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A227955
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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].
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2
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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
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MAPLE
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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);
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PROG
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(Sage)
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]
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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