A119441 Distribution of A063834 in Abramowitz and Stegun order.

1, 2, 1, 3, 2, 1, 5, 3, 4, 2, 1, 7, 5, 6, 3, 4, 2, 1, 11, 7, 10, 9, 5, 6, 8, 3, 4, 2, 1, 15, 11, 14, 15, 7, 10, 9, 12, 5, 6, 8, 3, 4, 2, 1, 22, 15, 22, 21, 25, 11, 14, 15, 20, 18, 7, 10, 9, 12, 16, 5, 6, 8, 3, 4, 2, 1, 30, 22, 30, 33, 35, 15, 22, 21

Offset: 1

Author

Alford Arnold, May 19 2006

Keywords

• easy
• nonn
• tabf

Examples

			1;
2, 1;
3, 2, 1;
5, 3, 4, 2, 1;
7, 5, 6, 3, 4, 2, 1;
T(5,2) = 5 because the second partition of 5 is 1+4 and 4 can be repartitioned in 5 different ways.
T(5,3) = 6 because the third partition of 5 is 2+3, where the 2 can be partitioned in 2 ways (2, 1+1) and the 3 can be partitioned in 3 ways (3, 1+2, 1+1+1), 6=2*3.
T(5,4) = 3 because the fourth partition of 5 is 1+1+3 and 3 can be partitioned in 3 different ways.
		

Links

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Crossrefs

Cf. A063834, A119442, A000041 (row lengths and also first column)

Programs

Maple

# Compare two partitions (list) in AS order.
AScompare := proc(p1,p2)
    if nops(p1) > nops(p2) then
        return 1;
    elif nops(p1) < nops(p2) then
        return -1;
    else
        for i from 1 to nops(p1) do
            if op(i,p1) > op(i,p2) then
                return 1;
            elif op(i,p1) < op(i,p2) then
                return -1;
            end if;
        end do:
        return 0 ;
    end if;
end proc:
# Produce list of partitions in AS order
ASPrts := proc(n)
    local pi,insrt,p,ex ;
    pi := [] ;
    for p in combinat[partition](n) do
        insrt := 0 ;
        for ex from 1 to nops(pi) do
            if AScompare(p, op(ex,pi)) > 0 then
                insrt := ex ;
            end if;
        end do:
        if nops(pi) = 0 then
            pi := [p] ;
        elif insrt = 0 then
            pi := [p,op(pi)] ;
        elif insrt = nops(pi) then
            pi := [op(pi),p] ;
        else
            pi := [op(1..insrt,pi),p,op(insrt+1..nops(pi),pi)] ;
        end if;
    end do:
    return pi ;
end proc:
A119441 := proc(n,k)
    local pi,a,p ;
    pi := ASPrts(n)[k] ;
    a := 1 ;
    for p in pi do
        a := a*combinat[numbpart](p) ;
    end do:
    a ;
end proc:
for n from 1 to 10 do
    for k from 1 to A000041(n) do
        printf("%d,",A119441(n,k)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Jul 12 2013

Formula

T(n,k) = product_{p=1..A036043(n,k)} A000041(c), 1<=k<=A000041(n), where c are the parts in the k-th partition of n. - R. J. Mathar, Jul 12 2013