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A184167
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Irregular triangle read by rows: T(n,k) is the number of vertices having escape distance k>=0 in the rooted tree having Matula-Goebel number n.
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4
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1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 3, 1, 2, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 2, 2, 2, 1, 3, 2, 2, 2, 2, 4, 1, 2, 1, 1, 1, 3, 3, 3, 1, 1, 3, 2, 1, 3, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 4, 2, 2, 2, 2, 1, 3, 3, 3, 3, 1, 4, 2, 2, 2, 2, 3, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 2, 2, 2, 1, 3, 2, 1, 3, 2, 2, 4, 3, 3, 2, 1, 4, 2, 3, 3, 1, 4, 2, 1
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OFFSET
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1,7
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COMMENTS
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The escape distance of a vertex v in a rooted tree T is the distance from v to the nearest leaf of T that is a descendant of v. For the rooted tree ARBCDEF, rooted at R, the escape distance of B is 4 (the leaf A is not a descendant of B).
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.
Each row is nonincreasing (each vertex with escape distance k (k>=1) is the parent of some vertex with escape distance k-1).
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REFERENCES
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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 Review, 10, 1968, 273.
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LINKS
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FORMULA
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We give the recursive construction of the row generating polynomials P(n)=P(n,x): if n = p(t) (=the t-th prime), then P(n)=P(t)+x^{1+LLL(t)}; if n=rs (r,s>=2), then P(n)=P(r)+P(s)-x^{max(LLL(r),LLL(s))}; LLL denotes the level of the lowest leaf (computed recursively and programmed in A184166) (2nd Maple program yields P(n)).
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EXAMPLE
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Row n=7 is [2,1,1] because the rooted tree with Matula-Goebel number 7 is the rooted tree Y, having 2 leaves and 1 (1) vertex at distance 1 (2) from either of the leaves.
Triangle starts:
1;
1,1;
1,1,1;
2,1;
1,1,1,1;
2,2;
2,1,1;
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MAPLE
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with(numtheory): P := proc (n) local r, s, LLL: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: LLL := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+LLL(pi(n)) else min(LLL(r(n)), LLL(s(n))) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then P(pi(n))+x^(1+LLL(pi(n))) else P(r(n))+P(s(n))-x^max(LLL(r(n)), LLL(s(n))) end if end proc: for n to 30 do seq(coeff(P(n), x, k), k = 0 .. degree(P(n))) end do; # yields sequence in triangular form
with(numtheory): P := proc (n) local r, s, LLL: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: LLL := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+LLL(pi(n)) else min(LLL(r(n)), LLL(s(n))) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then P(pi(n))+x^(1+LLL(pi(n))) else P(r(n))+P(s(n))-x^max(LLL(r(n)), LLL(s(n))) end if end proc: P(998877665544);
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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