A368099
Triangle read by rows where T(n,k) is the number of non-isomorphic k-element sets of finite nonempty multisets with cardinalities summing to n, or strict multiset partitions of weight n and length k.
Original entry on oeis.org
1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 5, 12, 5, 1, 0, 7, 28, 22, 5, 1, 0, 11, 66, 83, 31, 5, 1, 0, 15, 134, 252, 147, 34, 5, 1, 0, 22, 280, 726, 620, 203, 35, 5, 1, 0, 30, 536, 1946, 2283, 1069, 235, 35, 5, 1, 0, 42, 1043, 4982, 7890, 5019, 1469, 248, 35, 5, 1
Offset: 0
Triangle begins:
1
0 1
0 2 1
0 3 4 1
0 5 12 5 1
0 7 28 22 5 1
0 11 66 83 31 5 1
0 15 134 252 147 34 5 1
0 22 280 726 620 203 35 5 1
0 30 536 1946 2283 1069 235 35 5 1
0 42 1043 4982 7890 5019 1469 248 35 5 1
...
Row n = 4 counts the following representatives:
. {{1,1,1,1}} {{1},{1,1,1}} {{1},{2},{1,1}} {{1},{2},{3},{4}}
{{1,1,1,2}} {{1},{1,1,2}} {{1},{2},{1,2}}
{{1,1,2,2}} {{1},{1,2,2}} {{1},{2},{1,3}}
{{1,1,2,3}} {{1},{1,2,3}} {{1},{2},{3,3}}
{{1,2,3,4}} {{1},{2,2,2}} {{1},{2},{3,4}}
{{1},{2,2,3}}
{{1},{2,3,4}}
{{1,1},{1,2}}
{{1,1},{2,2}}
{{1,1},{2,3}}
{{1,2},{1,3}}
{{1,2},{3,4}}
Counting connected components instead of edges gives
A321194.
For set multipartitions we have
A334550.
Cf.
A255903,
A296122,
A302545,
A306005,
A317532,
A317775,
A317794,
A317795,
A319560,
A368094,
A368095.
-
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
Table[Length[Union[brute /@ Select[mpm[n],UnsameQ@@#&&Length[#]==k&]]], {n,0,5},{k,0,n}]
-
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
G(n)={my(s=0); forpart(q=n, my(p=sum(t=1, n, y^t*subst(x*Ser(K(q, t, n\t))/t, x, x^t))); s+=permcount(q)*exp(p-subst(subst(p, x, x^2), y, y^2))); s/n!}
T(n)={[Vecrev(p) | p <- Vec(G(n))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 11 2024
A357873
Numbers whose multiset of prime factors has all non-isomorphic multiset partitions.
Original entry on oeis.org
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 72, 73
Offset: 1
The multiset partitions of the prime indices of 12 are: {{1,1,2}}, {{1},{1,2}}, {{1,1},{2}}, {{1},{1},{2}}, all of which are non-isomorphic, so 12 is in the sequence.
The multiset partitions of the prime indices of 30 are: {{1,2,3}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1},{2},{3}}, of which the middle 3 are isomorphic, so 30 is not in the sequence.
A001055 counts multiset partitions of prime indices, non-isomorphic
A317791.
-
brute[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],brute[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[brute[m,1]]]];brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@brute/@mps[primeMS[#]]&]
A357874
Numbers whose multiset of prime factors has at least two multiset partitions that are isomorphic.
Original entry on oeis.org
30, 36, 42, 60, 66, 70, 78, 84, 90, 100, 102, 105, 110, 114, 120, 126, 130, 132, 138, 140, 150, 154, 156, 165, 168, 170, 174, 180, 182, 186, 190, 195, 196, 198, 204, 210, 216, 220, 222, 225, 228, 230, 231, 234, 238, 240, 246, 252, 255, 258, 260, 264, 266, 270
Offset: 1
The terms together with their prime indices begin:
30: {1,2,3}
36: {1,1,2,2}
42: {1,2,4}
60: {1,1,2,3}
66: {1,2,5}
70: {1,3,4}
78: {1,2,6}
84: {1,1,2,4}
90: {1,2,2,3}
100: {1,1,3,3}
For example, the multiset partitions of the prime indices of 36 include {{1},{1,2,2}} and {{2},{1,1,2}}, which are isomorphic, so 36 is in the sequence.
A001055 counts multiset partitions of prime indices, non-isomorphic
A317791.
-
brute[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],brute[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[brute[m,1]]]];brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!UnsameQ@@brute/@mps[primeMS[#]]&]
A321484
Number of non-isomorphic self-dual connected multiset partitions of weight n.
Original entry on oeis.org
1, 1, 1, 2, 3, 6, 9, 20, 35, 78, 141
Offset: 0
Non-isomorphic representatives of the a(1) = 1 through a(6) = 9 multiset partitions:
{{1}} {{11}} {{111}} {{1111}} {{11111}} {{111111}}
{{2}{12}} {{12}{12}} {{11}{122}} {{112}{122}}
{{2}{122}} {{12}{122}} {{12}{1222}}
{{2}{1222}} {{2}{12222}}
{{2}{13}{23}} {{22}{1122}}
{{3}{3}{123}} {{12}{13}{23}}
{{2}{13}{233}}
{{3}{23}{123}}
{{3}{3}{1233}}
Cf.
A007718,
A056156,
A138178,
A316983,
A319565,
A319616,
A319647,
A319719,
A321194,
A321585,
A321680,
A321681.
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