All Questions
5
questions
0
votes
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19
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Estimate the order of restricted number partitions
There are $k$ integers $m_l,1\leq l\leq k
$(maybe negetive), satisfiying $|m_l|\leq M$ and $\sum_l m_l=s$.
I want to get an order estimate of the number of solutions for $k, M$ when fixing $s$.
I came ...
- 91
3
votes
1
answer
81
views
In how many ways can I express a positive integer as a sum of elements in a subset of $\mathbb Z^+$?
Let $S\subseteq \mathbb Z^+$ be set of positive integers. Given $n\in\mathbb Z^+$, how can I find the number of ways in which we can express $n$ as a sum of elements in $S$? ($S$ can be infinite.)
$$
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- 7,534
6
votes
1
answer
228
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Recently proposed problem by George Andrews on partitions in Mathstudent Journal (India)
Show that the number of parts having odd multiplicities in all partitions of $n$ is equal to difference between the number of odd parts in all partitions of $n$ and the number of even parts in all ...
0
votes
1
answer
248
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Number of pairwise non-isomorphic spanning trees of the wheel $W_n$, with restrictions
I recently encountered this problem. Frankly I'm stuck; would be nice for some help. Here it is:
Let $N,k$ be positive integers. By $p_k(N)$ we denote the number of integer partitions of $N$ with ...
- 2,003
8
votes
2
answers
249
views
Minimizing over partitions $f(\lambda) = \sum \limits_{i = 1}^N |\lambda_i|^4/(\sum \limits_{i = 1}^N |\lambda_i|^2)^2$
I'm trying to characterize the behavior of the the quantity:
$$A = \frac{\sum \limits_{i = 1}^N x_i^4}{(\sum \limits_{i = 1}^N x_i^2)^2},$$
subject to the constraints that
$$ \sum \limits_{i = 1}^N ...
- 664