cp's OEIS Frontend

A345196 Number of integer partitions of n with reverse-alternating sum equal to the reverse-alternating sum of their conjugate.

%I A345196 #8 Jun 27 2021 07:52:21
%S A345196 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 A345196 172,238,314,345,372,500,649,721,782,1019,1307,1451,1577,2015,2552,
%U A345196 2841,3098,3885,4867,5418,5914,7318,9071,10109,11050
%N A345196 Number of integer partitions of n with reverse-alternating sum equal to the reverse-alternating sum of their conjugate.
%C A345196 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 A345196 The a(5) = 1 through a(12) = 11 partitions:
%e A345196   (311)  (321)  (43)    (44)    (333)    (541)    (65)      (66)
%e A345196                 (2221)  (332)   (531)    (4321)   (4322)    (552)
%e A345196                 (4111)  (2222)  (32211)  (32221)  (4331)    (4332)
%e A345196                         (4211)  (51111)  (52111)  (4421)    (4422)
%e A345196                                                   (6311)    (4431)
%e A345196                                                   (222221)  (6411)
%e A345196                                                   (422111)  (33222)
%e A345196                                                   (611111)  (53211)
%e A345196                                                             (222222)
%e A345196                                                             (422211)
%e A345196                                                             (621111)
%t A345196 sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
%t A345196 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A345196 Table[Length[Select[IntegerPartitions[n],sats[#]==sats[conj[#]]&]],{n,0,15}]
%Y A345196 The non-reverse version is A277103.
%Y A345196 Comparing even parts to odd conjugate parts gives A277579.
%Y A345196 Comparing signs only gives A340601.
%Y A345196 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A345196 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A345196 A120452 counts partitions of 2n with rev-alt sum 2 (negative: A344741).
%Y A345196 A124754 gives alternating sums of standard compositions (reverse: A344618).
%Y A345196 A316524 is the alternating sum of the prime indices of n (reverse: A344616).
%Y A345196 A325534 counts separable partitions, ranked by A335433.
%Y A345196 A325535 counts inseparable partitions, ranked by A335448.
%Y A345196 A344610 counts partitions by sum and positive reverse-alternating sum.
%Y A345196 A344611 counts partitions of 2n with reverse-alternating sum >= 0.
%Y A345196 Cf. A000070, A000097, A006330, A027187, A027193, A236559, A239829, A257991, A344607, A344608, A344651, A344654.
%K A345196 nonn
%O A345196 0,8
%A A345196 _Gus Wiseman_, Jun 26 2021