Tagged Questions

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No simple closed form for Bell numbers

The Bell number $B_n$ is the number of partitions of $[n]$. Unlike other basic combinatorial quantities, $B_n$ has no simple finite closed form. This seems surprising to me. Can anyone explain why ...
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Integral of products of cosines

Given $m+1$ integers $\alpha_0,\ldots,\alpha_m\geq 1$, I was trying to get a nice closed formula for the integral $$\int_0^\pi\cos(\alpha_1\theta)\cdots\cos(\alpha_m\theta)d\theta.$$ More precisely, ...
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Book recommendation for Integer partitions and q series

I have been studying number theory for a little while now, and I would like to learn about integer partitions and q series, but I have never studied anything in the field of combinatorics, so are ...
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Partitions of an integer into k parts.

I am interested in knowing whether an exact formula (analogous to the Hardy-Ramanujan-Rademacher formula for $p(n)$) for the number of partitions of a positive integer into k parts is known. I tried ...
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Shifted Young tableaux & Hook numbers & Bulgarian Solitaire

I would like to find articles or documentation regarding this process: Starting from what ever integer partition, e.g. 5,2 for the number 7. Construct his Young tableaux and then fill it with Hook ...
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The “theory of compound partitions”

I was reading a sort of mini-bio on Sylvester the other day and a "Theory of Compound Partitions" was mentioned in the discussion of his research interests. I wanted to ask, is this the same or the ...
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Closed-form Expression of the Partition Function $p(n)$

I feel like I have seen news that a paper was recently published, at most a few months ago, that solved the well-known problem of finding a closed-form expression for the partition function $p(n)$ ...
Let $t \geq 1$ and $\pi$ be an integer partition of $t$. Then the number of set partitions $Q$ of $\{1,2,\ldots,t\}$ for which the multiset $\{|q|:q \in Q\}=\pi$ is given by \[\frac{t!}{\prod_{i \geq ...