A182911 Prime encoded sequence of generic integer partitions of n in the antilexicographic order of the partitions.

1, 2, 1, 1, 36, 1, 216, 900, 1, 1296, 5400, 44100, 27000, 7776, 32400, 264600, 5336100, 162000, 1323000, 46656, 194400, 810000, 1587600, 9261000, 32016600, 901800900, 972000, 7938000, 160083000, 279936, 1166400, 4860000, 9525600, 39690000, 55566000, 192099600, 1120581000

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

Cf. A046056, A053445.

Sequence in context: A147802 A093076 A132454 * A362226 A058293 A172092

Adjacent sequences: A182908 A182909 A182910 * A182912 A182913 A182914

Keyword

nonn

Author

Peter Luschny, Jan 26 2011

Status

approved