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

%I A323301 #17 May 14 2021 08:08:12
%S A323301 1,1,1,5,5,9,21,25,37,53,137,153,249,337,505,845,1085,1497,2061,2785,
%T A323301 3661,7589,8849,13329,18033,26017,34225,48773,70805,91977,123765,
%U A323301 164761,216373,283205,367913,470889,758793,913825,1264105,1651613,2251709,2894793,3927837
%N A323301 Number of ways to fill a matrix with the parts of a strict integer partition of n.
%H A323301 Alois P. Heinz, <a href="/A323301/b323301.txt">Table of n, a(n) for n = 0..5000</a>
%F A323301 a(n) = Sum_{y1 + ... + yk = n, y1 > ... > yk} k! * A000005(k) for n > 0, a(0) = 1.
%e A323301 The a(6) = 21 matrices:
%e A323301   [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 A323301 .
%e A323301   [1] [5] [2] [4]
%e A323301   [5] [1] [4] [2]
%e A323301 .
%e A323301   [1] [1] [2] [2] [3] [3]
%e A323301   [2] [3] [1] [3] [1] [2]
%e A323301   [3] [2] [3] [1] [2] [1]
%p A323301 b:= proc(n, i, t) option remember;
%p A323301       `if`(n>i*(i+1)/2, 0, `if`(n=0, t!*numtheory[tau](t),
%p A323301        b(n, i-1, t)+b(n-i, min(n-i, i-1), t+1)))
%p A323301     end:
%p A323301 a:= n-> `if`(n=0, 1, b(n$2, 0)):
%p A323301 seq(a(n), n=0..50);  # _Alois P. Heinz_, Jan 15 2019
%t A323301 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A323301 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A323301 ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
%t A323301 Table[Sum[Length[ptnmats[k]],{k,Select[Times@@Prime/@#&/@IntegerPartitions[n],SquareFreeQ]}],{n,20}]
%t A323301 (* Second program: *)
%t A323301 b[n_, i_, t_] := b[n, i, t] = If[n > i(i+1)/2, 0,
%t A323301      If[n == 0, t!*DivisorSigma[0, t], b[n, i - 1, t] +
%t A323301      b[n - i, Min[n - i, i - 1], t + 1]]];
%t A323301 a[n_] := If[n == 0, 1, b[n, n, 0]];
%t A323301 a /@ Range[0, 50] (* _Jean-François Alcover_, May 13 2021, after _Alois P. Heinz_ *)
%Y A323301 Cf. A000005, A000009, A000142, A053529, A294617, A321654, A323295, A323300, A323307, A323351.
%K A323301 nonn
%O A323301 0,4
%A A323301 _Gus Wiseman_, Jan 12 2019
%E A323301 a(0)=1 prepended by _Alois P. Heinz_, Jan 15 2019