cp's OEIS Frontend

A382428 Number of normal multiset partitions of weight n into sets with distinct sizes.

%I A382428 #14 Mar 31 2025 13:38:47
%S A382428 1,1,1,6,8,35,292,673,2818,16956,219772,636748,3768505,20309534,
%T A382428 183403268,3227600747,12272598308,81353466578,561187259734,
%U A382428 4416808925866,50303004612136,1238783066956740,5566249468690291,44970939483601100,330144217684933896,3131452652308459402
%N A382428 Number of normal multiset partitions of weight n into sets with distinct sizes.
%C A382428 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.
%H A382428 Andrew Howroyd, <a href="/A382428/b382428.txt">Table of n, a(n) for n = 0..200</a>
%e A382428 The a(1) = 1 through a(4) = 8 multiset partitions:
%e A382428   {{1}}  {{1,2}}  {{1,2,3}}    {{1,2,3,4}}
%e A382428                   {{1},{1,2}}  {{1},{1,2,3}}
%e A382428                   {{1},{2,3}}  {{1},{2,3,4}}
%e A382428                   {{2},{1,2}}  {{2},{1,2,3}}
%e A382428                   {{2},{1,3}}  {{2},{1,3,4}}
%e A382428                   {{3},{1,2}}  {{3},{1,2,3}}
%e A382428                                {{3},{1,2,4}}
%e A382428                                {{4},{1,2,3}}
%t A382428 allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
%t A382428 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A382428 mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
%t A382428 Table[Length[Join@@(Select[mps[#],UnsameQ@@Length/@#&&And@@UnsameQ@@@#&]&/@allnorm[n])],{n,0,5}]
%o A382428 (PARI) R(n, k)={Vec(prod(j=1, n, 1 + binomial(k, j)*x^j + O(x*x^n)))}
%o A382428 seq(n)={sum(k=0, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)))} \\ _Andrew Howroyd_, Mar 31 2025
%Y A382428 For distinct sums instead of sizes we have A116539, see A050326.
%Y A382428 Without distinct lengths we have A116540 (normal set multipartitions).
%Y A382428 Without strict blocks we have A326517, for sum instead of size A326519.
%Y A382428 For equal instead of distinct sizes we have A331638.
%Y A382428 Twice-partitions of this type are counted by A358830.
%Y A382428 For distinct sums instead of sizes we have A381718.
%Y A382428 For equal instead of distinct sizes we have A382429.
%Y A382428 A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
%Y A382428 A001055 count factorizations, strict A045778.
%Y A382428 Normal multiset partitions: A034691, A035310, A255906.
%Y A382428 Set multipartitions: A089259, A270995, A296119, A318360.
%Y A382428 Cf. A000110, A007716, A050320, A255903, A275780, A279785, A317532, A326518, A381633, A382214, A382216.
%K A382428 nonn
%O A382428 0,4
%A A382428 _Gus Wiseman_, Mar 29 2025
%E A382428 a(10) onwards from _Andrew Howroyd_, Mar 31 2025