Cyril Banderier and Sylviane Schwer have written a paper addressing the question, `Why Delannoy's numbers?''. The paper is available at the following:
Cyril Banderier's web page, See under "Articles & Preprints": Why Delannoy's numbers?, C. Banderier and S. Schwer (May 2002), for the ps file. (See the last sentence of page 9.)
In June of 2002, after reading ``Why Delannoy's numbers?'', I became interested in collecting objects counted by the Delannoy numbers.
Presently, there are 29 different configurations in the collection. As an early reader of this collection, I invite you to contribute examples to the list or to lead me to other sources that might contain examples for the list. Your comments are also welcome. This collection is to share some of the spirit of Stanley's Catalan list.( Enumerative Combinatorics II, See the ``Catalan addendum''.)