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A323301
Number of ways to fill a matrix with the parts of a strict integer partition of n.
9
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
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
Sequence in context: A058584 A324792 A147197 * A147047 A173322 A097910
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 12 2019
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Jan 15 2019
STATUS
approved

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Last modified September 19 23:07 EDT 2024. Contains 376015 sequences. (Running on oeis4.)