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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].
2
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
0,2
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.
LINKS
Wikipedia, Young's lattice
EXAMPLE
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.
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);
PROG
(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)
CROSSREFS
Reversed rows: A036035, row sums: A074140.
Sequence in context: A322792 A253588 A228099 * A324922 A329886 A064538
KEYWORD
nonn,tabf
AUTHOR
Peter Luschny, Aug 01 2013
STATUS
approved