[seqfan] Re: A question about partitions

[William Keith]

Take the generating function given in A003056 and the Ramanujan form Sum_{k
>= 1} x^(k*(k+1)/2)/(1-x^k) for A001227.

Now as for the RHS, notice that such partitions are simply a triangle -- of
say, k parts -- plus 1 on the largest part, and then optional plus-1s on
anywhere from 0 to k-2 of the intermediate parts, having to be placed from
the second largest to the next-to-smallest part; plus any amount added to
all parts, which constitutes adding a multiple of k to the number being
partitioned.  For instance, such a partition into 4 parts is of one of the
forms:

(k+4) (k+3) (k+2) (k)
(k+4) (k+3) (k+1) (k)
(k+4) (k+2) (k+1) (k)

This gives the generating function \sum_{k \geq 1} q^{\binom{k+1}{2}+1} (1
+ q + \dots + q^{k-2}) \cdot \frac{1}{1-q^k} .   A little algebra gives
that this is the difference of the previous two generating functions.

I looked for a bijective proof but didn't find one immediately.

Best,
William Keith