cp's OEIS Frontend

A344608 Number of integer partitions of n with reverse-alternating sum < 0.

%I A344608 #16 Jun 09 2021 06:22:38
%S A344608 0,0,0,1,1,3,3,7,7,14,15,27,29,49,54,86,96,146,165,242,275,392,449,
%T A344608 623,716,973,1123,1498,1732,2274,2635,3411,3955,5059,5871,7427,8620,
%U A344608 10801,12536,15572,18065,22267,25821,31602,36617,44533,51560,62338,72105,86716
%N A344608 Number of integer partitions of n with reverse-alternating sum < 0.
%C A344608 The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
%C A344608 Also the number of reversed of integer partitions of n with alternating sum < 0.
%C A344608 No integer partitions have alternating sum < 0, so the non-reversed version is all zeros.
%C A344608 Is this sequence weakly increasing? Note: a(2n + 2) = A236914(n), a(2n) = A344743(n).
%C A344608 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 of even length whose conjugate parts are not all odd. Partitions of the latter type are counted by A086543. By conjugation, a(n) is also the number of integer partitions of n of even maximum whose parts are not all odd.
%e A344608 The a(3) = 1 through a(9) = 14 partitions:
%e A344608   (21)  (31)  (32)    (42)    (43)      (53)      (54)
%e A344608               (41)    (51)    (52)      (62)      (63)
%e A344608               (2111)  (3111)  (61)      (71)      (72)
%e A344608                               (2221)    (3221)    (81)
%e A344608                               (3211)    (4211)    (3222)
%e A344608                               (4111)    (5111)    (3321)
%e A344608                               (211111)  (311111)  (4221)
%e A344608                                                   (4311)
%e A344608                                                   (5211)
%e A344608                                                   (6111)
%e A344608                                                   (222111)
%e A344608                                                   (321111)
%e A344608                                                   (411111)
%e A344608                                                   (21111111)
%t A344608 sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
%t A344608 Table[Length[Select[IntegerPartitions[n],sats[#]<0&]],{n,0,30}]
%Y A344608 The opposite version (rev-alt sum > 0) is A027193, ranked by A026424.
%Y A344608 The strict case (for n > 2) is A067659 (odd bisection: A344650).
%Y A344608 The Heinz numbers of these partitions are A119899 (complement: A344609).
%Y A344608 The bisections are A236914 (odd) and A344743 (even).
%Y A344608 The ordered version appears to be A294175 (even bisection: A008549).
%Y A344608 The complement is counted by A344607 (even bisection: A344611).
%Y A344608 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A344608 A027187 counts partitions with alternating sum <= 0, ranked by A028260.
%Y A344608 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A344608 A120452 counts partitions with rev-alternating sum 2 (negative: A344741).
%Y A344608 A316524 is the alternating sum of the prime indices of n (reverse: A344616).
%Y A344608 A325534/A325535 count separable/inseparable partitions.
%Y A344608 A344604 counts wiggly compositions with twins.
%Y A344608 A344610 counts partitions by sum and positive reverse-alternating sum.
%Y A344608 A344618 gives reverse-alternating sums of standard compositions.
%Y A344608 Cf. A000070, A000097, A000346, A003242, A006330, A071322, A239829, A239830, A344649, A344651, A344654/A344740, A344739.
%K A344608 nonn
%O A344608 0,6
%A A344608 _Gus Wiseman_, May 30 2021