All Questions
5
questions
0
votes
0
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48
views
Find generating series on set of descending sequences, with weight function as taking sum of sequence
Given the set of all sequences of length k with descending (not strictly, so $3,3,2,1,0$ is allowed) terms of natural numbers (including $0$), $S_k$, and the weight function $w(x)$ as taking the sum ...
- 183
1
vote
1
answer
57
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What is the 11th unordered combination of natural numbers that add upto 6 in the partition function?
So, I was making unordered combinations of natural numbers which add upto a certain natural number. I was able to go till 6 when I got to know about the partition function. I was pleased to see that ...
- 177
2
votes
2
answers
81
views
Show that series converges by estimating number of partitions into distinct parts
I need some help with solving the following problem: Let $Q(n)$ be the number of partitions of $n$ into distinct parts. Show that $$\sum_{n=1}^\infty\frac{Q(n)}{2^n}$$ is convergent by estimating $Q(n)...
- 155
4
votes
0
answers
151
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Evaluate $ \frac{1}{(q)_\infty} \sum_{m \in \mathbb{Z}} q^{\frac{m^2}{2}} (-q^{-\frac{1}{2}}x)^m y^m(q^{1-m}y^{-1};q)_\infty $
This identity is taken from a physics paper [1] stated without proof, on page 43.
$$ \frac{1}{(q)_\infty} \sum_{m \in \mathbb{Z}} q^{\frac{m^2}{2}} (-q^{-\frac{1}{2}}x)^m y^m(q^{1-m}y^{-1};q)_\infty
=...
- 24.5k
2
votes
3
answers
2k
views
Algorithm for the number of partitions of $n$ into distinct parts
I am looking for an algorithm to find the number of ways of writing $n$ as a sum of positive integers without regard to order with the constraint that all integers in a given partition are distinct. ...
- 65