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• 31.3k
Accepted

Which of these sums appear most often?

In fact this question was already asked at MO, although in disguise: see here. Richard Stanley answered it wonderfully. The champions are the nearest integers to $n(n+1)/4$. For a quick proof, see ...
• 101k
Accepted

Identity involving a sum over all partitions of $n$

Here's a quick sketch (since I'm pressed for time). Multiply both sides of the identity by $t^n$ and sum over $n$ from $0$ to infinity. From the cycle decomposition identity (Polya's formula) the ...
• 43.5k
Accepted

Are the Fourier coefficients of $\eta(q^m)^m / \eta(q)$ non-negative?

Yes this is true and these Fourier coefficients actually enumerate some combinatorial objects called $m$-cores. These are partitions with no hooklength divisible by $m$. In fact for $m\geq 4$ these ...
• 85.2k
Accepted

• 10.8k
Accepted

"strange" diophantine and parity of the partition function

Your conjecture is true! Here is one way to get it using some mod 4 generatingfunctionology. The answer got a bit long, so I divided it into two parts, as an attempt to improve readability. Part1: ...
• 85.2k
Accepted

• 85.2k
Accepted

A binary hook-length formula?

I am afraid they are not always integers. Take large $p$ and $n=2^{2p}-1$. Then $[n]!_b$ is divisible by $p^N$ for $N={2p\choose p}+1$. And $[2n+1]!_b$ by $p^K$ for $K={2p+1\choose p}+2p+1<2N$. ...
• 104k
There's a well-known bijective proof that the number of self-conjugate partitions of $n$ is the same as the number of partitions of $n$ into distinct odd parts (see https://en.wikipedia.org/wiki/...