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Always look for the counterexample would be a worthwhile motto it seems. For $\alpha = 1/3$ (so that $e^x$ is a complex cube root of unity), $n = 6$ and $m = 9$ provide one. This is because $s(6) = …

As noted in the comments (but with not quite the right reference) the game is a first player win for $n \geq 4$. The question here is about the misere form, so this is a combination of Proposition 7, …

Suppose that we have two probability distributions, $f$ and $g$ on the subsets of a finite set $X$, i.e. $f, g: P(X) \to [0,1]$, with $$ \sum_{A \subseteq X} f(A) = \sum_{A \subseteq X} g(A) = 1. $$ …

This is a problem about the practicalities of removing singularities in multivariable complex functions. In trying to derive the generating function (in two variables) for a certain problem in combin …

I think there's a counterexample. Consider rectangles that look something like this: 10000000 10000000 11000000 01100000 00110000 ... 00000011 00000001 00000001 where we increase the size of the "m …

Square free words exist over all alphabet sizes greater than 2, and cube free words exist over all alphabet sizes greater than or equal to 2 (same article).

There are indeed (relatively) efficient algorithms to do this (computing canonical form in particular, deciding equality etc.) They are implemented in the CGSuite software package written by Aaron Sie …

Randomly permute $n$ and then divide into blocks of size $n/p$.

I am going to try to answer the version of the Observation which seems to imply that the original "orthogonal" basis condition really is meant to mean "positive multiples of the standard basis vectors …

This is basically covered in section VII.8.1 of Flajolet and Sedgewick's "Analytic Combinatorics". You're looking at the generating function for bridges in their terminology and the form of the genera …

Suppose that $G$ is obtained from the complete bipartite graph with parts of size $n/2$ by adding some edges in each part. Then a maximum independent set lies in one part or the other, so determining …

This is a follow up to my recent question on the asymptotics of A003238. Lucia gave a fine answer to that question, but as I hinted the 'real' problem I have in mind is slightly different, and I've no …

I'd like to promote Lucia's comment to an answer if I could but apparently I can't. I'll just fill in a few of the details. The basic idea is to pretend that $S(x) = C x^{\alpha}$. Plug in to the recu …

There is an explicit determinental formula for these numbers due to Gessel in Symmetric functions and P-recursiveness (JCTA, 1990). Asymptotics were known much earlier and appear in a paper by Amitai …

Sequence A003238 of the OEIS counts ``rooted trees with $n$ vertices in which vertices at the same level have the same degree.'' The sequence, $a$, begins 1, 1, 2, 3, 5, 6, 10, 11, 16, ... and it is …