OFFSET
0,8
COMMENTS
The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(m-1) times the number of odd parts in the conjugate partition, where m is the number of parts. By conjugation, this is also (-1)^(r-1) times the number of odd parts, where r is the greatest part. So a(n) is the number of integer partitions of n of even rank with the same number of odd parts as their conjugate.
EXAMPLE
The a(5) = 1 through a(12) = 11 partitions:
(311) (321) (43) (44) (333) (541) (65) (66)
(2221) (332) (531) (4321) (4322) (552)
(4111) (2222) (32211) (32221) (4331) (4332)
(4211) (51111) (52111) (4421) (4422)
(6311) (4431)
(222221) (6411)
(422111) (33222)
(611111) (53211)
(222222)
(422211)
(621111)
MATHEMATICA
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]], {i, Length[y]}];
conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
Table[Length[Select[IntegerPartitions[n], sats[#]==sats[conj[#]]&]], {n, 0, 15}]
CROSSREFS
The non-reverse version is A277103.
Comparing even parts to odd conjugate parts gives A277579.
Comparing signs only gives A340601.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 26 2021
STATUS
approved