cp's OEIS Frontend

A347443 Number of integer partitions of n with reverse-alternating product <= 1.

%I A347443 #13 Dec 13 2022 07:56:03
%S A347443 1,1,1,2,3,5,6,10,12,19,22,34,40,60,69,101,118,168,195,272,317,434,
%T A347443 505,679,793,1050,1224,1599,1867,2409,2811,3587,4186,5290,6168,7724,
%U A347443 9005,11186,13026,16062,18692,22894,26613,32394,37619,45535,52815,63593,73680
%N A347443 Number of integer partitions of n with reverse-alternating product <= 1.
%C A347443 Includes all partitions of even length (A027187).
%C A347443 Also the number of integer partitions of n with reverse-alternating sum <= 1.
%C A347443 Also the number of integer partitions of n having either even length (A027187) or having exactly one odd part in the conjugate partition (A100824).
%C A347443 We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). The reverse-alternating product is the alternating product of the reversed sequence.
%F A347443 a(n) = A027187(n) + A035363(n-1) for n >= 1. [Corrected by _Georg Fischer_, Dec 13 2022]
%F A347443 a(n) = A119620(n) + A344608(n).
%e A347443 The a(1) = 1 through a(8) = 12 partitions:
%e A347443   (1)  (11)  (21)   (22)    (32)     (33)      (43)       (44)
%e A347443              (111)  (31)    (41)     (42)      (52)       (53)
%e A347443                     (1111)  (221)    (51)      (61)       (62)
%e A347443                             (2111)   (2211)    (331)      (71)
%e A347443                             (11111)  (3111)    (2221)     (2222)
%e A347443                                      (111111)  (3211)     (3221)
%e A347443                                                (4111)     (3311)
%e A347443                                                (22111)    (4211)
%e A347443                                                (211111)   (5111)
%e A347443                                                (1111111)  (221111)
%e A347443                                                           (311111)
%e A347443                                                           (11111111)
%t A347443 altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t A347443 Table[Length[Select[IntegerPartitions[n],altprod[Reverse[#]]<=1&]],{n,0,30}]
%Y A347443 The odd-length case is A035363 (shifted).
%Y A347443 The strict case is A067661.
%Y A347443 The non-reverse version is counted by A119620, ranked by A347466.
%Y A347443 The even bisection is A236913.
%Y A347443 The opposite version (>= instead of <=) is A344607.
%Y A347443 The case of < 1 instead of <= 1 is A344608.
%Y A347443 The multiplicative version (factorizations) is A347438, non-reverse A339846.
%Y A347443 Allowing any integer reverse-alternating product gives A347445.
%Y A347443 The complement (> 1 instead of <= 1) is counted by A347449.
%Y A347443 Ranked by A347465, non-reverse A347450.
%Y A347443 A000041 counts partitions.
%Y A347443 A027187 counts partitions of even length.
%Y A347443 A027193 counts partitions of odd length.
%Y A347443 A058622 counts compositions with alternating sum <= 0 (A294175 for < 0).
%Y A347443 A100824 counts partitions with alternating sum <= 1.
%Y A347443 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A347443 A347461 counts possible alternating products of partitions.
%Y A347443 A347462 counts possible reverse-alternating products of partitions.
%Y A347443 Cf. A000070, A038548, A086543, A116406, A325534, A325535, A344611, A344654, A344740, A347440, A347442, A347446, A347448.
%K A347443 nonn
%O A347443 0,4
%A A347443 _Gus Wiseman_, Sep 14 2021