A198333 Irregular triangle read by rows: row n is the pruning partition of the rooted tree with Matula-Goebel number n.

1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 3, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 2, 1, 4, 2, 3, 1, 2, 3, 1, 4, 1, 2, 3, 1, 2, 3, 2, 2, 2, 1, 2, 3, 2, 4, 1, 2, 2, 2, 1, 2, 3, 1, 3, 3, 2, 4, 1, 2, 3, 2, 2, 3, 2, 2, 2, 1, 5, 1, 2, 2, 2, 1, 2, 3, 2

Offset: 1

Author

Emeric Deutsch, Nov 25 2011

Keywords

• nonn
• tabf

Comments

The pruning partition of a tree is the reverse sequence of the number of vertices of degree 1, deleted at the successive prunings. By pruning we mean the deletion of vertices of degree 1 and of their incident edges. See the Balaban reference (p. 360) and/or the Todeschini-Consonni reference (p. 42).

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

			Row 7 is 1,3 because the rooted tree with Matula-Goebel number 7 is Y, having 3 vertices of degree 1 and after the first pruning we obtain the 1-vertex tree.
The triangle starts:  |   Squared    |  Sum of squares (= A198334(n)).
1;                           1;             1
2;                           4;             4
1,2;                         1,4;           5
1,2;                         1,4;           5
2,2;                         4,4;           8
2,2;                         4,4;           8
1,3;                         1,9;          10
1,3;                         1,9;          10
1,2,2;                       1,4,4;         9
- edited by _Antti Karttunen_, Mar 07 2017
		

References

• A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355-375, 1979.
• R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, 2000.

Links

• F. Göbel, 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.
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A061775, A198329, A198334.

Programs

Maple

with(numtheory): N := 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 1+N(pi(n)) else N(r(n))+N(s(n))-1 end if end proc: a := proc (n) local r, s, b: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: b := proc (n) if n = 1 then 1 elif n = 2 then 1 elif bigomega(n) = 1 then ithprime(b(pi(n))) else b(r(n))*b(s(n)) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then b(pi(n)) else b(r(n))*b(s(n)) end if end proc: MS := proc (m) local A, i: A[m, 1] := m: for i from 2 while 1 < A[m, i-1] do A[m, i] := a(A[m, i-1]) end do: if A[m, i-2] = 2 then [seq(A[m, j], j = 1 .. i-2)] else [seq(A[m, j], j = 1 .. i-1)] end if end proc: PP := proc (n) local NVP, q: q := nops(MS(n)): NVP := map(N, MS(n)): NVP[q], seq(NVP[q-j]-NVP[q-j+1], j = 1 .. nops(NVP)-1) end proc: for n to 21 do PP(n) end do; # for the rooted tree with Matula-Goebel number n, N(n)=A061775(n) is the number of vertices, a(n) (=A198329(n)) is the Matula-Goebel number of the tree obtained after one pruning, MS(n) is the sequence of Matula-Goebel numbers of the trees obtained after 0,1,2,... prunings, PP(n) is the pruning partition, i.e. the number of vertices of degree 1 deleted at the successive prunings, given in reverse order.

Formula

A198329(n) is the Matula-Goebel number of the rooted tree obtained by removing from the rooted tree with Matula-Goebel number n the vertices of degree one, together with their incident edges. Repeated application of this yields the Matula-Goebel numbers of the trees obtained by successive prunings. Finding the number of vertices of these trees and taking differences lead to the pruning partition (see the Maple program and the explanation given there).