A228099 Triangle read by rows, T(n, k) = prime(1)^p(k,1)*...*prime(n)^p(k,n) where p(k,j) is the j-th part of the k-th partition of n, additionally T(0,0) = 1. The partitions of n are ordered such that partitions of n into r parts appear in lexicographic order previous to the partitions of n into s parts if s < r. (Fenner-Loizou tree).

1, 2, 6, 4, 30, 12, 8, 210, 60, 36, 24, 16, 2310, 420, 180, 120, 72, 48, 32, 30030, 4620, 1260, 840, 900, 360, 240, 216, 144, 96, 64, 510510, 60060, 13860, 9240, 6300, 2520, 1680, 1800, 1080, 720, 480, 432, 288, 192, 128, 9699690, 1021020, 180180, 120120

Offset: 0

Author

Peter Luschny, Aug 10 2013

Keywords

• nonn
• tabf
• look

Comments

The partitions' representation (A228100) is a weakly decreasing list of parts.

The rows (read from left to right) are strongly decreasing. The T(n, 0) are the primorial numbers A002110(n). The right side of the triangle are the powers of 2,

T(n, A000041(n)) = A000079(n). The row sums are A074140.

The partition corresponding to a(n), n > 0, can be recovered as the exponents of the primes in the canonical prime factorization of a(n).

Examples

			The six-th row is:
[1, 1, 1, 1, 1, 1] -> 30030
[2, 1, 1, 1, 1] -> 4620
[2, 2, 1, 1] -> 1260
[3, 1, 1, 1] -> 840
[2, 2, 2] -> 900
[3, 2, 1] -> 360
[4, 1, 1] -> 240
[3, 3] -> 216
[4, 2] -> 144
[5, 1] -> 96
[6] -> 64
		

References

• D. E. Knuth: The Art of Computer Programming. Generating all combinations and partitions, vol. 4, fasc. 3, 7.2.1.4, exercise 10.

Links

• Peter Luschny, Rows n = 0..25, flattened
• T. I. Fenner, G. Loizou, A binary tree representation and related algorithms for generating integer partitions, The Computer J. 23(4), 332-337 (1980).
• Peter Luschny, Integer Partition Trees

Crossrefs

Cf. A228100.

Programs

Maple

b:= proc(n, i) b(n, i):= `if`(n=0 or i=1, [[1$n]], [b(n, i-1)[],
      `if`(i>n, [], map(x-> [i, x[]], b(n-i, i)))[]])
    end:
T:= n-> map(h-> mul(ithprime(j)^h[j], j=1..nops(h)), sort(b(n$2),
        proc(x, y) local i; if nops(x)<>nops(y) then return
        nops(x)>nops(y) else for i to nops(x) do if x[i]<>y[i]
        then return x[i]Alois P. Heinz, Aug 13 2013

Mathematica

b[n_, i_] := If[n == 0 || i == 1, {Array[1&, n]}, Join[b[n, i-1], If[i>n, {}, Map[Function[x, Prepend[x, i]], b[n-i, i]]]]]; T[n_] := Map[Function[h, Times @@ ((Prime /@ Range[Length[h]])^h)], Sort[b[n, n], Which[Length[#1] > Length[#2], True, Length[#1] < Length[#2], False, True, OrderedQ[#1, #2]]&]]; Table[T[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jan 27 2014, after Maple *)

Sage

from collections import deque
def Partitions_Fenner_Loizou(n):
    p = ([], 0, n)
    queue = deque()
    queue.append(p)
    yield p
    while len(queue) > 0 :
        (phead, pheadLen, pnum1s) = queue.popleft()
        if pnum1s != 1 :
            head = phead[:pheadLen] + [2]
            q = (head, pheadLen + 1, pnum1s - 2)
            if 1 <= q[2] : queue.append(q)
            yield q
        if pheadLen == 1 or (pheadLen > 1 and \
                    (phead[pheadLen - 1] != phead[pheadLen - 2])) :
            head = phead[:pheadLen]
            head[pheadLen - 1] += 1
            q = (head, pheadLen, pnum1s - 1)
            if 1 <= q[2] : queue.append(q)
            yield q
def A228099_row(n):
    if n == 0: return [1]
    L = []
    P = primes_first_n(n)
    for p in Partitions_Fenner_Loizou(n):
        e = p[0] + [1 for i in range(p[2])]
        c = mul(P[i]^e[i] for i in range(len(e)))
        L.append(c)
    return L
for n in (0..7): A228099_row(n)