%I #6 Nov 20 2023 08:14:06

%S 1,1,1,1,2,1,3,1,1,3,3,1,5,3,2,1,4,7,2,1,1,6,7,6,2,1,6,10,6,7,1,7,12,

%T 11,8,3,1,6,16,11,17,3,2,1,10,14,20,19,10,2,1,1,7,22,17,31,14,7,2,1,9,

%U 22,27,37,22,11,6,1,10,24,27,51,32,16,15

%N Triangle read by rows with all zeros removed where T(n,k) is the number of integer partitions of n with k different semi-sums.

%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 Triangle begins:

%e 1

%e 1 1

%e 1 2

%e 1 3 1

%e 1 3 3

%e 1 5 3 2

%e 1 4 7 2 1

%e 1 6 7 6 2

%e 1 6 10 6 7

%e 1 7 12 11 8 3

%e 1 6 16 11 17 3 2

%e 1 10 14 20 19 10 2 1

%e 1 7 22 17 31 14 7 2

%e 1 9 22 27 37 22 11 6

%e 1 10 24 27 51 32 16 15

%e 1 11 27 39 57 43 27 22 4

%e 1 9 33 34 79 57 36 39 7 2

%e 1 13 31 51 86 77 45 62 14 4 1

%e Row n = 9 counts the following partitions:

%e (9) (81) (711) (621) (5211)

%e (72) (6111) (531) (4311)

%e (63) (522) (432) (4221)

%e (54) (51111) (33111) (42111)

%e (333) (441) (222111) (3321)

%e (111111111) (411111) (2211111) (32211)

%e (3222) (321111)

%e (3111111)

%e (22221)

%e (21111111)

%t DeleteCases[Table[Length[Select[IntegerPartitions[n], Length[Union[Total/@Subsets[#, {2}]]]==k&]], {n,10},{k,0,n}],0,2]

%Y Row sums are A000041.

%Y Column k = 1 is A088922.

%Y The non-binary version (with zeros) is A365658.

%Y The strict non-binary version (with zeros) is A365832.

%Y The corresponding rank statistic is A366739.

%Y A001358 lists semiprimes, squarefree A006881, conjugate A065119.

%Y A126796 counts complete partitions, ranks A325781, strict A188431.

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

%Y A365924 counts incomplete partitions, ranks A365830, strict A365831.

%Y A366738 counts semi-sums of partitions, non-binary A304792.

%Y A366741 counts semi-sums of strict partitions, non-binary A365925.

%Y Cf. A046663, A117855, A122768, A238628, A299701, A365543, A366753, A367095.

%K nonn,tabf

%O 0,5

%A _Gus Wiseman_, Nov 19 2023