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A367412 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.

%I A367412 #6 Nov 20 2023 08:14:06
%S A367412 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 A367412 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 A367412 22,27,37,22,11,6,1,10,24,27,51,32,16,15
%N A367412 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 A367412 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 A367412 Triangle begins:
%e A367412   1
%e A367412   1  1
%e A367412   1  2
%e A367412   1  3  1
%e A367412   1  3  3
%e A367412   1  5  3  2
%e A367412   1  4  7  2  1
%e A367412   1  6  7  6  2
%e A367412   1  6 10  6  7
%e A367412   1  7 12 11  8  3
%e A367412   1  6 16 11 17  3  2
%e A367412   1 10 14 20 19 10  2  1
%e A367412   1  7 22 17 31 14  7  2
%e A367412   1  9 22 27 37 22 11  6
%e A367412   1 10 24 27 51 32 16 15
%e A367412   1 11 27 39 57 43 27 22  4
%e A367412   1  9 33 34 79 57 36 39  7  2
%e A367412   1 13 31 51 86 77 45 62 14  4  1
%e A367412 Row n = 9 counts the following partitions:
%e A367412   (9)  (81)         (711)       (621)      (5211)
%e A367412        (72)         (6111)      (531)      (4311)
%e A367412        (63)         (522)       (432)      (4221)
%e A367412        (54)         (51111)     (33111)    (42111)
%e A367412        (333)        (441)       (222111)   (3321)
%e A367412        (111111111)  (411111)    (2211111)  (32211)
%e A367412                     (3222)                 (321111)
%e A367412                     (3111111)
%e A367412                     (22221)
%e A367412                     (21111111)
%t A367412 DeleteCases[Table[Length[Select[IntegerPartitions[n], Length[Union[Total/@Subsets[#, {2}]]]==k&]], {n,10},{k,0,n}],0,2]
%Y A367412 Row sums are A000041.
%Y A367412 Column k = 1 is A088922.
%Y A367412 The non-binary version (with zeros) is A365658.
%Y A367412 The strict non-binary version (with zeros) is A365832.
%Y A367412 The corresponding rank statistic is A366739.
%Y A367412 A001358 lists semiprimes, squarefree A006881, conjugate A065119.
%Y A367412 A126796 counts complete partitions, ranks A325781, strict A188431.
%Y A367412 A276024 counts positive subset-sums of partitions, strict A284640.
%Y A367412 A365924 counts incomplete partitions, ranks A365830, strict A365831.
%Y A367412 A366738 counts semi-sums of partitions, non-binary A304792.
%Y A367412 A366741 counts semi-sums of strict partitions, non-binary A365925.
%Y A367412 Cf. A046663, A117855, A122768, A238628, A299701, A365543, A366753, A367095.
%K A367412 nonn,tabf
%O A367412 0,5
%A A367412 _Gus Wiseman_, Nov 19 2023