# Questions tagged [partitions]

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### Equality of two $q$-series. Proof?

Recall the notation $(z;q)_n=(1-z)(1-zq)(1-zq^2)\cdots(1-zq^{n-1})$. My earlier MO question did not find enough interest or yield an answer. Perhaps the modulo $2$ part might have thrown people off. ...
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### Congruence modulo 2 for q-series

This quest arose from certain calculations with integer partitions (having distinct parts) and the corresponding values of their Dyson ranks. I would like to ask: QUESTION. Is this congruence true ...
196 views

### Agreement between two sets of data on partitions

Let $\lambda=(\lambda_1,\lambda_2,\dotsc,\lambda_{\ell(\lambda)})$ be an integer partition of positive numbers where $\ell(\lambda)$ is the length of the partition. One may associate a Ferrer diagram ...
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### Factorizable partition polynomials

Let $p(n)$ denote the number of (unrestricted) integer partition of $n$ which has the product generating function $$\sum_{n\geq0}p(n)\,x^n=\prod_{j\geq1}\frac1{1-x^j}.$$ On the other hand, for the ...
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### A refinment of Beck's conjecture

Let $\mathcal{O}(n)$ and $\mathcal{D}(n)$ denote the set of all integer partitions of $n$ into odd parts and distinct parts, respectively. Let $o(n)=\#\mathcal{O}(n)$ and $d(n)=\#\mathcal{D}(n)$. ...
1 vote
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