Ω is weak-geodetically closed whenever Ω is weak-geodetically closed with respect to w for any w ∈ X. ∗A manuscript for algebraic combinatorics workshop 7/2-4 in National Hsinchu University of Education, an extension of 2007 International Conference on Graph Theory and Combinatorics & Fourth Cross-strait Conference on Graph Theory and Combinatorics. †Department of Applied Mathematics, National ...

A possible upgrade of a curriculum in undergraduate course in combinatorics is presented by giving more bijective proofs in the standard (or primary) combinatorics and by adding some topics on more refined (or secondary) combinatorics, including Dyck and Motzkin paths, Catalan, Narayana and Motzkin numbers and secondary structures coming from biology. Some log–convexity properties and asymptoti...