cp's OEIS Frontend

A344607 Number of integer partitions of n with reverse-alternating sum >= 0.

%I A344607 #26 Jun 09 2021 06:22:18
%S A344607 1,1,2,2,4,4,8,8,15,16,27,29,48,52,81,90,135,151,220,248,352,400,553,
%T A344607 632,859,985,1313,1512,1986,2291,2969,3431,4394,5084,6439,7456,9357,
%U A344607 10836,13479,15613,19273,22316,27353,31659,38558,44601,53998,62416,75168
%N A344607 Number of integer partitions of n with reverse-alternating sum >= 0.
%C A344607 The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
%C A344607 Also the number of reversed integer partitions of n with alternating sum >= 0.
%C A344607 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 A344607 All integer partitions have alternating sum >= 0, so the non-reversed version is A000041.
%C A344607 Is this sequence weakly increasing? In particular, is A344611(n) <= A160786(n)?
%F A344607 a(n) + A344608(n) = A000041(n).
%F A344607 a(2n+1) = A160786(n).
%e A344607 The a(1) = 1 through a(8) = 15 partitions:
%e A344607   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A344607        (11)  (111)  (22)    (221)    (33)      (322)      (44)
%e A344607                     (211)   (311)    (222)     (331)      (332)
%e A344607                     (1111)  (11111)  (321)     (421)      (422)
%e A344607                                      (411)     (511)      (431)
%e A344607                                      (2211)    (22111)    (521)
%e A344607                                      (21111)   (31111)    (611)
%e A344607                                      (111111)  (1111111)  (2222)
%e A344607                                                           (3311)
%e A344607                                                           (22211)
%e A344607                                                           (32111)
%e A344607                                                           (41111)
%e A344607                                                           (221111)
%e A344607                                                           (2111111)
%e A344607                                                           (11111111)
%t A344607 sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
%t A344607 Table[Length[Select[IntegerPartitions[n],sats[#]>=0&]],{n,0,30}]
%Y A344607 The non-reversed version is A000041.
%Y A344607 The opposite version (rev-alt sum <= 0) is A027187, ranked by A028260.
%Y A344607 The strict case for n > 0 is A067659 (even bisection: A344650).
%Y A344607 The ordered version appears to be A116406 (even bisection: A114121).
%Y A344607 The odd bisection is A160786.
%Y A344607 The complement is counted by A344608.
%Y A344607 The Heinz numbers of these partitions are A344609 (complement: A119899).
%Y A344607 The even bisection is A344611.
%Y A344607 A000070 counts partitions with alternating sum 1 (reversed: A000004).
%Y A344607 A000097 counts partitions with alternating sum 2 (reversed: A120452).
%Y A344607 A035363 counts partitions with alternating sum 0, ranked by A000290.
%Y A344607 A103919 counts partitions by sum and alternating sum.
%Y A344607 A316524 is the alternating sum of prime indices of n (reversed: A344616).
%Y A344607 A325534/A325535 count separable/inseparable partitions.
%Y A344607 A344610 counts partitions by sum and positive reverse-alternating sum.
%Y A344607 A344612 counts partitions by sum and reverse-alternating sum.
%Y A344607 A344618 gives reverse-alternating sums of standard compositions.
%Y A344607 Cf. A006330, A071321, A071322, A124754, A239829, A239830, A344604, A344651, A344654, A344739, A344742.
%K A344607 nonn
%O A344607 0,3
%A A344607 _Gus Wiseman_, May 29 2021