10
votes
Accepted
Ordinary partitions vs partitions into odd parts
The g.f. for the right-hand is
$$e^{2x}\cdot e^{2x^2} \cdots = e^{2\frac{x}{1-x}}.$$
For the left-hand side, additionally introducing variable $y$ to account for partition length, we get
$$\sum_{n\...
- 29.1k
9
votes
Ordinary partitions vs partitions into odd parts
Yes, they are the same. It is more convenient to write this in terms of compositions instead of partitions. Writing $\mathcal{OC}(n)$ for the set of compositions of $n$ with odd parts, the LHS is $$\...
- 1,432
7
votes
Ordinary partitions vs partitions into odd parts
For what it worth, here is a combinatorial proof.
We start with a known
Lemma 1. Let $m$ be an even positive integer. Then the number of permutations of $[m]$ with only odd cycles equals to the number ...
- 96.8k
6
votes
Accepted
Reference request for a preprint by Effros-Ruan
I am not sure that requests for filesharing are appropriate for MO, but in this case I think I can justify it to myself, because the preprint you refer to was (probably) the one that became the ...
3
votes
Integration against Eisenstein series can be regarded as a cup product
Yes, that does indeed sound like something I might have said :)
I was referring to some extremely powerful theorems, originally due to Michael Harris, which show that:
The cohomology groups of ...
- 34.9k
2
votes
Reference request for combinatorial problem related to $\max$ relation
I also don't have a reference, but I think one should be able to do this fairly explicitly as follows [it appears that Peter Taylor alludes to this strategy in their comment above]:
First, note that ...
- 305
2
votes
Apéry's constant $\zeta(3)$ fastest convergent series
To answer your question, a google search of part of the formula shows no other results but this one, suggesting it is original
https://www.google.com/search?q=%22376698240%22
It' possible to make ...
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2
votes
Accepted
I am looking for a paper by Zalgaller
There is an English PDF on Springer, and Google Scholar gives this for the Russian version.
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1
vote
I am looking for a paper by Zalgaller
Here is the URL of this pdf file:
https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=znsl&paperid=2499&option_lang=rus
- 86.4k
1
vote
Extending Hölder functions
I think that for the $C^{k, \alpha}(A)$ space, where $A$ is bounded and the regularity of its boundary is $C^{k, \alpha}$ as well, an extension theorem holds: see for example Theorem 4 in Brian Krumme'...
- 11
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