OFFSET
2,2
COMMENTS
The old name (referring to the Chen-Goyal article) was "[Number of] essential complementary partitions of [an] n-set."
This sequence was obtained using the deletion-contraction recursions satisfied by the number of spanning trees for graphs. It is readily seen that the number of spanning trees in K_{n}-e (the complete graph K_{n} with an edge e deleted) is (n-2)*(n^{n-3}). Since the number of spanning trees in K_{n} is n^{n-2}, we see that (n-2)*(n^{n-3})+f(n)=n^{n-2} by the deletion-contraction recursion. Hence it follows that f(n)=2*n^{n-3}. - N. Eaton, W. Kook and L. Thoma (andrewk(AT)math.uri.edu), Jan 17 2004
With offset 0, the number of acyclic functions from {1,...,n} to {1,...,n+2}. See link below. - Dennis P. Walsh, Nov 27 2011
With offset 0, a(n) is the number of forests of rooted labeled trees on n nodes in which some (possibly all or none) of the trees have been specially designated. a(n) = Sum_{k=1..n} A061356(n,k)*2^k. E.g.f. is exp(T(x))^2 where T(x) is the e.g.f for A000169. The expected number of trees in each forest approaches 3 as n gets large. Cf. A225497. - Geoffrey Critzer, May 10 2013
REFERENCES
J. Oxley, Matroid Theory, Oxford University Press, 1992.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Michael De Vlieger, Table of n, a(n) for n = 2..388
Alin Bostan, Frédéric Chyzak, Bérénice Delcroix-Oger, Guillaume Laplante-Anfossi, Vincent Pilaud, and Kurt Stoeckl, Diagonals of permutahedra and associahedra, Sém. Lotharingien Comb., 37th Formal Power Series Alg. Comb. (FPSAC 2025). See p. 7.
Wai-Kai Chen and I. C. Goyal, Tables of essential complementary partitions, IEEE Trans. Circuit Theory, 18 (1971), 562-563.
Wai-Kai Chen and I. C. Goyal, Tables of essential complementary partitions, IEEE Trans. Circuit Theory, 18 (1971), 562-563. (Annotated scanned copy)
Spencer Daugherty, Pamela E. Harris, Ian Klein, and Matt McClinton, Metered Parking Functions, arXiv:2406.12941 [math.CO], 2024. See pp. 19, 22.
Enrica Duchi, Adrián Lillo, Pablo Puerto, Mercedes Rosas, and Stefan Trandafir, The genesis sequence, tree records and endofunctions, arXiv:2601.07938 [math.CO], 2026. See pp. 17, 25.
Nancy Eaton, W. Kook and L. Thoma, Monotonicity for complete graphs and symmetric complete bipartite graphs, preprint, 2003.
Jeff Ledford, Kevin Rivera-Ayala, and Emma Schroeder, A note concerning frames and geometric inequalities, arXiv:2509.05611 [math.MG], 2025. See p. 12.
Jean-Baptiste Priez and Aladin Virmaux, Non-commutative Frobenius characteristic of generalized parking functions: Application to enumeration, arXiv:1411.4161 [math.CO], 2014-2015.
Dennis Walsh, Notes on acyclic functions
FORMULA
a(n) = 2*n^{n-3} (n>=2).
E.g.f.: (-W(-x)/x)*exp(-W(-x)). - Paul Barry, Nov 19 2010 [With offset 0, and W = LambertW. Equals (W(-x)/(-x))^2 = (exp(-W(-x)))^2 (see a comment above). - Wolfdieter Lang, Nov 11 2022]
G.f.: Sum_{n>=1} a(n+1) * x^n / (1 + n*x)^n = x/(1-x). - Paul D. Hanna, Jan 17 2013
EXAMPLE
a(3)=2 because K_{3}/e consists of two vertices and two parallel edges, where each edge is a spanning tree.
MATHEMATICA
nn = 17; tx = Sum[n^(n - 1) x^n/n!, {n, 1, nn}];
Range[0, nn]! CoefficientList[Series[Exp[ tx]^2, {x, 0, nn}], x] (* Geoffrey Critzer, May 10 2013 *)
PROG
(PARI) {a(n)=if(n==2, 1, 1-polcoeff(sum(k=2, n-1, a(k)*x^k/(1+(k-1)*x+x*O(x^n))^(k-1)), n))} /* Paul D. Hanna, Jan 17 2013 */
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
a(6) corrected and more terms from Sean A. Irvine, Dec 19 2017
After correction, this became identical (except for the offset) with A089104, contributed by N. Eaton, W. Kook and L. Thoma (andrewk(AT)math.uri.edu), Jan 17 2004. The two entries have been merged using the older A-number. - N. J. A. Sloane, Dec 19 2017
STATUS
approved