OFFSET
1,2
COMMENTS
It is observed by Gus Wiseman in A368596 and A368730 that this sequence appears to be the complement of those sequences. If this is the case, then a(n) is the number of labeled graphs with loops allowed in which each connected component has an equal number of vertices and edges and the conjectured formula holds. Terms for n >= 9 are expected to be 167341283, 4191140394, 116425416531, ... - Andrew Howroyd, Jan 10 2024
From Gus Wiseman, Mar 22 2024: (Start)
An equivalent conjecture is that a(n) is the number of loop-graphs with n vertices and n edges such that it is possible to choose a different vertex from each edge. I call these graphs choosable. For example, the a(3) = 17 choosable loop-graphs are the following (loops shown as singletons):
{{1},{2},{3}} {{1},{2},{1,3}} {{1},{1,2},{1,3}} {{1,2},{1,3},{2,3}}
{{1},{2},{2,3}} {{1},{1,2},{2,3}}
{{1},{3},{1,2}} {{1},{1,3},{2,3}}
{{1},{3},{2,3}} {{2},{1,2},{1,3}}
{{2},{3},{1,2}} {{2},{1,2},{2,3}}
{{2},{3},{1,3}} {{2},{1,3},{2,3}}
{{3},{1,2},{1,3}}
{{3},{1,2},{2,3}}
{{3},{1,3},{2,3}}
(End)
REFERENCES
R. P. Stanley (Proposer), Problem 12191, Amer. Math. Monthly, 127:6 (2020), 563.
LINKS
Thomas Selig, The stochastic sandpile model on complete graphs, arXiv:2209.07301 [math.PR], 2022.
FORMULA
Conjectured e.g.f.: exp(-log(1-T(x))/2 + T(x)/2 - T(x)^2/4) where T(x) = -LambertW(-x) is the e.g.f. of A000169. - Andrew Howroyd, Jan 10 2024
EXAMPLE
For n=2 the parking functions are (1,1), (1,2), (2,1). They are the only integer points in their convex hull. For n=3, in addition to the 16 parking functions, there is the additional point (2,2,2).
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Richard Stanley, Mar 15 2020
STATUS
approved