OFFSET
1,2
COMMENTS
LINKS
Alois P. Heinz, Rows n = 1..19, flattened
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 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Jan 26 2011
STATUS
approved