A227955 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. The partitions of n are ordered in reversed lexicographic order read from left-to-right, starting with [1,1,...1] going down to [n].

1, 2, 6, 4, 30, 12, 8, 210, 60, 36, 24, 16, 2310, 420, 180, 120, 72, 48, 32, 30030, 4620, 1260, 900, 840, 360, 216, 240, 144, 96, 64, 510510, 60060, 13860, 6300, 9240, 2520, 1800, 1080, 1680, 720, 432, 480, 288, 192, 128, 9699690, 1021020, 180180, 69300, 44100

Offset: 0

Author

Peter Luschny, Aug 01 2013

Keywords

• nonn
• tabf

Comments

The sequence can be seen as an encoding of Young's lattice (see the links).

The ordering of Young's lattice is such that for two Young diagrams s, t, we have s <= t if and only if the Young diagram of s fits entirely inside the Young diagram of t (when the two diagrams are arranged so their lower-left corners coincide.) This order translates to our encoding as the divisibility relation. The number corresponding to s divides the number corresponding to t if and only if s <= t.

The partition corresponding to a number can be recovered as the exponents of the primes in the prime factorization of the number.

Examples

			For instance the partitions of 4 are ordered [1,1,1,1], [2,1,1,0], [2,2,0,0], [3,1,0,0], [4,0,0,0]. Consider the partition P = (3,2,1,1) written as a Young diagram (in French notation):
    [ ]
    [ ]
    [ ][ ]
    [ ][ ][ ]
Next replace the boxes at the bottom line by the sequence of primes and write the number of boxes in the same column as exponents; then multiply. 2^4*3^2*5^1 = 720. 720 will appear in line 7 of the triangle (because P is a partition of 7) at position 10 (because the sequence of exponents [4, 2, 1] is the 10th partition in the order of partitions which we assume).
[0]     1,
[1]     2,
[2]     6,    4,
[3]    30,   12,    8,
[4]   210,   60,   36,  24,  16,
[5]  2310,  420,  180, 120,  72,  48,  32,
[6] 30030, 4620, 1260, 900, 840, 360, 216, 240, 144, 96, 64.
		

Links

• Peter Luschny, Rows n = 0..25, flattened
• Peter Luschny, Young's lattice (diagram)
• Peter Luschny, Integer partition trees
• Wikipedia, Young's lattice

Crossrefs

Reversed rows: A036035, row sums: A074140.

Programs

Maple

with(combinat):
A227955_row := proc(n) local e, w, p;
p := [seq(ithprime(i), i=1..n)];
w := e -> mul(p[i]^e[nops(e)-i+1], i=1..nops(e));
seq(w(e), e = partition(n)) end:
seq(print(A227955_row(i)), i=0..8);

Sage

def A227955_row(n):
    L = []
    P = primes_first_n(n)
    for p in Partitions(n):
        L.append(mul(P[i]^p[i] for i in range(len(p))))
    return L[::-1]
for n in (0..8): A227955_row(n)