A305936
Irregular triangle whose n-th row is the multiset spanning an initial interval of positive integers with multiplicities equal to the n-th row of A296150 (the prime indices of n in weakly decreasing order).
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 2, 2, 1
Offset: 1
Row 90 is {1,1,1,2,2,3,3,4} because 90 = prime(3)*prime(2)*prime(2)*prime(1).
Triangle begins:
1:
2: 1
3: 1 1
4: 1 2
5: 1 1 1
6: 1 1 2
7: 1 1 1 1
8: 1 2 3
9: 1 1 2 2
10: 1 1 1 2
11: 1 1 1 1 1
12: 1 1 2 3
13: 1 1 1 1 1 1
Row lengths are
A056239. Number of distinct elements in row n is
A001222(n). Number of distinct multiplicities in row n is
A001221(n).
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nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Array[nrmptn,30]
A318361
Number of strict set multipartitions (sets of sets) of a multiset whose multiplicities are the prime indices of n.
Original entry on oeis.org
1, 1, 0, 2, 0, 1, 0, 5, 1, 0, 0, 4, 0, 0, 0, 15, 0, 5, 0, 1, 0, 0, 0, 16, 0, 0, 8, 0, 0, 2, 0, 52, 0, 0, 0, 23, 0, 0, 0, 7, 0, 0, 0, 0, 5, 0, 0, 68, 0, 1, 0, 0, 0, 40, 0, 1, 0, 0, 0, 14, 0, 0, 1, 203, 0, 0, 0, 0, 0, 0, 0, 111, 0, 0, 4, 0, 0, 0, 0, 41, 80, 0, 0
Offset: 1
The a(24) = 16 sets of sets with multiset union {1,1,2,3,4}:
{{1},{1,2,3,4}}
{{1,2},{1,3,4}}
{{1,3},{1,2,4}}
{{1,4},{1,2,3}}
{{1},{2},{1,3,4}}
{{1},{3},{1,2,4}}
{{1},{4},{1,2,3}}
{{1},{1,2},{3,4}}
{{1},{1,3},{2,4}}
{{1},{1,4},{2,3}}
{{2},{1,3},{1,4}}
{{3},{1,2},{1,4}}
{{4},{1,2},{1,3}}
{{1},{2},{3},{1,4}}
{{1},{2},{4},{1,3}}
{{1},{3},{4},{1,2}}
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nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[sqfacs[Times@@Prime/@nrmptn[n]]],{n,90}]
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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}
sig(n)={my(f=factor(n)); concat(vector(#f~, i, vector(f[i, 2], j, primepi(f[i, 1]))))}
count(sig)={my(r=0, A=O(x*x^vecmax(sig))); for(n=1, vecsum(sig)+1, my(s=0); forpart(p=n, my(q=prod(i=1, #p, 1 + x^p[i] + A)); s+=prod(i=1, #sig, polcoef(q, sig[i]))*(-1)^#p*permcount(p)); r+=(-1)^n*s/n!); r/2}
a(n)={if(n==1, 1, my(s=sig(n)); if(#s==1, s[1]==1, count(sig(n))))} \\ Andrew Howroyd, Dec 18 2018
A318285
Number of non-isomorphic 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, 3, 7, 7, 7, 9, 11, 12, 16, 5, 15, 17, 22, 16, 29, 19, 30, 16, 21, 30, 23, 29, 42, 52, 56, 7, 47, 45, 57, 43, 77, 67, 77, 31, 101, 98, 135, 47, 85, 97, 176, 29, 66, 64, 118, 77, 231, 69, 97, 57, 181, 139, 297, 137, 385, 195, 166, 11, 162, 171, 490, 118
Offset: 1
Non-isomorphic representatives of the a(12) = 9 multiset partitions of {1,1,2,3}:
{{1,1,2,3}}
{{1},{1,2,3}}
{{2},{1,1,3}}
{{1,1},{2,3}}
{{1,2},{1,3}}
{{1},{1},{2,3}}
{{1},{2},{1,3}}
{{2},{3},{1,1}}
{{1},{1},{2},{3}}
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\\ See links in A339645 for combinatorial species functions.
sig(n)={my(f=factor(n), sig=vector(primepi(vecmax(f[,1])))); for(i=1, #f~, sig[primepi(f[i,1])]=f[i,2]); sig}
C(sig)={my(n=sum(i=1, #sig, i*sig[i]), A=Vec(symGroupSeries(n)-1), B=O(x*x^n), c=prod(i=1, #sig, if(sig[i], sApplyCI(A[sig[i]], sig[i], A[i], i), 1))); polcoef(OgfSeries(sCartProd(c*x^n + B, sExp(x*Ser(A) + B))), n)}
a(n)={if(n==1, 1, C(sig(n)))} \\ Andrew Howroyd, Jan 17 2023
A318287
Number of non-isomorphic strict multiset partitions of a multiset whose multiplicities are the prime indices of n.
Original entry on oeis.org
1, 1, 1, 2, 2, 3, 2, 3, 4, 5, 3, 7, 4, 7, 9, 5, 5, 12, 6, 12, 14, 10, 8, 13, 12, 14, 14, 18, 10, 34
Offset: 1
Non-isomorphic representatives of the a(20) = 12 strict multiset partitions of {1,1,1,2,3}:
{{1,1,1,2,3}}
{{1},{1,1,2,3}}
{{2},{1,1,1,3}}
{{1,1},{1,2,3}}
{{1,2},{1,1,3}}
{{2,3},{1,1,1}}
{{1},{2},{1,1,3}}
{{1},{1,1},{2,3}}
{{1},{1,2},{1,3}}
{{2},{3},{1,1,1}}
{{2},{1,1},{1,3}}
{{1},{2},{3},{1,1}}
A318362
Number of non-isomorphic set multipartitions (multisets of sets) of a multiset whose multiplicities are the prime indices of n.
Original entry on oeis.org
1, 1, 1, 2, 1, 2, 1, 3, 3, 2, 1, 5, 1, 2, 3, 5, 1, 7, 1, 5, 3, 2, 1, 9, 4, 2, 8, 5, 1, 10
Offset: 1
Non-isomorphic representatives of the a(12) = 5 set multipartitions of {1,1,2,3}:
{{1},{1,2,3}}
{{1,2},{1,3}}
{{1},{1},{2,3}}
{{1},{2},{1,3}}
{{1},{1},{2},{3}}
Showing 1-5 of 5 results.