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

1,6

Comments

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

Links

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

Formula

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

Example

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

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}]

Prog

(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

Crossrefs

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

Sequence in context: A326135 A318305 A306264 * A323410 A182665 A131299

Adjacent sequences: A321644 A321645 A321646 * A321648 A321649 A321650

Keyword

nonn

Author

Gus Wiseman, Nov 15 2018

Extensions

More terms from Antti Karttunen, Feb 09 2019

Status

approved