A198339 - OEIS

%I #13 Mar 07 2017 11:27:05

%S 1,1,1,2,1,1,1,2,2,3,1,1,1,2,2,4,1,1,1,1,2,2,2,3,3,5,1,1,1,1,2,2,2,3,

%T 4,6,1,1,1,1,2,2,2,3,3,4,7,1,1,1,1,2,2,2,4,4,4,8,1,1,1,1,1,2,2,2,2,3,

%U 3,4,6,6,9,1,1,1,1,1,2,2,2,2,3,3,4,5,6,10

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

%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 A184161(n). Row n>=2 can be easily identified: its first entry is the entry 1 following the first occurrence of n-1 and its last entry is the first occurrence of n.

%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

%H F. Goebel, <a href="http://dx.doi.org/10.1016/0095-8956(80)90049-0">On a 1-1-correspondence between rooted trees and natural numbers</a>, J. Combin. Theory, B 29 (1980), 141-143.

%H I. Gutman and A. Ivic, <a href="http://dx.doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a> Discrete Math., 150 (1996),  131-142.

%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>

%F We consider separately the subtrees that contain the root (root subtrees) and those that do not contain the root (non-root subtrees). A root subtree of a rooted tree T is a subtree of T containing the root. The Matula numbers of the root subtrees of the rooted tree with Matula-Goebel number n are described in A198338. The non-root subtrees are the following: if n=1, then there is no non-root subtree; if n = p(t) (= the t-th prime), then the non-root subtrees corresoponding to n are all the subtrees corresponding to t; if n=rs (r,s >=2), then the non-root subtrees consist of the non-root subtrees corresponding to r and those corresponding to s. The Maple program, based on this recursive procedure, yields row n (<=2000; adjustable) with the command MST(n).

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

%e 1;

%e 1,1,2;

%e 1,1,1,2,2,3;

%e 1,1,1,2,2,4;

%e 1,1,1,1,2,2,2,3,3,5;

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

%p m2union := proc (x, y) sort([op(x), op(y)]) end proc:

%p with(numtheory):

%p MRST := proc (n) local r, s:

%p    r := proc (n) options operator, arrow: op(1, factorset(n)) end proc:

%p    s := proc (n) options operator, arrow; n/r(n) end proc:

%p    if n = 1 then [1] elif bigomega(n) = 1 then [1, seq(ithprime(mrst[pi(n)][i]), i = 1 .. nops(mrst[pi(n)]))] else [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:

%p MNRST := proc (n) local r, s:

%p    r := proc (n) options operator, arrow: op(1, factorset(n)) end proc:

%p    s := proc (n) options operator, arrow: n/r(n) end proc:

%p    if n = 1 then [] elif bigomega(n) = 1 then m2union(mrst[pi(n)], mnrst[pi(n)]) else m2union(mnrst[r(n)], mnrst[s(n)]) end if end proc:

%p MST := proc (n) m2union(mrst[n], mnrst[n]) end proc:

%p for n to 2000 do mrst[n] := MRST(n): mnrst[n] := MNRST(n): mst[n] := MST(n) end do;

%Y Cf. A184161, A198338, A198341.

%K nonn,tabf

%O 1,4

%A _Emeric Deutsch_, Dec 04 2011