A325510
Number of non-isomorphic multiset partitions of the multiset of prime indices of n!.
Original entry on oeis.org
1, 1, 1, 2, 7, 16, 98, 269, 1397, 7582, 70520, 259906, 1677259, 5229112, 44726100, 666355170, 4917007185, 18459879921
Offset: 0
Non-isomorphic representatives of the a(2) = 1 through a(5) = 16 multiset partitions:
{{1}} {{12}} {{1222}} {{12333}}
{{1}{2}} {{1}{222}} {{1}{2333}}
{{12}{22}} {{12}{333}}
{{2}{122}} {{13}{233}}
{{1}{2}{22}} {{3}{1233}}
{{2}{2}{12}} {{33}{123}}
{{1}{2}{2}{2}} {{1}{2}{333}}
{{1}{23}{33}}
{{1}{3}{233}}
{{3}{12}{33}}
{{3}{13}{23}}
{{3}{3}{123}}
{{1}{1}{1}{23}}
{{1}{2}{3}{33}}
{{1}{3}{3}{23}}
{{1}{2}{3}{3}{3}}
Cf.
A000142,
A001055,
A007716,
A011371,
A022559,
A076716,
A115627,
A317791,
A318285,
A322583,
A325272,
A325276,
A325508,
A325509,
A325511.
-
\\ Requires C(sig) from A318285.
a(n)={if(n<2, 1, my(f=factor(n!)[,2], sig=vector(vecmax(f))); for(i=1, #f, sig[f[i]]++); C(sig))} \\ Andrew Howroyd, Jan 17 2023
A324325
Number of non-crossing multiset partitions of a multiset whose multiplicities are the prime indices of n.
Original entry on oeis.org
1, 1, 2, 2, 3, 4, 5, 5, 9, 7, 7, 11, 11, 12, 16, 14, 15, 26, 22, 21, 29, 19, 30, 33, 31, 30, 66, 38, 42, 52, 56, 42, 47, 45, 57, 82, 77, 67, 77, 67, 101, 98, 135, 64, 137, 97, 176, 104, 109, 109, 118, 105, 231, 213, 97, 127, 181, 139, 297, 173, 385, 195, 269
Offset: 1
The a(16) = 14 non-crossing multiset partitions of the multiset {1,2,3,4}:
{{1,2,3,4}}
{{1},{2,3,4}}
{{2},{1,3,4}}
{{3},{1,2,4}}
{{4},{1,2,3}}
{{1,2},{3,4}}
{{1,4},{2,3}}
{{1},{2},{3,4}}
{{1},{3},{2,4}}
{{1},{4},{2,3}}
{{2},{3},{1,4}}
{{2},{4},{1,3}}
{{3},{1,2},{4}}
{{1},{2},{3},{4}}
Missing from this list is {{1,3},{2,4}}.
Cf.
A000108,
A001055,
A001970,
A016098,
A054726,
A099947,
A181821,
A305936,
A306438,
A318284,
A318285.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
nonXQ[stn_]:=!MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x
A321188
Number of set systems with no singletons whose multiset union is row n of A305936 (a multiset whose multiplicities are the prime indices of n).
Original entry on oeis.org
1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 4, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 1, 0, 0, 0, 0, 11, 0, 0, 0, 4, 0, 0, 0, 1
Offset: 1
The a(36) = 4 set systems with no singletons whose multiset union is {1,1,2,2,3,4}:
{{1,2},{1,2,3,4}}
{{1,2,3},{1,2,4}}
{{1,2},{1,3},{2,4}}
{{1,2},{1,4},{2,3}}
Cf.
A000070,
A000296,
A000569,
A050326,
A056239,
A112798,
A283877,
A292444,
A305936,
A306005,
A318285,
A318361,
A320922,
A320923,
A320924.
-
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
hyp[m_]:=Select[mps[m],And[And@@UnsameQ@@@#,UnsameQ@@#,Min@@Length/@#>1]&];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Table[Length[hyp[nrmptn[n]]],{n,30}]
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