This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A382429 #17 Apr 06 2025 14:05:06
%S A382429 1,1,2,3,5,7,13,26,57,113,283,854,2401,6998,24072,85061,308956,
%T A382429 1190518,4770078,19949106,87059592
%N A382429 Number of normal multiset partitions of weight n into sets with a common sum.
%C A382429 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 A382429 The a(1) = 1 through a(6) = 13 partitions:
%e A382429 {1} {12} {123} {1234} {12345} {123456}
%e A382429 {1}{1} {3}{12} {12}{12} {24}{123} {123}{123}
%e A382429 {1}{1}{1} {14}{23} {34}{124} {125}{134}
%e A382429 {3}{3}{12} {3}{12}{12} {135}{234}
%e A382429 {1}{1}{1}{1} {5}{14}{23} {145}{235}
%e A382429 {3}{3}{3}{12} {12}{12}{12}
%e A382429 {1}{1}{1}{1}{1} {14}{14}{23}
%e A382429 {14}{23}{23}
%e A382429 {16}{25}{34}
%e A382429 {3}{3}{12}{12}
%e A382429 {5}{5}{14}{23}
%e A382429 {3}{3}{3}{3}{12}
%e A382429 {1}{1}{1}{1}{1}{1}
%e A382429 The corresponding factorizations:
%e A382429 2 6 30 210 2310 30030
%e A382429 2*2 5*6 6*6 21*30 30*30
%e A382429 2*2*2 14*15 35*42 6*6*6
%e A382429 5*5*6 5*6*6 66*70
%e A382429 2*2*2*2 5*5*5*6 110*105
%e A382429 11*14*15 154*165
%e A382429 2*2*2*2*2 5*5*6*6
%e A382429 14*14*15
%e A382429 14*15*15
%e A382429 26*33*35
%e A382429 5*5*5*5*6
%e A382429 11*11*14*15
%e A382429 2*2*2*2*2*2
%t A382429 allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
%t A382429 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A382429 mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
%t A382429 Table[Length[Join@@(Select[mps[#],SameQ@@Total/@#&&And@@UnsameQ@@@#&]&/@allnorm[n])],{n,0,5}]
%Y A382429 Without the common sum we have A116540 (normal set multipartitions).
%Y A382429 Twice-partitions of this type are counted by A279788.
%Y A382429 For common sizes instead of sums we have A317583.
%Y A382429 Without strict blocks we have A326518, non-strict blocks A326517.
%Y A382429 For a common length instead of sum we have A331638.
%Y A382429 For distinct instead of equal block-sums we have A381718.
%Y A382429 Factorizations of this type are counted by A382080.
%Y A382429 For distinct block-sums and constant blocks we have A382203.
%Y A382429 For constant instead of strict blocks we have A382204.
%Y A382429 A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
%Y A382429 A001055 count multiset partitions of prime indices, strict A045778.
%Y A382429 A321469 counts multiset partitions with distinct block-sums, ranks A326535.
%Y A382429 Normal multiset partitions: A035310, A255906, A304969, A317532.
%Y A382429 Set multipartitions: A089259, A116539, A270995, A296119, A318360.
%Y A382429 Set multipartitions with distinct sums: A279785, A381806, A381870.
%Y A382429 Constant blocks with distinct sums: A381635, A381636, A381716.
%Y A382429 Cf. A000110, A034691, A038041, A050320, A255903, A326520, A381633, A381996, A382214, A382216.
%K A382429 nonn,more
%O A382429 0,3
%A A382429 _Gus Wiseman_, Mar 26 2025
%E A382429 a(11) from _Robert Price_, Mar 30 2025
%E A382429 a(12)-a(20) from _Christian Sievers_, Apr 06 2025