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%I #7 May 08 2019 12:31:54
%S 1,1,1,1,1,2,3,1,1,3,2,1,6,1,3,7,2,5,9,2,1,8,11,2,1,13,14,1,3,19,15,3,
%T 5,27,19,1,11,34,22,2,1,15,49,23,2,1,27,59,28,3,3,39,78,30,1,5,60,93,
%U 34,3,11,82,118,36,1,18,115,140,41,3,1,30,155,170,42,2,1,48
%N Triangle T(n,k) = number of conjugacy classes of partitions of n using only k types of piles, read by rows.
%C Lengths of rows are 1 1 2 2 2 3 3 3 3 4 4 4 4 4 ... (A003056).
%e Triangle turned on its side begins:
%e 1.1.1.2.1.2.1.2.2..2..1..3..1..2..2....etc A038548
%e ....1.1.3.3.6.7.9.11.14.15.19.22.23....etc A270060
%e ..........1.1.3.5..8.13.19.27.34.49....etc
%e ...................1..1..3..5.11.15....etc
%p compareL := proc(L1,L2)
%p if nops(L1) < nops(L2) then
%p -1 ;
%p elif nops(L1) > nops(L2) then
%p 1;
%p else
%p for i from 1 to nops(L1) do
%p if op(i,L1) > op(i,L2) then
%p return 1 ;
%p elif op(i,L1) < op(i,L2) then
%p return -1 ;
%p end if;
%p end do:
%p 0 ;
%p end if;
%p end proc:
%p A067627 := proc(n,k)
%p local a,p,s,pc ;
%p a := 0 ;
%p for p in combinat[partition](n) do
%p s := convert(p,set) ;
%p if nops(s) = k then
%p pc := combinat[conjpart](p) ;
%p if compareL(p,pc) <= 0 then
%p a := a+1 ;
%p end if;
%p end if;
%p end do:
%p a ;
%p end proc:
%p for n from 1 to 30 do
%p for k from A003056(n) to 1 by -1 do
%p printf("%4d,",A067627(n,k)) ;
%p end do:
%p printf("\n") ;
%p end do: # _R. J. Mathar_, May 08 2019
%Y Cf. A000700, A000701, A046682, A060177. Diagonals give A038548. row sums give A046682.
%K easy,nonn,tabf
%O 1,6
%A _Naohiro Nomoto_, Feb 02 2002
%E More terms from _R. J. Mathar_, May 08 2019
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