cp's OEIS Frontend

A381718 Number of normal multiset partitions of weight n into sets with distinct sums.

%I A381718 #17 Apr 05 2025 18:13:56
%S A381718 1,1,2,6,23,106,549,3184,20353,141615,1063399,8554800,73281988,
%T A381718 665141182,6369920854,64133095134,676690490875,7462023572238,
%U A381718 85786458777923,1025956348473929,12739037494941490
%N A381718 Number of normal multiset partitions of weight n into sets with distinct sums.
%C A381718 We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The weight of a multiset partition is the sum of sizes of its blocks.
%e A381718 The a(1) = 1 through a(3) = 6 multiset partitions:
%e A381718   {{1}}  {{1,2}}    {{1,2,3}}
%e A381718          {{1},{2}}  {{1},{1,2}}
%e A381718                     {{1},{2,3}}
%e A381718                     {{2},{1,2}}
%e A381718                     {{2},{1,3}}
%e A381718                     {{1},{2},{3}}
%e A381718 The a(4) = 23 factorizations:
%e A381718   2*3*6  5*30    3*30    2*30    210
%e A381718          10*15   6*15    6*10    2*105
%e A381718          2*5*15  2*3*15  2*3*10  3*70
%e A381718          3*5*10                  5*42
%e A381718                                  7*30
%e A381718                                  6*35
%e A381718                                  10*21
%e A381718                                  2*3*35
%e A381718                                  2*5*21
%e A381718                                  2*7*15
%e A381718                                  3*5*14
%e A381718                                  2*3*5*7
%t A381718 allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
%t A381718 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A381718 mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
%t A381718 Table[Length[Join@@(Select[mps[#],UnsameQ@@Total/@#&&And@@UnsameQ@@@#&]&/@allnorm[n])],{n,0,5}]
%Y A381718 For distinct blocks instead of sums we have A116539, see A050326.
%Y A381718 Without distinct sums we have A116540 (normal set multipartitions).
%Y A381718 Twice-partitions of this type are counted by A279785.
%Y A381718 Without strict blocks we have A326519.
%Y A381718 Factorizations of this type are counted by A381633.
%Y A381718 For constant instead of strict blocks we have A382203.
%Y A381718 For distinct sizes instead of sums we have A382428, non-strict blocks A326517.
%Y A381718 For equal instead of distinct block-sums we have A382429, non-strict blocks A326518.
%Y A381718 A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
%Y A381718 A001055 count factorizations, strict A045778.
%Y A381718 Normal multiset partitions: A034691, A035310, A255906.
%Y A381718 Set multipartitions: A089259, A270995, A296119, A318360.
%Y A381718 Cf. A000110, A007716, A050320, A255903, A275780, A317532, A317583, A321469, A381635, A382204, A382214, A382216.
%K A381718 nonn
%O A381718 0,3
%A A381718 _Gus Wiseman_, Mar 26 2025
%E A381718 a(10)-a(11) from _Robert Price_, Mar 31 2025
%E A381718 a(12)-a(20) from _Christian Sievers_, Apr 05 2025