Combinatorial constructions found by computer In preparation for a talk I am giving to our undergraduate mathematics society, I am trying to collect examples of combinatorial constructions that were found by computer. Thus my question is the following.
What are the most notable examples of combinatorial constructions found by computer search?
For example, one fertile area for such results is lower bounds on Ramsey numbers, for example this recent paper by Exoo and Tatarevic. (But since I am looking for constructions, computer-assisted upper bounds on Ramsey numbers don't count.)
Another non-example would be Aubrey de Grey's construction of a unit-distance graph on 1581 vertices with chromatic number 5 (thereby showing that the chromatic number of the plane is at least 5), since the graph was constructed by hand.
 A: I don't know how notable this would be but here is another example from my paper "16,051 formulas for Ottaviani's invariant of cubic threefolds" with Christian Ikenmeyer and Gordon Royle, published in J. Algebra 2016. It is this graph:

It represents the $SL_5$-invariant of degree 15 of cubics in 5 variables.
It was found by a computer search of $1-(15,5,3)$ design with certain desired chromatic properties.
A: There is a beautiful story involving the faithfulness question of the Burau representation of the braid group.
Moody showed that the Burau representation is unfaithful if $n$ (the number of strands) is at least 10. The method involved a beautiful reduction to an existence question for a simple closed curve, with certain unusual intersecton properties. Long and Patton improved this to $n \ge 6$ by finding more curves.
Bigelow did extensive computer calculations and found such an exceptional curve for $n=5$, thus showing unfaithfulness for this $n$. He also found a ridiculously simple example for $n = 6$:
https://arxiv.org/pdf/math/9904100.pdf
A: there are results in graph theory of this sort, essentially enumerating all the graphs on at most N vertices with such and such properties, and seeing that there are none/some.
E.g. fullerens (modeled by planar graphs with certain  properties) might nice topic, see
https://doi.org/10.1006/jagm.1996.0806
the most famous computer-based result in graph theory is probably 4-color theorem
https://en.m.wikipedia.org/wiki/Four_color_theorem
My humble self constructed few interesting graphs and digraphs on computer (although then publishing (mostly) "computer-free" proofs).
https://doi.org/10.1016/j.aim.2017.03.029
https://doi.org/10.1016/0097-3165(95)90062-4
similarly, in finite group theory (which is quite combinatorial subject in a way)  one might look in a  complete database and reach some conclusion...
A: I would suggest browsing through the issues of Experimental Mathematics, which has many examples of the kind you seek. (Full disclosure: I am an editor, so you may construe this as tooting my own horn).
A: Not sure whether this counts as combinatorial: there are several equations for generating functions that were found by computer.  I think
Bostan, Alin; Kauers, Manuel; Van Hoeij, Mark, The complete generating function for Gessel walks is algebraic, Proc. Am. Math. Soc. 138, No. 9, 3063-3078 (2010). ZBL1206.05013.
and its predecessor
Kauers, Manuel; Koutschan, Christoph; Zeilberger, Doron, Proof of Ira Gessel’s lattice path conjecture, Proc. Natl. Acad. Sci. USA 106, No. 28, 11502-11505 (2009). ZBL1203.05010.
is very nice.
Personally, I am very fond of Exact formulas for the partition function?
A: My group, The Permuta Triangle, has been working on automating the discovery of combinatorial specifications. Some of the discovered results for pattern avoiding permutations, set partitions and (soon) Motzkin paths can be viewed at the ComboPal website. Click "Random" in the left hand corner to go on a tour of specifications!
A: Many coverings in La Jolla Covering Repository were found by computer.
A: The bijection in
Rubey, Martin; Stump, Christian, Double deficiencies of Dyck paths via the Billey-Jockusch-Stanley bijection, J. Integer Seq. 20, No. 9, Article 17.9.6, 9 p. (2017). ZBL1384.05048.
was essentially found by computer.
A: Emmanuel Jeandel, Michael Rao, An aperiodic set of 11 Wang tiles,
https://arxiv.org/abs/1506.06492
