ linearly independent vectors in $n$-dimensional Euclidean space, $v_1,...,v_m \in \mathbb R^n$ where $1\leq m<n$, and let $U := {\rm span}(v_1,...,v_m)$ denote the $m$-dimensional ...
- 686
1
vote
2
answers
111
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A two-parameter inequality on product of linear terms
I would like to ask about a certain inequality that I need and which came out of some work in here.
Question. For integers $n\geq1$ and $k\geq3$, is this true? If so, any proof?
$$6\prod_{j=1}^k(...
- 43.2k