A333331 Number of integer points in the convex hull in R^n of parking functions of length n.

1, 3, 17, 144, 1623, 22804, 383415, 7501422

Offset: 1

Author

Richard Stanley, Mar 15 2020

Keywords

• nonn
• more

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)

Examples

			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).
		

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.

Crossrefs

All of the following relative references pertain to the conjecture:

The case of unique choice A000272.

The version without the choice condition is A014068, covering A368597.

The case of just pairs A137916.

For any number of edges of any positive size we have A367902.

The complement A368596, covering A368730.

Allowing edges of any positive size gives A368601, complement A368600.

Counting by singletons gives A368924.

For any number of edges we have A368927, complement A369141.

The connected case is A368951.

The unlabeled version is A368984, complement A368835.

A000085, A100861, A111924 count set partitions into singletons or pairs.

A006125 counts graphs, unlabeled A000088.

A058891 counts set-systems (without singletons A016031), unlabeled A000612.

Cf. A000169, A000666, A057500, A062740, A129271, A133686, A367863, A367903, A369146, A369199.

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