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%I #8 Jan 26 2024 08:40:08
%S 1,1,1,2,1,2,3,2,2,3,5,3,2,3,5,7,5,4,4,5,7,11,7,6,3,6,7,11,15,11,8,7,
%T 7,8,11,15,22,15,12,10,4,10,12,15,22,30,22,16,14,12,12,14,16,22,30,42,
%U 30,22,17,17,6,17,17,22,30,42,56,42,30,25,23,20,20,23,25,30,42,56
%N Triangle read by rows where T(n,k) is the number of integer partitions of n with a unique submultiset summing to k.
%F A367094(n,1) = A108917(n).
%e Triangle begins:
%e 1
%e 1 1
%e 2 1 2
%e 3 2 2 3
%e 5 3 2 3 5
%e 7 5 4 4 5 7
%e 11 7 6 3 6 7 11
%e 15 11 8 7 7 8 11 15
%e 22 15 12 10 4 10 12 15 22
%e 30 22 16 14 12 12 14 16 22 30
%e 42 30 22 17 17 6 17 17 22 30 42
%e 56 42 30 25 23 20 20 23 25 30 42 56
%e 77 56 40 31 30 27 7 27 30 31 40 56 77
%e Row n = 5 counts the following partitions:
%e (5) (41) (32) (32) (41) (5)
%e (41) (311) (311) (311) (311) (41)
%e (32) (221) (221) (221) (221) (32)
%e (311) (2111) (11111) (11111) (2111) (311)
%e (221) (11111) (11111) (221)
%e (2111) (2111)
%e (11111) (11111)
%e Row n = 6 counts the following partitions:
%e (6) (51) (42) (33) (42) (51) (6)
%e (51) (411) (411) (2211) (411) (411) (51)
%e (42) (321) (321) (111111) (321) (321) (42)
%e (411) (3111) (3111) (3111) (3111) (411)
%e (33) (2211) (222) (222) (2211) (33)
%e (321) (21111) (111111) (111111) (21111) (321)
%e (3111) (111111) (111111) (3111)
%e (222) (222)
%e (2211) (2211)
%e (21111) (21111)
%e (111111) (111111)
%t Table[Length[Select[IntegerPartitions[n], Count[Total/@Union[Subsets[#]], k]==1&]], {n,0,10}, {k,0,n}]
%Y Columns k = 0 and k = n are A000041(n).
%Y Column k = 1 and k = n-1 are A000041(n-1).
%Y Column k = 2 appears to be 2*A027336(n).
%Y The version for non-subset-sums is A046663, strict A365663.
%Y Diagonal n = 2k is A108917, complement A366754.
%Y Row sums are A304796, non-unique version A304792.
%Y The non-unique version is A365543.
%Y Cf. A002219, A122768, A275972, A299702, A299729, A301854, A364272, A364911, A365658, A365661, A367094.
%K nonn,tabl
%O 1,4
%A _Gus Wiseman_, Nov 18 2023