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A382203 Number of normal multiset partitions of weight n into constant multisets with distinct sums.

%I A382203 #13 Apr 04 2025 23:42:07
%S A382203 1,1,2,4,9,19,37,76,159,326,671,1376,2815,5759,11774,24083,49249,
%T A382203 100632,205490,419420,855799,1745889,3561867,7268240,14836127,
%U A382203 30295633,61888616
%N A382203 Number of normal multiset partitions of weight n into constant multisets with distinct sums.
%C A382203 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 A382203 The a(1) = 1 through a(4) = 9 multiset partitions:
%e A382203   {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
%e A382203          {{1},{2}}  {{1},{1,1}}    {{1},{1,1,1}}
%e A382203                     {{1},{2,2}}    {{1,1},{2,2}}
%e A382203                     {{1},{2},{3}}  {{1},{2,2,2}}
%e A382203                                    {{2},{1,1,1}}
%e A382203                                    {{1},{2},{2,2}}
%e A382203                                    {{1},{2},{3,3}}
%e A382203                                    {{1},{3},{2,2}}
%e A382203                                    {{1},{2},{3},{4}}
%e A382203 The a(5) = 19 factorizations:
%e A382203   32  2*16  2*3*27   2*3*5*25  2*3*5*7*11
%e A382203       4*8   2*4*9    2*3*5*9
%e A382203       2*81  2*3*8    2*3*5*49
%e A382203       4*27  2*3*125  2*3*7*25
%e A382203       9*8   2*9*25
%e A382203       3*16  2*5*27
%e A382203             5*4*9
%t A382203 allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
%t A382203 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A382203 mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
%t A382203 Table[Length[Join@@(Select[mps[#],UnsameQ@@Total/@#&&And@@SameQ@@@#&]&/@allnorm[n])],{n,0,5}]
%Y A382203 Without distinct sums we have A055887.
%Y A382203 Twice-partitions of this type are counted by A279786.
%Y A382203 For distinct blocks instead of sums we have A304969.
%Y A382203 Without constant blocks we have A326519.
%Y A382203 Factorizations of this type are counted by A381635.
%Y A382203 For strict instead of constant blocks we have A381718.
%Y A382203 For equal instead of distinct block-sums we have A382204.
%Y A382203 For equal block-sums and strict blocks we have A382429.
%Y A382203 A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
%Y A382203 A001055 count multiset partitions of prime indices, strict A045778.
%Y A382203 A089259 counts set multipartitions of integer partitions.
%Y A382203 A321469 counts multiset partitions with distinct block-sums, ranks A326535.
%Y A382203 Normal multiset partitions: A035310, A116540, A255906, A317532.
%Y A382203 Set multipartitions with distinct sums: A279785, A381806, A381870.
%Y A382203 Cf. A007716, A116539, A255903, A275780, A317583, A326517, A326518, A381633, A381636, A382216, A382428.
%K A382203 nonn,more
%O A382203 0,3
%A A382203 _Gus Wiseman_, Mar 26 2025
%E A382203 a(14)-a(26) from _Christian Sievers_, Apr 04 2025