A198338 - OEIS

%I #9 Jan 07 2013 13:05:55

%S 1,1,2,1,2,3,1,2,2,4,1,2,3,5,1,2,2,3,4,6,1,2,3,3,7,1,2,2,2,4,4,4,8,1,

%T 2,2,3,3,4,6,6,9,1,2,2,3,4,5,6,10,1,2,3,5,11,1,2,2,2,3,4,4,4,6,6,8,12,

%U 1,2,3,3,4,6,6,7,15,1,2,2,3,3,4,5,6,6

%N Irregular triangle read by rows: row n is the sequence of Matula numbers of the rooted subtrees of the rooted tree with Matula-Goebel number n. A root subtree of a rooted tree T is a subtree of T containing the root.

%C The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.

%C Number of entries in row n is A184160(n). Row n>=2 can be easily identified: it starts with the entry following the first occurrence of n-1 and it ends with the first occurrence of n.

%D F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.

%D I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.

%D I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.

%D D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.

%H E. Deutsch, <a href="http://arxiv.org/abs/1111.4288"> Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288

%F Row 1 is [1]; if n = p(t) (= the t-th prime), then row n is [1, p(a), p(b), ... ], where [a,b,...] is row t; if n=rs (r,s >=2), then row n consists of the numbers r[i]*s[j], where [r[1], r[2],...] is row r and [s[1], s[2], ...] is row s. The Maple program, based on this recursive procedure, yields row n (<=1000; adjustable) with the command MRST(n).

%e Row 4 is [1,2,2,4] because the rooted tree with Matula-Goebel number 4 is V and its root subtrees are *, |, |, and V.

%e Triangle starts:

%e 1;

%e 1,2;

%e 1,2,3;

%e 1,2,2,4;

%e 1,2,3,5;

%e 1,2,2,3,4,6;

%p with(numtheory): MRST := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow; n/r(n) end proc: if n = 1 then [1] elif bigomega(n) = 1 then sort([1, seq(ithprime(mrst[pi(n)][i]), i = 1 .. nops(mrst[pi(n)]))]) else sort([seq(seq(mrst[r(n)][i]*mrst[s(n)][j], i = 1 .. nops(mrst[r(n)])), j = 1 .. nops(mrst[s(n)]))]) end if end proc: for n to 1000 do mrst[n] := MRST(n) end do;

%Y Cf. A198339.

%K nonn,tabf

%O 1,3

%A _Emeric Deutsch_, Dec 04 2011