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%I #5 Nov 18 2023 18:07:58
%S 0,0,0,0,0,0,0,1,2,2,4,5,8,10,14,16,23,27,35,42,52,61,75,89,106,126,
%T 149,173,204,237,274,319,369,424,490,560,642,734,838,952,1085,1231,
%U 1394,1579,1784,2011,2269,2554,2872,3225,3619,4054,4540,5077,5671,6332
%N Number of strict 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 partition y = (4,2,1) has semi-sums {3,5,6} which are missing 4, so y is counted under a(7).
%e The a(7) = 1 through a(13) = 10 partitions:
%e (4,2,1) (4,3,1) (5,3,1) (5,3,2) (5,4,2) (6,4,2) (6,4,3)
%e (5,2,1) (6,2,1) (5,4,1) (6,3,2) (6,5,1) (6,5,2)
%e (6,3,1) (6,4,1) (7,3,2) (7,4,2)
%e (7,2,1) (7,3,1) (7,4,1) (7,5,1)
%e (8,2,1) (8,3,1) (8,3,2)
%e (9,2,1) (8,4,1)
%e (5,4,2,1) (9,3,1)
%e (6,3,2,1) (10,2,1)
%e (6,4,2,1)
%e (7,3,2,1)
%t Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&(d=Total/@Subsets[#, {2}];If[d=={},{}, Range[Min@@d,Max@@d]]!=Union[d])&]], {n,0,30}]
%Y For parts instead of sums we have A238007:
%Y - complement A001227
%Y - non-strict complement A034296, ranks A073491
%Y - non-strict A239955, ranks A073492
%Y The non-strict version is A367403.
%Y The non-strict complement is A367402.
%Y The complement is counted by A367410.
%Y The non-binary version is A365831:
%Y - non-strict complement A126796, ranks A325781
%Y - complement A188431
%Y - non-strict A365924, ranks A365830
%Y A000009 counts partitions covering an initial interval, ranks A055932.
%Y A046663 counts partitions w/o submultiset summing to k, strict A365663.
%Y A365543 counts partitions w/ submultiset summing to k, strict A365661.
%Y Cf. A000041, A002033, A261036, A264401, A276024, A284640, A304792, A364272.
%K nonn
%O 0,9
%A _Gus Wiseman_, Nov 17 2023