A323301 Number of ways to fill a matrix with the parts of a strict integer partition of n.

1, 1, 1, 5, 5, 9, 21, 25, 37, 53, 137, 153, 249, 337, 505, 845, 1085, 1497, 2061, 2785, 3661, 7589, 8849, 13329, 18033, 26017, 34225, 48773, 70805, 91977, 123765, 164761, 216373, 283205, 367913, 470889, 758793, 913825, 1264105, 1651613, 2251709, 2894793, 3927837

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Formula

a(n) = Sum_{y1 + ... + yk = n, y1 > ... > yk} k! * A000005(k) for n > 0, a(0) = 1.

Example

The a(6) = 21 matrices:
  [6] [1 5] [5 1] [2 4] [4 2] [1 2 3] [1 3 2] [2 1 3] [2 3 1] [3 1 2] [3 2 1]
.
  [1] [5] [2] [4]
  [5] [1] [4] [2]
.
  [1] [1] [2] [2] [3] [3]
  [2] [3] [1] [3] [1] [2]
  [3] [2] [3] [1] [2] [1]

Maple

b:= proc(n, i, t) option remember;
      `if`(n>i*(i+1)/2, 0, `if`(n=0, t!*numtheory[tau](t),
       b(n, i-1, t)+b(n-i, min(n-i, i-1), t+1)))
    end:
a:= n-> `if`(n=0, 1, b(n$2, 0)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 15 2019

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/@#&];
Table[Sum[Length[ptnmats[k]], {k, Select[Times@@Prime/@#&/@IntegerPartitions[n], SquareFreeQ]}], {n, 20}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n > i(i+1)/2, 0,
     If[n == 0, t!*DivisorSigma[0, t], b[n, i - 1, t] +
     b[n - i, Min[n - i, i - 1], t + 1]]];
a[n_] := If[n == 0, 1, b[n, n, 0]];
a /@ Range[0, 50] (* Jean-François Alcover, May 13 2021, after Alois P. Heinz *)

Crossrefs

Cf. A000005, A000009, A000142, A053529, A294617, A321654, A323295, A323300, A323307, A323351.

Sequence in context: A058584 A324792 A147197 * A147047 A173322 A097910

Adjacent sequences: A323298 A323299 A323300 * A323302 A323303 A323304

Keyword

nonn

Author

Gus Wiseman, Jan 12 2019

Extensions

a(0)=1 prepended by Alois P. Heinz, Jan 15 2019

Status

approved