A182911 - OEIS

%I #23 Jul 21 2023 14:05:08

%S 1,2,1,1,36,1,216,900,1,1296,5400,44100,27000,7776,32400,264600,

%T 5336100,162000,1323000,46656,194400,810000,1587600,9261000,32016600,

%U 901800900,972000,7938000,160083000,279936,1166400,4860000,9525600,39690000,55566000,192099600,1120581000

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

%C 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.

%C 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'.

%C 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).

%C 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.

%H Alois P. Heinz, <a href="/A182911/b182911.txt">Table of n, a(n) for n = 0..19200</a>

%H Peter Luschny, Integer partition trees, <a href="http://oeis.org/wiki/User:Peter_Luschny/IntegerPartitionTrees">OEIS wiki</a>.

%e 0:  {}                   -> 1

%e 1:  {[1]}                -> 2^1 = 2

%e 2:  {}                   -> 1

%e 3:  {}                   -> 1

%e 4:  {[22]}               -> 2^2*3^2 = 36

%e 5:  {}                   -> 1

%e 6:  {[33],[222]}         -> 2^3*3^3 = 216; 2^2*3^2*5^2 = 900

%e 7:  {}                   -> 1

%e 8:  {[44],[332],[2222]}  -> 1296, 5400, 44100

%e 9:  {[333]}              -> 27000

%p a:= proc(n) local b, ll; b:=

%p       proc(n,i,l) local nl; nl:= nops(l);

%p         if n<0 then

%p       elif n=0 then ll:= ll,

%p                `if`(nl=0 or nl=1 and l[1]=1 or

%p                     nl>1 and l[-1]<>1 and l[1]=l[2],

%p                     mul(ithprime(t)^l[t], t=1..nl), NULL)

%p       elif i=0 then

%p       else b(n-i, i, [l[], i]), b(n, i-1, l)

%p         fi

%p       end;

%p       ll:= NULL; b(n,n,[]);

%p      `if`(ll=NULL,1,ll)

%p     end:

%p seq(a(n), n=0..15);

%Y Cf. A046056, A053445.

%K nonn

%O 0,2

%A _Peter Luschny_, Jan 26 2011