(Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and arxiv.org)

Written: May 20, 2020.

    In memory of our hero Richard Guy (1916-2020)

In his classic essay ``The Strong Law of Small Numbers", Richard Guy gave numerous cautionary tales where one can't `jump to conclusions' from the first few terms of a sequence. But if you are cautious enough you can find many families of enumeration problems where it is very safe to deduce the general pattern from the first few cases, obviating the need for either the human or the computer to think too hard, and by using the `Keep It Simple Stupid' principle ( KISS for short), one can easily derive many deep enumeration theorems by doing exactly what Richard Guy told us not to do, computing the first few terms of the sequence and deducing the formula for the general term. We admit that often "few" should be replaced by "quite a few", but it is still much less painful than trying to figure out the intricate combinatorial structure by `conceptual' means.

Maple package

• Dyck.txt

Sample Input and Output Files for Dyck.txt

• If you want to see an article with 16 theorems enumerating Dyck paths of semi-length n obeying certain restrictions about the heights of peaks and valleys and length of upward and downward runs where the sets of restictions are finite
  The input gives you the output.

• If you want to see a much longer such article with 256 theorems enumerating Dyck paths of semi-length n obeying certain restrictions about the heights of peaks and valleys and length of upward and downward runs
  The input gives you the output.

• If you want to see an article where the set of restrictions is avoiding the infinite arithmetical progression 2r+3 . Note that Theorem 9 (featuring the Motzkin numbers) answers Vladimir Retakh's question that inspired this whole project.
  The input gives you the output.

• If you want to see an article where the set of restrictions is avoiding the infinite arithmetical progression 3r+1
  The input gives you the output.

• If you want to see an article where the set of restrictions is avoiding the infinite arithmetical progression 4r+3
  The input gives you the output.

• If you want to see an article where the set of restrictions is avoiding the infinite arithmetical progression 5r+1
  The input gives you the output.

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