NCTS(South)/ NCKU Math Colloquium | |
| DATE | 2009-10-01¡@15:10-16:00 |
| PLACE | R204, 2F, NCTS, NCKU |
| SPEAKER | Dr. Jun Ma °¨«T ³Õ¤h¡]Institute of Mathematics, Academia Sinica¡^ |
| TITLE | Generalizations of Chung-Feller Theorems |
| ABSTRACT |
In this talk, let A be the set of integers k with k ≤ 1, N the set of nonnegative integers and P the
set of positive integers. Let £f : P¡ÑA ¡÷ N be a mapping. We call £f a length function. We first
focus on (n,m)-lattice paths. We consider two parameters ¡§£f -nonpositive length¡¨ and ¡§£f -rightmost
minimum length¡¨ for (n,m)-lattice paths. Fix a length function £f such that £f (x, y) = x. We find
that there are incomplete Chung-Feller phenomenons for (n,m)-lattice paths. Then we turn to rooted (n,m)-lattice paths. We study two parameters ¡§£f -nonpositive root length¡¨ and ¡§£f -rightmost minimum rooted length¡¨ for rooted (n,m)-lattice paths. By considering £f -cyclic permutations of a rooted (n,m)- lattice paths, we give generalizations of Theorem 2 in the page 70 of Mohanty¡¦s book [Lattice path counting and applications, New York, Academic Press, 1979]. Using generalizations in this paper, we prove Chung-Feller theorems for rooted (n,m)-lattice paths. Let S be a finite set of vectors in P¡ÑA. As applications of generalizations in this paper, we prove Chung-Feller theorems for rooted (S,n,m)- lattice paths. Hence, Chung-Feller theorems for Dyck paths, Motzkin paths and schrAoder paths can be derived as special cases of our results. |