A321647 Number of distinct row/column permutations of the Ferrers diagram of the integer partition with Heinz number n.

1, 1, 1, 1, 1, 4, 1, 1, 1, 6, 1, 6, 1, 8, 6, 1, 1, 6, 1, 9, 12, 10, 1, 8, 1, 12, 1, 12, 1, 36, 1, 1, 20, 14, 8, 12, 1, 16, 30, 12, 1, 72, 1, 15, 9, 18, 1, 10, 1, 9, 42, 18, 1, 8, 20, 16, 56, 20, 1, 72, 1, 22, 18, 1, 40, 120, 1, 21, 72, 72, 1, 20, 1, 24, 9, 24, 10, 180, 1, 15, 1, 26, 1, 144, 70, 28, 90, 20, 1, 72, 30, 27, 110, 30, 112, 12, 1, 12

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(10) = 6 permutations:
  o o   o o   o     o       o     o
  o       o   o o   o     o o     o
  o       o   o     o o     o   o o
The a(21) = 12 permutations:
  o o   o o   o o   o o   o o   o o   o     o     o       o     o     o
  o o   o o   o     o       o     o   o o   o o   o     o o   o o     o
  o       o   o o   o     o o     o   o o   o     o o   o o     o   o o
  o       o   o     o o     o   o o   o     o o   o o     o   o o   o o
		

Links

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

Crossrefs

Cf. A000219, A008480, A049311, A056239, A068313, A101370, A112798, A122111, A296150, A321645, A321646, A321648, A321655.

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[primeMS[n]]]*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)));
A321647(n) = (A008480(n) * A008480(A122111(n))); \\ Antti Karttunen, Feb 09 2019

Formula

a(n) = A008480(n) * A008480(A122111(n)) = A008480(n) * A321648(n).

Extensions

More terms from Antti Karttunen, Feb 09 2019