A182907 Triangle read by rows: row n is the degree sequence (written in nondecreasing order) of the rooted tree with Matula-Goebel number n.

0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 3, 1, 1, 1, 2, 3, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 4, 1, 1, 1, 2, 2, 3, 1, 1, 1, 2, 2, 3, 1, 1, 2, 2, 2, 2, 1, 1, 1, 2, 2, 3, 1, 1, 1, 1, 2, 4, 1, 1, 2, 2, 2, 2, 2

Offset: 1

Author

Emeric Deutsch, Oct 05 2011

Keywords

• nonn
• tabf

Comments

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.

The number of entries in row n is A061775(n) (= number of vertices of the rooted tree).

Examples

			Row 7 is 1,1,1,3 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.
Row 32 is 1,1,1,1,1,5 because the rooted tree with Matula-Goebel number 32 is a star with 5 edges.
		

Links

• Emeric Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
• F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
• I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
• I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
• D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.

Crossrefs

Cf. A061775.

Programs

Maple

with(numtheory): g := 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(expand(g(pi(n))+x^bigomega(pi(n))*(x-1)+x)) else sort(expand(g(r(n))+g(s(n))-x^bigomega(r(n))-x^bigomega(s(n))+x^bigomega(n))) end if end proc: S := proc (n) if n = 1 then 0 else seq(seq(j, i = 1 .. coeff(g(n), x, j)), j = 1 .. degree(g(n))) end if end proc: for n to 25 do S(n) end do; # yields sequence in triangular form

Mathematica

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
g[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, g[PrimePi[n]] + x^PrimeOmega[PrimePi[n]]*(x - 1) + x, True, g[r[n]] + g[s[n]] - x^PrimeOmega[r[n]] - x^PrimeOmega[s[n]] + x^PrimeOmega[n]];
S[n_] := If[n == 1, {0}, Table[t = Table[j, {i, 1, Coefficient[g[n], x, j]}]; If[t == {}, Nothing, t], {j, 1, Exponent[g[n], x]}]] // Flatten;
Table[S[n], {n, 1, 25}] // Flatten (* Jean-François Alcover, Jun 20 2024, after Maple code *)

Formula

For a graph with degree sequence a,b,c,..., define the degree sequence polynomial to be x^a + x^b + x^c + ... . The degree sequence polynomial g(n)=g(n,x) of the rooted tree with Matula-Goebel number n can be obtained recursively in the following way: g(1)=1; if n=prime(t) (=the t-th prime), then g(n)=g(t)+x^G(t)*(x-1)+x; if n=r*s (r,s>=2), then g(n)=g(r)+g(s)-x^G(r)-x^G(s)+x^G(n); G(m) is the number of prime divisors of m counted with multiplicities. The Maple program, based on this recursive procedure, finds for an arbitrary n the polynomial g(n,x) and then extracts from this polynomial the degree sequence S(n).