  We obtain a combinatorial proof of a surprising weighted partition equality
of Berkovich and Uncu. Our proof naturally leads to a formula for the number of
partitions with a given parity of the smallest part, in terms of S(i), the
number of partitions of i into distinct parts with even rank minus the number
with odd rank, for which there is an almost closed formula by Andrews, Dyson
and Hickerson. This method of calculating the number of partitions of n with a
given parity of the smallest part is practical and efficient.
