A067627 Triangle T(n,k) = number of conjugacy classes of partitions of n using only k types of piles, read by rows.

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, 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, 34, 3, 11, 82, 118, 36, 1, 18, 115, 140, 41, 3, 1, 30, 155, 170, 42, 2, 1, 48

Offset: 1

Author

Naohiro Nomoto, Feb 02 2002

Keywords

• easy
• nonn
• tabf

Comments

Lengths of rows are 1 1 2 2 2 3 3 3 3 4 4 4 4 4 ... (A003056).

Examples

			Triangle turned on its side begins:
1.1.1.2.1.2.1.2.2..2..1..3..1..2..2....etc A038548
....1.1.3.3.6.7.9.11.14.15.19.22.23....etc A270060
..........1.1.3.5..8.13.19.27.34.49....etc
...................1..1..3..5.11.15....etc
		

Crossrefs

Cf. A000700, A000701, A046682, A060177. Diagonals give A038548. row sums give A046682.

Programs

Maple

compareL := proc(L1,L2)
    if nops(L1) < nops(L2) then
        -1 ;
    elif nops(L1) > nops(L2) then
        1;
    else
        for i from 1 to nops(L1) do
            if op(i,L1) > op(i,L2) then
                return 1 ;
            elif op(i,L1) < op(i,L2) then
                return -1 ;
            end if;
        end do:
        0 ;
    end if;
end proc:
A067627 := proc(n,k)
    local a,p,s,pc ;
    a := 0 ;
    for p in combinat[partition](n) do
        s := convert(p,set) ;
        if nops(s) = k then
            pc := combinat[conjpart](p) ;
            if compareL(p,pc) <= 0 then
                a := a+1 ;
            end if;
        end if;
    end do:
    a ;
end proc:
for n from 1 to 30 do
for k from A003056(n) to 1 by -1 do
    printf("%4d,",A067627(n,k)) ;
end do:
printf("\n") ;
end do: # R. J. Mathar, May 08 2019

Extensions

More terms from R. J. Mathar, May 08 2019