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EXAMPLE
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The a(2) = 1 through a(8) = 14 partitions:
(211) (321) (422) (532) (633) (743) (844)
(3111) (431) (541) (642) (752) (853)
(41111) (5221) (651) (761) (862)
(5311) (4332) (7322) (871)
(511111) (5331) (7331) (5443)
(6222) (7421) (7441)
(6411) (7511) (7531)
(33222) (72221) (8332)
(6111111) (74111) (8521)
(71111111) (8611)
(82222)
(83311)
(85111)
(811111111)
For example, the partition (7,5,3,1) has submultisets (), (1), (3), (5), (7), (3,1), (5,1), (5,3), (7,1), (7,3), (7,5), (5,3,1), (7,3,1), (7,5,1), (7,5,3), (7,5,3,1), all of which have different sums except for (5,3) and (7,1), which both sum to 8, so (7,5,3,1) is counted under a(8).
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