%I #10 Oct 16 2015 23:00:59
%S 1,2,4,3,8,9,9,9,5,16,27,27,6,25,27,25,6,27,25,25,25,25,7,32,81,81,18,
%T 125,81,125,18,18,15,125,15,15,49,81,125,125,15,49,18,15,18,81,125,15,
%U 125,15,49,125,49,15,125,49,15,15,125,49,49,49,49,49,11,64,243,243,54
%N Prime-factorization encoded partition codes for the Łukasiewicz-words in A071153.
%C If the signature-permutation of a Catalan automorphism SP satisfies the condition A129593(SP(n)) = A129593(n) for all n, then it is called a Łukasiewicz-word permuting automorphism. In addition to all the automorphisms whose signature permutation satisfies the more restricted condition A127301(SP(n)) = A127301(n) for all n, this includes also certain automorphisms like *A072797 that do not preserve the non-oriented form of the general tree. A000041(n) distinct values occur in each range [A014137(n-1)..A014138(n-1)]. All natural numbers occur. Cf. A129599.
%H A. Karttunen, <a href="/A129593/b129593.txt">Table of n, a(n) for n = 0..2055</a>
%H OEIS Wiki, <a href="/wiki/Łukasiewicz_words">Łukasiewicz words</a>
%H <a href="/index/Lu#Lukasiewicz">Index entries for sequences related to Łukasiewicz</a>
%F Construction: remove zeros from the Łukasiewicz-word of a general plane tree encoded by A014486(n) (i.e. A071153(n)), sort the numbers into ascending order and interpreting it as a partition of a natural number, encode it in the manner explained in A129595.
%e The terms A071153(5..7) are 201, 210 and 120. After discarding zero and sorting, each produces partition 1+2. Converting it to prime-exponents like explained in A129595, we get 2^0 * 3^2 = 9, thus a(5) = a(6) = a(7) = 9.
%Y a(n) = a(A072797(n)).
%Y Variant: A129599. To be computed: the position of the first and the last occurrence of n, the number of occurrences of each n.
%K nonn
%O 0,2
%A _Antti Karttunen_, May 01 2007