login
Number of integer partitions of n whose semi-sums do not cover an interval of positive integers.
5

%I #6 Nov 18 2023 18:18:46

%S 0,0,0,0,0,1,2,5,9,13,22,30,46,63,91,118,167,216,290,374,490,626,810,

%T 1022,1297,1628,2051,2551,3176,3929,4845,5963,7311,8932,10892,13227,

%U 16035,19395,23397,28156,33803,40523,48439,57832,68876,81903,97212,115198

%N Number of integer partitions of n whose semi-sums do not cover an interval of positive integers.

%C We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

%e The a(0) = 0 through a(9) = 13 partitions:

%e . . . . . (311) (411) (331) (422) (441)

%e (3111) (421) (431) (522)

%e (511) (521) (531)

%e (4111) (611) (621)

%e (31111) (3311) (711)

%e (4211) (4311)

%e (5111) (5211)

%e (41111) (6111)

%e (311111) (33111)

%e (42111)

%e (51111)

%e (411111)

%e (3111111)

%t Table[Length[Select[IntegerPartitions[n], (d=Total/@Subsets[#,{2}];If[d=={}, {}, Range[Min@@d,Max@@d]]!=Union[d])&]], {n,0,15}]

%Y The complement for parts instead of sums is A034296, ranks A073491.

%Y The complement for all sub-sums is A126796, ranks A325781, strict A188431.

%Y For parts instead of sums we have A239955, ranks A073492.

%Y For all subset-sums we have A365924, ranks A365830, strict A365831.

%Y The complement is counted by A367402.

%Y The strict case is A367411, complement A367410.

%Y A000009 counts partitions covering an initial interval, ranks A055932.

%Y A086971 counts semi-sums of prime indices.

%Y A261036 counts complete partitions by maximum.

%Y A276024 counts positive subset-sums of partitions, strict A284640.

%Y Cf. A000041, A002033, A046663, A108917, A264401, A304792, A365543, A365658.

%K nonn

%O 0,7

%A _Gus Wiseman_, Nov 17 2023