OFFSET
0,2
COMMENTS
By definition [1] is a generic partition and 0 has no generic partitions. For n > 1 a partition p of n is generic if it does not have the form [1+r_1,r_2,...,r_k] or [r_1,r_2,...,r_k,1] for some partition [r_1,r_2,...,r_k] of n-1.
Encoding: The partition p = [p_1,...,p_k] is represented by Product_{i=1..k} prime(i) ^ p_i. If n has generic partitions then these encodings are listed in the antilexicographic order of the partitions; if n has no generic partitions then this fact is represented by '1'.
Starting from generic partitions a table of all partitions can be built by two operations: appending '1' at the tail of a partition or adding 1 to the head of a partition (see the table at the link given).
A generic partition is a partition of the form [x,x,p_2,...,p_k-1,y] with y > 1; in addition [1] is a generic partition by definition.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..19200
Peter Luschny, Integer partition trees, OEIS wiki.
EXAMPLE
0: {} -> 1
1: {[1]} -> 2^1 = 2
2: {} -> 1
3: {} -> 1
4: {[22]} -> 2^2*3^2 = 36
5: {} -> 1
6: {[33],[222]} -> 2^3*3^3 = 216; 2^2*3^2*5^2 = 900
7: {} -> 1
8: {[44],[332],[2222]} -> 1296, 5400, 44100
9: {[333]} -> 27000
MAPLE
a:= proc(n) local b, ll; b:=
proc(n, i, l) local nl; nl:= nops(l);
if n<0 then
elif n=0 then ll:= ll,
`if`(nl=0 or nl=1 and l[1]=1 or
nl>1 and l[-1]<>1 and l[1]=l[2],
mul(ithprime(t)^l[t], t=1..nl), NULL)
elif i=0 then
else b(n-i, i, [l[], i]), b(n, i-1, l)
fi
end;
ll:= NULL; b(n, n, []);
`if`(ll=NULL, 1, ll)
end:
seq(a(n), n=0..15);
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Jan 26 2011
STATUS
approved