On Refining Partitions

3. A lower bound To obtain a lower bound we count only those sequences of refinements which include the partition 1 .2.3 . . . d . r of n into d or d + 1 parts, d of which are of different size, where 0 < r = n-$d(d + 1) < d, so that J(2n) > d > J(2n)-3. Moreover we only count sequences in which we split off 1 from each of the d 1 parts of different size greater than 1. These d 1 steps can be made in (d l ) ! ways and result in the partition ldf 2 .3 . . . (d 1) .r of n into 2d 1 or 2d parts, d 1 (or possibly d ) of which are of different size. We deal with this in the same way, making splits of size 1 from each of the d-2 parts of different size greater than 1, in (d-2)! possible ways. If we continue, we see that the number of sequences of refinements is at least