cp's OEIS Frontend

A323307 Number of ways to fill a matrix with the parts of a multiset whose multiplicities are the prime indices of n.

%I A323307 #5 Jan 13 2019 08:50:20
%S A323307 1,1,2,4,2,6,3,12,18,12,2,36,4,10,20,72,2,60,4,40,60,24,3,120,80,14,
%T A323307 360,120,4,240,2,240,42,32,70,720,6,27,112,480,2,210,4,84,420,40,4,
%U A323307 1440,280,280,108,224,5,1260,224,420,180,22,2,840,6,72,1680,2880
%N A323307 Number of ways to fill a matrix with the parts of a multiset whose multiplicities are the prime indices of n.
%C A323307 This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.
%H A323307 Gus Wiseman, <a href="/A323307/a323307.txt">The a(27) = 360 matrix-arrangements of (1,1,2,2,3,3).</a>
%F A323307 a(n) = A318762(n) * A000005(A056239(n)).
%e A323307 The a(22) = 24 matrices:
%e A323307   [111112] [111121] [111211] [112111] [121111] [211111]
%e A323307 .
%e A323307   [111] [111] [111] [112] [121] [211]
%e A323307   [112] [121] [211] [111] [111] [111]
%e A323307 .
%e A323307   [11] [11] [11] [11] [12] [21]
%e A323307   [11] [11] [12] [21] [11] [11]
%e A323307   [12] [21] [11] [11] [11] [11]
%e A323307 .
%e A323307   [1] [1] [1] [1] [1] [2]
%e A323307   [1] [1] [1] [1] [2] [1]
%e A323307   [1] [1] [1] [2] [1] [1]
%e A323307   [1] [1] [2] [1] [1] [1]
%e A323307   [1] [2] [1] [1] [1] [1]
%e A323307   [2] [1] [1] [1] [1] [1]
%t A323307 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A323307 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A323307 ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
%t A323307 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A323307 Array[Length[ptnmats[Times@@Prime/@nrmptn[#]]]&,30]
%Y A323307 Cf. A007716, A056239, A112798, A120733, A181821, A305936, A318283, A318284, A323295, A323300, A323351.
%K A323307 nonn
%O A323307 1,3
%A A323307 _Gus Wiseman_, Jan 13 2019