A345196 - OEIS

%I #8 Jun 27 2021 07:52:21

%S 1,1,0,1,1,1,1,3,4,4,4,8,11,11,11,20,27,29,31,48,65,70,74,109,145,160,

%T 172,238,314,345,372,500,649,721,782,1019,1307,1451,1577,2015,2552,

%U 2841,3098,3885,4867,5418,5914,7318,9071,10109,11050

%N Number of integer partitions of n with reverse-alternating sum equal to the reverse-alternating sum of their conjugate.

%C 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.

%e The a(5) = 1 through a(12) = 11 partitions:

%e   (311)  (321)  (43)    (44)    (333)    (541)    (65)      (66)

%e                 (2221)  (332)   (531)    (4321)   (4322)    (552)

%e                 (4111)  (2222)  (32211)  (32221)  (4331)    (4332)

%e                         (4211)  (51111)  (52111)  (4421)    (4422)

%e                                                   (6311)    (4431)

%e                                                   (222221)  (6411)

%e                                                   (422111)  (33222)

%e                                                   (611111)  (53211)

%e                                                             (222222)

%e                                                             (422211)

%e                                                             (621111)

%t sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];

%t conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];

%t Table[Length[Select[IntegerPartitions[n],sats[#]==sats[conj[#]]&]],{n,0,15}]

%Y The non-reverse version is A277103.

%Y Comparing even parts to odd conjugate parts gives A277579.

%Y Comparing signs only gives A340601.

%Y A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.

%Y A103919 counts partitions by sum and alternating sum (reverse: A344612).

%Y A120452 counts partitions of 2n with rev-alt sum 2 (negative: A344741).

%Y A124754 gives alternating sums of standard compositions (reverse: A344618).

%Y A316524 is the alternating sum of the prime indices of n (reverse: A344616).

%Y A325534 counts separable partitions, ranked by A335433.

%Y A325535 counts inseparable partitions, ranked by A335448.

%Y A344610 counts partitions by sum and positive reverse-alternating sum.

%Y A344611 counts partitions of 2n with reverse-alternating sum >= 0.

%Y Cf. A000070, A000097, A006330, A027187, A027193, A236559, A239829, A257991, A344607, A344608, A344651, A344654.

%K nonn

%O 0,8

%A _Gus Wiseman_, Jun 26 2021