[A058387] Series-Parallel Confusion

N. J. A. Sloane njas at research.att.com
Fri Nov 8 04:43:27 CET 2002

i just added the defn to the main sequence,
A000084, based on the Riordan-Shannon paper
(from Shannon's collected papers, which i edited)
and the new entry is the following

%I A000084 M1207 N0466
%S A000084 1,2,4,10,24,66,180,522,1532,4624,14136,43930,137908,437502,1399068,
%T A000084 4507352,14611576,47633486,156047204,513477502,1696305728,5623993944,
%U A000084 18706733128,62408176762,208769240140,700129713630,2353386723912
%N A000084 Series-parallel networks with n unlabeled edges. Also called yoke-chains by Cayley and MacMahon.
%C A000084 This is a series-parallel network: o-o; all other series-parallel networks are obtained by connecting two series-parallel networks in series or in parallel.
%D A000084 P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102.
%D A000084 Z. A. Lomnicki, Two-terminal series-parallel networks, Adv. Appl. Prob., 4 (1972), 109-150.
%D A000084 P. A. MacMahon, Yoke-trains and multipartite compositions in connexion with the analytical forms called "trees", Proc. London Math. Soc. 22 (1891), 330-346; reprinted in Coll. Papers I, pp. 600-616. Page 333 gives A000084 = 2*A000669.
%D A000084 P. A. MacMahon, The combination of resistances, The Electrician, 28 (1892), 601-602; reprinted in Coll. Papers I, pp. 617-619.
%D A000084 J. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226.
%D A000084 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 142.
%D A000084 J. Riordan and C. E. Shannon, The number of two-terminal series-parallel networks, J. Math. Phys., 21 (1942), 83-93. Reprinted in Claude Elwood Shannon: Collected Papers, edited by N. J. A. Sloane and A. D. Wyner, IEEE Press, NY, 1993, pp. 560-
%D A000084 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.40, notes on p. 133.
%H A000084 P. J. Cameron, <a href="http://www.math.uwaterloo.ca/JIS/index.html#P00.1.5">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H A000084 N. J. A. Sloane, <a href="http://www.research.att.com/~njas/sequences/a669.txt">First 1001 terms of A000669 and A000084</a>
%F A000084 Let b(1)=1, b(n)=a(n)/2 for n >= 2. Then sequence satisfies Product_{k=1..inf} 1/(1-x^k)^b(k) = 1 + Sum_{k=1..inf} a(k)*x^k.
%p A000084 (continue from A000669) A000084:=n-> if n=1 then 1 else 2*A000669(n); fi;
%p A000084 # N denotes all series-parallel networks, S = series networks, P = parallel networks; spec84:=[ N,{N=Union(Z,S,P),S=Set(Union(Z,P),card>=2),P=Set(Union(Z,S),card>=2)} ]: A000084:=n->combstruct[count](spec84,size=n);
%Y A000084 Apart from initial term, 2*A000669. Cf. A058351, A058352, A058353, A000311, A006351 (labeled version).
%e A000084 The series-parallel networks with 1, 2 and 3 edges are:
%e A000084 1 edge: o-o
%e A000084 2 edges: o-o-o o=o
%e A000084 ....................... /\
%e A000084 3 edges: o-o-o-o o-o=o o--o o-o-o
%e A000084 ....................... \/ ..\o/
%Y A000084 See also A058964, A058965.
%K A000084 nonn,nice,easy
%O A000084 1,2
%A A000084 njas

i'm certainly all in favor of giving
precise definitions for sequences

often i indicate this in the Index file
by putting a star next to the sequences with definitions


More information about the SeqFan mailing list