A212626 Number of largest independent vertex subsets of the rooted tree with Matula-Goebel number n.

1, 2, 1, 1, 3, 3, 1, 1, 1, 1, 1, 2, 2, 2, 4, 1, 2, 5, 1, 1, 1, 4, 5, 2, 1, 5, 1, 1, 1, 2, 4, 1, 1, 1, 3, 4, 2, 2, 2, 1, 5, 3, 1, 3, 6, 1, 2, 2, 1, 1, 3, 3, 1, 9, 5, 1, 1, 2, 1, 2, 4, 1, 1, 1, 7, 7, 2, 1, 6, 1, 1, 4, 3, 4, 2, 1, 1, 8, 3, 1, 1, 2, 1, 2, 1, 3, 1, 3, 2, 4, 2, 1, 5, 6, 3, 2, 1, 2, 1, 1

Offset: 1

Author

Emeric Deutsch, Jun 01 2012

Keywords

• nonn

Comments

A vertex subset in a tree is said to be independent if no pair of vertices is connected by an edge. The empty set is considered to be independent.

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.

Examples

			a(5)= 3 because the rooted tree with Matula-Goebel number 5 is the path tree R - A - B - C with independent vertex subsets: {}, {R}, {A}, {B}, {C}, {R,B}, {R,C}, {A,C}; the largest size (namely 2) is attained by 3 of them.
		

References

• 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 Y-N. 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 Review, 10, 1968, 273.

Links

• Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A212618, A212619, A212620, A212621, A212622, A212623, A212624, A212625, A212627, A212628, A212629, A212630, A212631, A212632.

Programs

Maple

with(numtheory): A := 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 [x, 1] elif bigomega(n) = 1 then [expand(x*A(pi(n))[2]), expand(A(pi(n))[1])+A(pi(n))[2]] else [sort(expand(A(r(n))[1]*A(s(n))[1]/x)), sort(expand(A(r(n))[2]*A(s(n))[2]))] end if end proc: P := proc (n) options operator, arrow: sort(A(n)[1]+A(n)[2]) end proc: a := proc (n) options operator, arrow: coeff(P(n), x, degree(P(n))) end proc: seq(a(n), n = 1 .. 120);

Mathematica

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
A[n_] := A[n] = Which[n == 1, {x, 1}, PrimeOmega[n] == 1, {x*A[PrimePi[n]][[2]], A[PrimePi[n]][[1]] + A[PrimePi[n]][[2]]}, True, {A[r[n]][[1]]* A[s[n]][[1]]/x, A[r[n]][[2]]*A[s[n]][[2]]}];
P[n_] := A[n] // Total;
a[n_] := Coefficient[P[n], x, Exponent[P[n], x]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 24 2024, after Maple code *)

Formula

In A212623 one finds the generating polynomial P(n,x) with respect to the number of vertices of the independent vertex subsets of the rooted tree with Matula-Goebel number n. We have a(n) = coefficient of the largest power of x in P(n,x).