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as doubtless many have done before me, i recently fell into wondering how many of the binary operations on a finite set are associative. the stackexchange software fortunately pointed me to this discussion Is the Ratio of Associative Binary Operations to All Binary Operations on a Set of $n$ Elements Generally Small? before i could irritate people by submitting a duplicate question.

on that location André Nicolas provided a neat answer to Doug Spoonwood's enquiry concerning the limit as $n \to \infty$ of the proportion ($r_n$) of operations on a set of $n$ elements which are associative.

as André points out his estimate is sufficient to show $r_n \to 0$ but there is a lot of scope to use similar arguments to tighten it. in fact $r_n$ appears to drop very rapidly with $n$. suppose we define: $$ \alpha_n = \sum_{j=1}^n r_j $$ Q1 is $\alpha_n$ a convergent sequence? (easy?)

Q2 (harder?) if the answer to Q1 is positive, and $\alpha = \lim_{n \to \infty} \alpha_n$, is $\alpha$ a Liouville number, hence transcendental?

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David Holden
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André arrives at a ratio $r_n=O(\frac1n)$ merely by checking if $1^2$ commutes with $1$. For "most" choices, we should be able to find $d$ elemends $k_1,\ldots, k_d$ that in most caes differe from their squares as soon as $n\gg 2d$. If for each of these we check if $k^2$ commutes with $k$, this should give us that $r_n=O(n^{-d})$. So Q1 is alredy answered positively with $d=2$ (which may even be not too hard to handle explicitly with all special cases).

Hagen von Eitzen
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