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Math Circler, ex-Googler and Dad. Now at Facebook.

comment Counting graphs up to isomorphism
Question 1 is unclear. Are you allowing nodes with a single child? What version of isomorphism do you have in mind? For full binary trees where L and R are distinct, the number is the Catalan number. There's certainly no "compact formula" for question 3.
comment out-trees and least upper boundness
I'm still very confused. Yes, lattices contain semicycles, that's just the point - you were asking if every LUB graph is a (kind of) tree, and it isn't. You now changed the question to ask if every connected digraph without semicycles (which further satisfies LUB) must be a (kind of) tree. Well, it is, by definition, right? The underlying undirected graph certainly is a tree (connected and cycle free). This is just an out-tree except that we haven't chosen a specific root.
comment Finite simple groups and conjugacy classes with 2p elements
Is it known that this can't occur with the alternating groups? Empirically it looks like the sizes of the conjugacy classes of $A_n$ all have at least 3 prime factors once $n>8$.
comment Spanning trees in 3 regular graphs.
What do "clip" and "half edges" mean?
comment Second eigenvalue of suspension of a graph
Shouldn't you be looking at the Laplacian, rather than the adjacency matrix? For non-regular graphs I'm not sure if the second largest eigenvalue is the thing that controls mixing. Also, the operation you're interested in seems more like taking the cone over $G$, rather than a suspension.
comment What are some proofs of Godel's Theorem which are *essentially different* from the original proof?
An awesome summary!
comment What should be learned in an introductory analytic number theory course?
I'm curious: what is "Pollack's new book"?
comment Gently falling functions
Sorry if that's a silly question but in example 1), a particle starting at (0,1) won't go anywhere unless you give it some initial horizontal velocity. Are you suggesting that the separation point tends to the indicated point as that velocity tends to 0?
comment Proof that any NP problem can be reduced (in P time) to any problem in NPC?
I don't understand your modified question. Of course every problem in NP can be reduced some subset of NP problems - it can be "reduced" to itself, and often to lots of other problems polynomially-equivalent to it. Why does that imply that NPC is empty?
comment Is there any sequence $a_n$ of nonnegative numbers for which $\sum_{n \geq 1}a_n^2 <\infty$ and $\sum_{n \geq 1}\left(\sum_{k \geq 1}\frac{a_{kn}}{k}\right)^2=\infty$?
In line 3, did you mean "the sequence whose $n$-th term is..."?
comment Examples of theorems misapplied to non-mathematical contexts
If he does, it's a slam dunk.
comment Proof of no rational point on Selmer's Curve 3x^3+4y^3+5z^3=0
@Ho Chung Siu, do you happen to have a copy of that paper anywhere accessible?
comment What is the high-concept explanation on why real numbers are useful in number theory?
This is totally tangential but I just want to say I strongly disagree with the sentiment of the first sentence. If statements and proofs were typically "in the same world", math would be so dull. Dystopia.
comment Motivation behind Tutte's 1-factor theorem
I changed the question's title to better correspond to its content. You don't appear to be interested in the historical development of the theorem, but rather the motivation behind this characterization. Perhaps a better way to phrase the question would be to ask, is there any algorithmic or theoretical advantage to this (rather daunting) condition.
comment Is “second-countable implies separable” equivalent to the Axiom of countable Choice?
Surely this requires only the axiom of countable choice?
comment How many finite simple groups of order $p+1$?
n!/2-1 is prime for n=5, 6, 9, 31, 41, 373 ... (sequence A082671 on the OEIS). Is it known if this sequence is finite?
comment Ingenuity in mathematics
To complete the triviality of the situation, after your audience proposes a bunch of complex strategies, propose an alternative game whereby after "STOP" it's the last card in the deck that gets turned over. Everyone sees that this game is strategy-indifferent, and then you deliver the coup de grace... I've used this several times in math circles. Gets them every time.
comment Proofs without words
Done! Thanks for the comment.
comment roots of permutations
True. My answer attempts to generalize further into equations in several variables.
comment 4-colorable graphs
It's even falser :-) In fact, there are graphs with high girth and high chromatic number. A graph with high girth looks like a tree around any vertex, so it avoids much more than just triangles. Proving that such graphs exist is perhaps even simpler than the explicit construction by Mycielski; it's basically a counting argument.