|
%I #26 Jun 09 2021 06:22:18
%S 1,1,2,2,4,4,8,8,15,16,27,29,48,52,81,90,135,151,220,248,352,400,553,
%T 632,859,985,1313,1512,1986,2291,2969,3431,4394,5084,6439,7456,9357,
%U 10836,13479,15613,19273,22316,27353,31659,38558,44601,53998,62416,75168
%N Number of integer partitions of n with reverse-alternating sum >= 0.
%C The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
%C Also the number of reversed integer partitions of n with alternating sum >= 0.
%C A formula for the reverse-alternating sum of a partition is: (-1)^(k-1) times the number of odd parts in the conjugate partition, where k is the number of parts. So a(n) is the number of integer partitions of n whose conjugate parts are all even or whose length is odd. By conjugation, this is also the number of integer partitions of n whose parts are all even or whose greatest part is odd.
%C All integer partitions have alternating sum >= 0, so the non-reversed version is A000041.
%C Is this sequence weakly increasing? In particular, is A344611(n) <= A160786(n)?
%F a(n) + A344608(n) = A000041(n).
%F a(2n+1) = A160786(n).
%e The a(1) = 1 through a(8) = 15 partitions:
%e (1) (2) (3) (4) (5) (6) (7) (8)
%e (11) (111) (22) (221) (33) (322) (44)
%e (211) (311) (222) (331) (332)
%e (1111) (11111) (321) (421) (422)
%e (411) (511) (431)
%e (2211) (22111) (521)
%e (21111) (31111) (611)
%e (111111) (1111111) (2222)
%e (3311)
%e (22211)
%e (32111)
%e (41111)
%e (221111)
%e (2111111)
%e (11111111)
%t sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
%t Table[Length[Select[IntegerPartitions[n],sats[#]>=0&]],{n,0,30}]
%Y The non-reversed version is A000041.
%Y The opposite version (rev-alt sum <= 0) is A027187, ranked by A028260.
%Y The strict case for n > 0 is A067659 (even bisection: A344650).
%Y The ordered version appears to be A116406 (even bisection: A114121).
%Y The odd bisection is A160786.
%Y The complement is counted by A344608.
%Y The Heinz numbers of these partitions are A344609 (complement: A119899).
%Y The even bisection is A344611.
%Y A000070 counts partitions with alternating sum 1 (reversed: A000004).
%Y A000097 counts partitions with alternating sum 2 (reversed: A120452).
%Y A035363 counts partitions with alternating sum 0, ranked by A000290.
%Y A103919 counts partitions by sum and alternating sum.
%Y A316524 is the alternating sum of prime indices of n (reversed: A344616).
%Y A325534/A325535 count separable/inseparable partitions.
%Y A344610 counts partitions by sum and positive reverse-alternating sum.
%Y A344612 counts partitions by sum and reverse-alternating sum.
%Y A344618 gives reverse-alternating sums of standard compositions.
%Y Cf. A006330, A071321, A071322, A124754, A239829, A239830, A344604, A344651, A344654, A344739, A344742.
%K nonn
%O 0,3
%A _Gus Wiseman_, May 29 2021
| 