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A181317
Triangle in which n-th row lists all partitions of n, in the order of increasing smallest numbers of prime signatures.
6
1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 4, 3, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 5, 4, 1, 3, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 6, 5, 1, 4, 2, 3, 3, 4, 1, 1, 3, 2, 1, 3, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 6, 1, 5, 2, 4, 3, 5, 1, 1, 4, 2, 1, 3, 3, 1, 4, 1, 1, 1, 3, 2, 2, 3, 2, 1, 1, 2
OFFSET
1,2
COMMENTS
The parts of each partition are listed in decreasing order.
Differs from A080577 at a(48) and from A036037 at a(56). A181087 uses the same order for all partitions.
LINKS
EXAMPLE
[3,1,1,1] and [2,2,2] are both partitions of 6, the smallest numbers having these prime signatures are 2^3*3^1*5^1*7^1=840 and 2^2*3^2*5^2=900, thus [3,1,1,1] < [2,2,2] in this order.
Triangle begins:
[1];
[2], [1,1];
[3], [2,1], [1,1,1];
[4], [3,1], [2,2], [2,1,1], [1,1,1,1];
[5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], [1,1,1,1,1];
[6], [5,1], [4,2], [3,3], [4,1,1], [3,2,1], [3,1,1,1], [2,2,2];
...
MAPLE
a:= proc(n) local b, ll; # gives all parts of partitions of row n
b:= proc(n, i, l)
if n<0 then
elif n=0 then ll:= ll, [mul(ithprime(t)^l[t], t=1..nops(l)), l]
elif i=0 then
else b(n-i, i, [l[], i]), b(n, i-1, l)
fi
end;
ll:= NULL; b(n, n, []);
map(h-> h[2][], sort([ll], (x, y)-> x[1]<y[1]))[]
end:
seq(a(n), n=1..7);
MATHEMATICA
f[P_] := Times @@ (Prime[Range[Length[P]]]^P);
row[n_] := SortBy[IntegerPartitions[n], f];
Array[row, 7] // Flatten (* Jean-François Alcover, Feb 16 2021 *)
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Jan 26 2011
STATUS
approved