[seqfan] A conjecture of George Beck about A008484

Allan Wechsler acwacw at gmail.com
Fri Aug 18 04:49:21 CEST 2023

https://oeis.org/A008484 counts the number of partitions of n into parts of
size at least 4.

The last comment in the comment field is a conjecture that I suspect could
quickly be proven or disproven by a motivated undergraduate, and I urge
anybody to try it.

The conjecture states that A(n) is also the number of partitions of n for
which three times the number of 1-parts is twice the total number of parts.

I'm almost certain this could be settled with generatingfunctionology, but
an ideal demonstration would be combinatorial.

If it resists proof for more than a few days, it might be false, and it
would be worth writing a program to look for values of n for which the
counts of the two kinds of partitions are different. If they are the same
up to n = 1000, then go back to trying to prove it.

It just feels like really low-hanging fruit, but I wasn't able to do it in
ten minutes, so here we are.

-- Allan

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