10 votes

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\...
Max Alekseyev's user avatar
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 $$\...
lambda's user avatar
  • 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 ...
Fedor Petrov's user avatar
  • 96.8k
6 votes

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 ...
David Loeffler's user avatar
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 ...
Christian Gorski's user avatar
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 ...
CarP24's user avatar
  • 187
2 votes

I am looking for a paper by Zalgaller

There is an English PDF on Springer, and Google Scholar gives this for the Russian version.
Command Master's user avatar
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
Alexandre Eremenko's user avatar
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'...
Sevenoseven's user avatar

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