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

Author

Peter Luschny, Jan 26 2011

Keywords

• nonn

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.

Examples

			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
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..19200
• Peter Luschny, Integer partition trees, OEIS wiki.

Crossrefs

Cf. A046056, A053445.

Programs

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);