A323300 Number of ways to fill a matrix with the parts of the integer partition with Heinz number n.

1, 1, 1, 2, 1, 4, 1, 2, 2, 4, 1, 6, 1, 4, 4, 3, 1, 6, 1, 6, 4, 4, 1, 12, 2, 4, 2, 6, 1, 12, 1, 2, 4, 4, 4, 18, 1, 4, 4, 12, 1, 12, 1, 6, 6, 4, 1, 10, 2, 6, 4, 6, 1, 12, 4, 12, 4, 4, 1, 36, 1, 4, 6, 4, 4, 12, 1, 6, 4, 12, 1, 20, 1, 4, 6, 6, 4, 12, 1, 10, 3, 4

Offset: 1

Author

Gus Wiseman, Jan 12 2019

Keywords

• nonn

Comments

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

Examples

			The a(24) = 12 matrices whose entries are (2,1,1,1):
  [1 1 1 2] [1 1 2 1] [1 2 1 1] [2 1 1 1]
.
  [1 1] [1 1] [1 2] [2 1]
  [1 2] [2 1] [1 1] [1 1]
.
  [1] [1] [1] [2]
  [1] [1] [2] [1]
  [1] [2] [1] [1]
  [2] [1] [1] [1]
		

Crossrefs

Positions of 1's are one and prime numbers A008578.

Positions of 2's are primes to prime powers A053810.

Cf. A000005, A001222, A008480, A056239, A063989, A112798, A120733.

Cf. A323295, A323305, A323307, A323351.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
Array[Length[ptnmats[#]]&,100]

Formula

a(n) = A008480(n) * A000005(A001222(n)).