I mentioned the discovery of a hypergeometric combinatorial enumeration formula with a useful decomposition that could be used to find a limit probability distribution for the objects I was enumerating. I edited out the following: Richard Stanley had previously derived a rational generating function for enumerating the almost-injective functions, however, I could find no obvious way to manipulate this to obtain the limit probability distribution, nor did it suggest the decomposition of a non-machine summable formula into machine summable, asymptotically significant and non-machine-summable, asymptotically insignificant parts. The two-term formula was obtained by classifying the almost-injective functions using a regular grammar; the closed term arose from a sub-language closed under concatenation. Computer experiments suggest that a class of enumeration problems arising from regular monoids have closed form hypergeometric enumeration formulas.
Finding an algorithm for computing closed forms of indefinite binomial coefficient sums was an open problem in Knuth's Art of Programming; it was solved by Wilf and Zeilberger. In the example I considered, there is no closed form solution; however there is a way of finding a decomposition of the sum with good properties. Finding the asymptotically significant term required an idea: the point is that the algorithms I used don't help you find this. One of those algorithms I used was due to R. W. Gosper, a legendary hacker.
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I mentioned the discovery of a hypergeometric combinatorial enumeration formula with a useful decomposition that could be used to find a limit probability distribution for the objects I was enumerating. I edited out the following: Richard Stanley had previously derived a rational generating function for enumerating the almost-injective functions, however, I could find no obvious way to manipulate this to obtain the limit probability distribution, nor did it suggest the decomposition of a non-machine summable formula into machine summable, asymptotically significant and non-machine-summable, asymptotically insignificant parts. The two-term formula was obtained by classifying the almost-injective functions using a regular grammar; the closed term arose from a sub-language closed under concatenation. Computer experiments suggest that a class of enumeration problems arising from regular monoids have closed form hypergeometric enumeration formulas.
Finding an algorithm for computing closed forms of indefinite binomial coefficient sums was an open problem in Knuth's Art of Programming; it was solved by Wilf and Zeilberger. In the example I considered, there is no closed form solution; however there is a way of finding a decomposition of the sum with good properties. Finding the asymptotically significant term required an idea: the point is that the algorithms I used don't help you find this. One of those algorithms I used was due to R. W. Gosper, a legendary hacker.