A348552 - OEIS

%I #5 Nov 03 2021 09:08:29

%S 0,1,1,2,2,4,5,7,8,12,14,19,23,31,36,46,55,69,83,100,122,144,175,203,

%T 249,284,348,393,484,536,661,725,898,975,1208,1297,1614,1715,2136,

%U 2251,2812,2939,3674,3813,4779,4920,6172,6315,7943,8070,10156,10263,12944

%N Number of integer partitions of n with the same alternating product as alternating sum.

%C The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. In the case of a partition, this equals the number of odd parts in the conjugate partition.

%C We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).

%e The a(1) = 1 through a(9) = 12 partitions:

%e   1   2   3     4     5       6       7         8         9

%e           111   211   221     42      322       332       333

%e                       311     222     331       422       441

%e                       11111   411     511       611       522

%e                               21111   22111     4211      711

%e                                       31111     22211     22221

%e                                       1111111   41111     32211

%e                                                 2111111   33111

%e                                                           51111

%e                                                           2211111

%e                                                           3111111

%e                                                           111111111

%e For example, we have 3 - 2 + 2 - 1 + 1 = 3 / 2 * 2 / 1 * 1 = 3, so the partition (3,2,2,1,1) is counted under a(9).

%t ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];

%t altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];

%t Table[Length[Select[IntegerPartitions[n],altprod[#]==ats[#]&]],{n,0,30}]

%Y The version for reverse-alternating sum (or product, or both) is A025065.

%Y Dominated by A347446.

%Y A000041 counts partitions with alternating sum 0.

%Y A027187 counts partitions of even length.

%Y A027193 counts partitions of odd length, ranked by A026424.

%Y A097805 counts compositions by sum and alternating sum.

%Y A103919 counts partitions by sum and alternating sum (reverse: A344612).

%Y A119620 counts partitions with alternating product 1, ranked by A028982.

%Y A124754 gives alternating sums of standard compositions.

%Y A277103 counts partitions with the same alternating sum as their conjugate.

%Y A345927 gives alternating sums of binary expansions.

%Y Cf. A000070, A000097, A001700, A025047, A236913, A325534, A325535, A339846, A344607, A345196, A347443, A347448.

%K nonn

%O 0,4

%A _Gus Wiseman_, Oct 30 2021