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

1,6

Comments

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

Links

Table of n, a(n) for n=1..78.

Example

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

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

Crossrefs

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

Sequence in context: A271566 A296659 A270823 * A077233 A282290 A178795

Adjacent sequences: A067624 A067625 A067626 * A067628 A067629 A067630

Keyword

easy,nonn,tabf

Author

Naohiro Nomoto, Feb 02 2002

Extensions

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

Status

approved