A323301 - OEIS

%I #17 May 14 2021 08:08:12

%S 1,1,1,5,5,9,21,25,37,53,137,153,249,337,505,845,1085,1497,2061,2785,

%T 3661,7589,8849,13329,18033,26017,34225,48773,70805,91977,123765,

%U 164761,216373,283205,367913,470889,758793,913825,1264105,1651613,2251709,2894793,3927837

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

%H Alois P. Heinz, <a href="/A323301/b323301.txt">Table of n, a(n) for n = 0..5000</a>

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

%e The a(6) = 21 matrices:

%e   [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]

%e .

%e   [1] [5] [2] [4]

%e   [5] [1] [4] [2]

%e .

%e   [1] [1] [2] [2] [3] [3]

%e   [2] [3] [1] [3] [1] [2]

%e   [3] [2] [3] [1] [2] [1]

%p b:= proc(n, i, t) option remember;

%p       `if`(n>i*(i+1)/2, 0, `if`(n=0, t!*numtheory[tau](t),

%p        b(n, i-1, t)+b(n-i, min(n-i, i-1), t+1)))

%p     end:

%p a:= n-> `if`(n=0, 1, b(n$2, 0)):

%p seq(a(n), n=0..50);  # _Alois P. Heinz_, Jan 15 2019

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];

%t Table[Sum[Length[ptnmats[k]],{k,Select[Times@@Prime/@#&/@IntegerPartitions[n],SquareFreeQ]}],{n,20}]

%t (* Second program: *)

%t b[n_, i_, t_] := b[n, i, t] = If[n > i(i+1)/2, 0,

%t      If[n == 0, t!*DivisorSigma[0, t], b[n, i - 1, t] +

%t      b[n - i, Min[n - i, i - 1], t + 1]]];

%t a[n_] := If[n == 0, 1, b[n, n, 0]];

%t a /@ Range[0, 50] (* _Jean-François Alcover_, May 13 2021, after _Alois P. Heinz_ *)

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

%K nonn

%O 0,4

%A _Gus Wiseman_, Jan 12 2019

%E a(0)=1 prepended by _Alois P. Heinz_, Jan 15 2019