A321647 - OEIS

A321647

Number of distinct row/column permutations of the Ferrers diagram of the integer partition with Heinz number n.

%I #10 Feb 09 2019 23:26:26

%S 1,1,1,1,1,4,1,1,1,6,1,6,1,8,6,1,1,6,1,9,12,10,1,8,1,12,1,12,1,36,1,1,

%T 20,14,8,12,1,16,30,12,1,72,1,15,9,18,1,10,1,9,42,18,1,8,20,16,56,20,

%U 1,72,1,22,18,1,40,120,1,21,72,72,1,20,1,24,9,24,10,180,1,15,1,26,1,144,70,28,90,20,1,72,30,27,110,30,112,12,1,12

%N Number of distinct row/column permutations of the Ferrers diagram of the integer partition with Heinz number n.

%C The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

%H Antti Karttunen, <a href="/A321647/b321647.txt">Table of n, a(n) for n = 1..16384</a>

%F a(n) = A008480(n) * A008480(A122111(n)) = A008480(n) * A321648(n).

%e The a(10) = 6 permutations:

%e o o o o o o o o

%e The a(21) = 12 permutations:

%e o o o o o o o o o o o o o o o o o o

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

%t conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];

%t Table[Length[Permutations[primeMS[n]]]*Length[Permutations[conj[primeMS[n]]]],{n,50}]

%o (PARI)

%o A008480(n) = {my(sig=factor(n)[, 2]); vecsum(sig)!/factorback(apply(k->k!, sig))}; \\ From A008480

%o A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};

%o A122111(n) = if(1==n,n,prime(bigomega(n))*A122111(A064989(n)));

%o A321647(n) = (A008480(n) * A008480(A122111(n))); \\ _Antti Karttunen_, Feb 09 2019

%Y Cf. A000219, A008480, A049311, A056239, A068313, A101370, A112798, A122111, A296150, A321645, A321646, A321648, A321655.

%K nonn

%O 1,6

%A _Gus Wiseman_, Nov 15 2018

%E More terms from _Antti Karttunen_, Feb 09 2019