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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 (list; graph; refs; listen; history; text; internal format)
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}]

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

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Last modified July 8 04:16 EDT 2020. Contains 335504 sequences. (Running on oeis4.)