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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).
+10
44
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
EXAMPLE
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
MATHEMATICA
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]
CROSSREFS
Row lengths are A056239. Number of distinct elements in row n is A001222(n). Number of distinct multiplicities in row n is A001221(n).
Number of strict set multipartitions (sets of sets) of a multiset whose multiplicities are the prime indices of n.
+10
18
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
EXAMPLE
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}}
MATHEMATICA
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}]
PROG
(PARI)
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
Number of non-isomorphic multiset partitions of a multiset whose multiplicities are the prime indices of n.
+10
14
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
EXAMPLE
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}}
PROG
(PARI) \\ 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)}
Number of non-isomorphic strict multiset partitions of a multiset whose multiplicities are the prime indices of n.
+10
7
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
EXAMPLE
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}}
Number of non-isomorphic set multipartitions (multisets of sets) of a multiset whose multiplicities are the prime indices of n.
+10
7
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
EXAMPLE
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}}
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