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

1,4

Comments

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

Links

Table of n, a(n) for n=1..82.

Formula

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

Example

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]

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]

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.

Sequence in context: A258127 A181982 A070194 * A349128 A366450 A105584

Adjacent sequences: A323297 A323298 A323299 * A323301 A323302 A323303

Keyword

nonn

Author

Gus Wiseman, Jan 12 2019

Status

approved