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 A358836 #16 Aug 22 2024 16:08:35
%S A358836 1,1,2,4,8,15,28,51,92,164,289,504,871,1493,2539,4290,7201,12017,
%T A358836 19939,32911,54044,88330,143709,232817,375640,603755,966816,1542776,
%U A358836 2453536,3889338,6146126,9683279,15211881,23830271,37230720,58015116,90174847,139820368,216286593
%N A358836 Number of multiset partitions of integer partitions of n with all distinct block sizes.
%C A358836 Also the number of integer compositions of n whose leaders of maximal weakly decreasing runs are strictly increasing. For example, the composition (1,2,2,1,3,1,4,1) has maximal weakly decreasing runs ((1),(2,2,1),(3,1),(4,1)), with leaders (1,2,3,4), so is counted under a(15). - _Gus Wiseman_, Aug 21 2024
%H A358836 Andrew Howroyd, <a href="/A358836/b358836.txt">Table of n, a(n) for n = 0..1000</a>
%H A358836 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%F A358836 G.f.: Product_{k>=1} (1 + [y^k]P(x,y)) where P(x,y) = 1/Product_{k>=1} (1 - y*x^k). - _Andrew Howroyd_, Dec 31 2022
%e A358836 The a(1) = 1 through a(5) = 15 multiset partitions:
%e A358836 {1} {2} {3} {4} {5}
%e A358836 {1,1} {1,2} {1,3} {1,4}
%e A358836 {1,1,1} {2,2} {2,3}
%e A358836 {1},{1,1} {1,1,2} {1,1,3}
%e A358836 {1,1,1,1} {1,2,2}
%e A358836 {1},{1,2} {1,1,1,2}
%e A358836 {2},{1,1} {1},{1,3}
%e A358836 {1},{1,1,1} {1},{2,2}
%e A358836 {2},{1,2}
%e A358836 {3},{1,1}
%e A358836 {1,1,1,1,1}
%e A358836 {1},{1,1,2}
%e A358836 {2},{1,1,1}
%e A358836 {1},{1,1,1,1}
%e A358836 {1,1},{1,1,1}
%e A358836 From _Gus Wiseman_, Aug 21 2024: (Start)
%e A358836 The a(0) = 1 through a(5) = 15 compositions whose leaders of maximal weakly decreasing runs are strictly increasing:
%e A358836 () (1) (2) (3) (4) (5)
%e A358836 (11) (12) (13) (14)
%e A358836 (21) (22) (23)
%e A358836 (111) (31) (32)
%e A358836 (112) (41)
%e A358836 (121) (113)
%e A358836 (211) (122)
%e A358836 (1111) (131)
%e A358836 (221)
%e A358836 (311)
%e A358836 (1112)
%e A358836 (1121)
%e A358836 (1211)
%e A358836 (2111)
%e A358836 (11111)
%e A358836 (End)
%t A358836 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A358836 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A358836 Table[Length[Select[Join@@mps/@IntegerPartitions[n],UnsameQ@@Length/@#&]],{n,0,10}]
%t A358836 (* second program *)
%t A358836 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Less@@First/@Split[#,GreaterEqual]&]],{n,0,15}] (* _Gus Wiseman_, Aug 21 2024 *)
%o A358836 (PARI)
%o A358836 P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
%o A358836 seq(n) = {my(g=P(n,y)); Vec(prod(k=1, n, 1 + polcoef(g, k, y) + O(x*x^n)))} \\ _Andrew Howroyd_, Dec 31 2022
%Y A358836 The version for set partitions is A007837.
%Y A358836 For sums instead of sizes we have A271619.
%Y A358836 For constant instead of distinct sizes we have A319066.
%Y A358836 These multiset partitions are ranked by A326533.
%Y A358836 For odd instead of distinct sizes we have A356932.
%Y A358836 The version for twice-partitions is A358830.
%Y A358836 The case of distinct sums also is A358832.
%Y A358836 Ranked by positions of strictly increasing rows in A374740, opposite A374629.
%Y A358836 A001970 counts multiset partitions of integer partitions.
%Y A358836 A011782 counts compositions.
%Y A358836 A063834 counts twice-partitions, strict A296122.
%Y A358836 A238130, A238279, A333755 count compositions by number of runs.
%Y A358836 A335456 counts patterns matched by compositions.
%Y A358836 Runs and leaders: A000041, A188900, A374634, A374635, A374637, A374679, A374742 (ranks A374744), A374743 (ranks A374701), A374746, A374747.
%Y A358836 Cf. A000009, A141199, A273873, A279375, A279785, A279790, A358334.
%Y A358836 Cf. A003242, A106356, A188920, A189076, A238343, A333213, A336342.
%K A358836 nonn
%O A358836 0,3
%A A358836 _Gus Wiseman_, Dec 05 2022
%E A358836 Terms a(11) and beyond from _Andrew Howroyd_, Dec 31 2022