A214573 Irregular triangle read by rows: the (n,k)-entry is the number of vertices with Strahler number k in the rooted tree with Matula-Goebel number n (n>=1, k>=1).
1, 2, 3, 2, 1, 4, 3, 1, 2, 2, 3, 1, 4, 1, 4, 1, 5, 4, 1, 3, 2, 3, 2, 5, 1, 4, 1, 2, 3, 5, 1, 3, 2, 5, 1, 4, 2, 5, 1, 4, 2, 5, 1, 6, 1, 4, 2, 6, 1, 4, 2, 4, 2, 6, 1, 6, 5, 1, 6, 1, 3, 3, 5, 2, 6, 1, 4, 2, 4, 2, 5, 2, 6, 1, 3, 3, 5, 2, 3, 3, 6, 1, 7, 1, 5, 2, 5, 2, 6, 1, 4, 2, 1, 7, 1, 4, 3, 5, 2, 4, 2, 7, 1, 7, 1, 5, 2, 5, 2, 5, 2, 2, 4, 7, 1, 5, 2, 6, 1, 6, 2, 6, 1, 6, 2, 7, 1, 3, 3, 4, 3, 6, 2, 6, 2, 5, 2, 7, 1, 4, 3, 5, 2
Offset: 1
Examples
Row 4 is 2,1. Indeed, the rooted tree with Matula-Goebel number 4 is V; the two leaves have Strahler numbers 1,1, and the root has Strahler number 2. Triangle starts: 1; 2; 3; 2,1; 4; 3,1; 2,2; ...
Links
- Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
- Emeric Deutsch, Rooted tree statistics from Matula numbers, Discrete Appl. Math., 160, 2012, 2314-2322.
- 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.
- Wikipedia, Strahler number
- Index entries for sequences related to Matula-Goebel numbers
Programs
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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 x elif bigomega(n) = 1 then sort(expand(x^degree(G(pi(n)))+G(pi(n)))) elif 1 < bigomega(n) and degree(G(r(n))) <> degree(G(s(n))) then sort(G(r(n))-x^degree(G(r(n)))+G(s(n))-x^degree(G(s(n)))+x^max(degree(G(r(n))), degree(G(s(n))))) else sort(G(r(n))-x^degree(G(r(n)))+G(s(n))-x^degree(G(s(n)))+x^(1+degree(G(r(n))))) end if end proc: for n to 100 do g[n] := G(n) end do: for n to 100 do seq(coeff(g[n], x, j), j = 1 .. degree(g[n])) end do: seq(seq(coeff(g[n], x, j), j = 1 .. degree(g[n])), n = 1 .. 100);
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Mathematica
r[n_] := FactorInteger[n][[1, 1]]; s[n_] := n/r[n]; degree = Exponent[#, x]&; G[n_] := G[n] = Which[n == 1, x, PrimeOmega[n] == 1, Sort[ Expand[ x^degree[G[PrimePi[n]]] + G[PrimePi[n]]]], 1 < PrimeOmega[n] && degree[G[r[n]], x] != degree[G[s[n]]], Sort[G[r[n]] - x^degree[G[r[n]]] + G[s[n]] - x^degree[G[s[n]]] + x^Max[degree[G[r[n]]], degree[G[s[n]]]]], True, Sort[G[r[n]] - x^degree[G[r[n]]] + G[s[n]] - x^degree[G[s[n]]] + x^(1 + degree[G[r[n]]])]]; Table[Table[Coefficient[G[n], x, j], {j, 1, degree[G[n], x]}], {n, 1, 100}] // Flatten (* Jean-François Alcover, Aug 11 2024, after Emeric Deutsch *)
Formula
Define the Strahler polynomial of a rooted tree T as the generating polynomial of the vertices of T with respect to their Strahler numbers. For example, it follows at once that the Strahler polynomial of the rooted tree V is 2x + x^2. Denote by G(n)=G(n;x) the Strahler polynomial of the rooted tree with Matula-Goebel number n. Clearly, A214573(n,k) is the coefficient of x^k in G(n). We have (i) G(1)= x; (ii) if n=p(t) (the t-th prime), then G(n) = x^{degree(G(t)} + G(t); (iii) if n=rs (r,s>=2), then G(n) = G(r) - degree (G(r)) + G(s) - degree(G(s)) + x^m, where m = 1+degree(G(r)) if degree(G(r))=degree(G(s)) and m = max(degree(G(r), G(s)) otherwise. The Maple program is based on these recursion relations.
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