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

%I A182911 #23 Jul 21 2023 14:05:08
%S A182911 1,2,1,1,36,1,216,900,1,1296,5400,44100,27000,7776,32400,264600,
%T A182911 5336100,162000,1323000,46656,194400,810000,1587600,9261000,32016600,
%U A182911 901800900,972000,7938000,160083000,279936,1166400,4860000,9525600,39690000,55566000,192099600,1120581000
%N A182911 Prime encoded sequence of generic integer partitions of n in the antilexicographic order of the partitions.
%C A182911 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 A182911 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 A182911 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 A182911 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 A182911 Alois P. Heinz, <a href="/A182911/b182911.txt">Table of n, a(n) for n = 0..19200</a>
%H A182911 Peter Luschny, Integer partition trees, <a href="http://oeis.org/wiki/User:Peter_Luschny/IntegerPartitionTrees">OEIS wiki</a>.
%e A182911 0:  {}                   -> 1
%e A182911 1:  {[1]}                -> 2^1 = 2
%e A182911 2:  {}                   -> 1
%e A182911 3:  {}                   -> 1
%e A182911 4:  {[22]}               -> 2^2*3^2 = 36
%e A182911 5:  {}                   -> 1
%e A182911 6:  {[33],[222]}         -> 2^3*3^3 = 216; 2^2*3^2*5^2 = 900
%e A182911 7:  {}                   -> 1
%e A182911 8:  {[44],[332],[2222]}  -> 1296, 5400, 44100
%e A182911 9:  {[333]}              -> 27000
%p A182911 a:= proc(n) local b, ll; b:=
%p A182911       proc(n,i,l) local nl; nl:= nops(l);
%p A182911         if n<0 then
%p A182911       elif n=0 then ll:= ll,
%p A182911                `if`(nl=0 or nl=1 and l[1]=1 or
%p A182911                     nl>1 and l[-1]<>1 and l[1]=l[2],
%p A182911                     mul(ithprime(t)^l[t], t=1..nl), NULL)
%p A182911       elif i=0 then
%p A182911       else b(n-i, i, [l[], i]), b(n, i-1, l)
%p A182911         fi
%p A182911       end;
%p A182911       ll:= NULL; b(n,n,[]);
%p A182911      `if`(ll=NULL,1,ll)
%p A182911     end:
%p A182911 seq(a(n), n=0..15);
%Y A182911 Cf. A046056, A053445.
%K A182911 nonn
%O A182911 0,2
%A A182911 _Peter Luschny_, Jan 26 2011