

A212625


Number of vertices in the largest independent vertex subset of the rooted tree with MatulaGoebel number n.


11



1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 4, 4, 3, 3, 4, 4, 3, 4, 4, 4, 4, 3, 5, 4, 4, 4, 4, 4, 4, 4, 5, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 4, 4, 5, 4, 4, 5, 5, 4, 4, 5, 4, 4, 5, 6, 4, 4, 4, 5, 4, 5, 5, 5, 4, 4, 5, 5, 5, 4, 4, 6, 5, 4, 4, 5, 5, 4, 5, 5, 5, 5, 5, 5, 4, 4, 5, 6, 5, 5
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OFFSET

1,3


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 MatulaGoebel number of a rooted tree can be defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the MatulaGoebel 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 MatulaGoebel numbers of the m branches of T.


REFERENCES

F. Goebel, On a 11 correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YN. Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.


LINKS

Table of n, a(n) for n=1..98.
E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288.
Index entries for sequences related to MatulaGoebel numbers


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 MatulaGoebel number n. We have a(n)=degree(P(n,x)).


EXAMPLE

a(5)=2 because the rooted tree with MatulaGoebel number 5 is the path tree R  A  B  C with independent vertex subsets: {}, {R}, {A}, {B}, {C}, {R,B}, {R,C}, {A,C}; their sizes are 0,1,and 2.


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: degree(P(n)) end proc: seq(a(n), n = 1 .. 120);


CROSSREFS

Cf. A212618, A212619, A212620, A212621, A212622, A212623, A212624, A212626, A212627, A212628, A212629, A212630, A212631, A212632.
Sequence in context: A156875 A066339 A052375 * A171626 A074279 A072750
Adjacent sequences: A212622 A212623 A212624 * A212626 A212627 A212628


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Jun 01 2012


STATUS

approved



