cp's OEIS Frontend

A356941 Number of multiset partitions of integer partitions of n such that all blocks are gapless.

%I A356941 #9 Dec 30 2022 21:38:23
%S A356941 1,1,3,6,13,24,49,88,166,297,534,932,1635,2796,4782,8060,13521,22438,
%T A356941 37080,60717,98979,160216,258115,413382,659177,1045636,1651891,
%U A356941 2597849,4069708,6349677,9871554,15290322,23604794,36318256,55705321,85177643,129865495
%N A356941 Number of multiset partitions of integer partitions of n such that all blocks are gapless.
%C A356941 A multiset is gapless if it covers an interval of positive integers. For example, {2,3,3,4} is gapless but {1,1,3,3} is not.
%H A356941 Andrew Howroyd, <a href="/A356941/b356941.txt">Table of n, a(n) for n = 0..1000</a>
%H A356941 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vR-C_picqWlu0KOguRGWaPjhS2HY7m43aGXGDcolDh4Qtyy-pu2lkq5mbHAbiMSyQoiIESG2mCGtc2j/pub">Counting and ranking classes of multiset partitions related to gapless multisets</a>
%F A356941 G.f.: 1/Product_{k>=1} (1 - x^k)^A034296(k). - _Andrew Howroyd_, Dec 30 2022
%e A356941 The a(1) = 1 through a(4) = 13 multiset partitions:
%e A356941   {{1}}  {{2}}      {{3}}          {{4}}
%e A356941          {{1,1}}    {{1,2}}        {{2,2}}
%e A356941          {{1},{1}}  {{1,1,1}}      {{1,1,2}}
%e A356941                     {{1},{2}}      {{1},{3}}
%e A356941                     {{1},{1,1}}    {{2},{2}}
%e A356941                     {{1},{1},{1}}  {{1,1,1,1}}
%e A356941                                    {{1},{1,2}}
%e A356941                                    {{2},{1,1}}
%e A356941                                    {{1},{1,1,1}}
%e A356941                                    {{1,1},{1,1}}
%e A356941                                    {{1},{1},{2}}
%e A356941                                    {{1},{1},{1,1}}
%e A356941                                    {{1},{1},{1},{1}}
%t A356941 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A356941 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A356941 nogapQ[m_]:=Or[m=={},Union[m]==Range[Min[m],Max[m]]];
%t A356941 Table[Length[Select[Join@@mps/@IntegerPartitions[n],And@@nogapQ/@#&]],{n,0,5}]
%o A356941 (PARI) \\ Here G(n) gives A034296 as vector
%o A356941 G(N) = Vec(sum(n=1, N, x^n/(1-x^n) * prod(k=1, n-1, 1+x^k+O(x*x^(N-n))) ));
%o A356941 seq(n) = {my(u=G(n)); Vec(1/prod(k=1, n-1, (1 - x^k + O(x*x^n))^u[k])) } \\ _Andrew Howroyd_, Dec 30 2022
%Y A356941 A000041 counts integer partitions, strict A000009.
%Y A356941 A000670 counts patterns, ranked by A333217, necklace A019536.
%Y A356941 A001055 counts factorizations.
%Y A356941 A011782 counts multisets covering an initial interval.
%Y A356941 A356069 counts gapless divisors, initial A356224 (complement A356225).
%Y A356941 Gapless multisets are counted by A034296, ranked by A073491.
%Y A356941 Other types: A356233, A356942, A356943, A356944.
%Y A356941 Other conditions: A001970, A006171, A007294, A089259, A107742, A356932.
%Y A356941 Cf. A055887, A072233, A270995.
%K A356941 nonn
%O A356941 0,3
%A A356941 _Gus Wiseman_, Sep 11 2022
%E A356941 Terms a(11) and beyond from _Andrew Howroyd_, Dec 30 2022