A321648 Number of permutations of the conjugate of the integer partition with Heinz number n.

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 4, 3, 1, 1, 2, 1, 3, 6, 5, 1, 2, 1, 6, 1, 4, 1, 6, 1, 1, 10, 7, 4, 2, 1, 8, 15, 3, 1, 12, 1, 5, 3, 9, 1, 2, 1, 3, 21, 6, 1, 2, 10, 4, 28, 10, 1, 6, 1, 11, 6, 1, 20, 20, 1, 7, 36, 12, 1, 2, 1, 12, 3, 8, 5, 30, 1, 3, 1, 13

Offset: 1

Author

Gus Wiseman, Nov 15 2018

Keywords

• nonn

Comments

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Examples

			The a(42) = 12 permutations: (3211), (3121), (3112), (2311), (2131), (2113), (1321), (1312), (1231), (1213), (1132), (1123).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A008480, A056239, A112798, A122111, A296150, A321645, A321646, A321647, A321649, A321650.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Permutations[conj[primeMS[n]]]],{n,50}]

PARI

A008480(n) = {my(sig=factor(n)[, 2]); vecsum(sig)!/factorback(apply(k->k!, sig))}; \\ From A008480
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));
A321648(n) = A008480(A122111(n)); \\ Antti Karttunen, Dec 23 2018

Formula

a(n) = A008480(A122111(n)).