Counting Depth Zero Patterns in Ballot Paths
-
Heinrich Niederhausen
and
Shaun Sullivan
Abstract.
The purpose of this work is to extend
the theory of finite operator calculus to the multivariate setting, and
apply it to the enumeration of certain lattice paths. The lattice paths we
consider are ballot paths. A ballot path is a path that stays weakly above
the diagonal 
, starts at the origin, and takes steps from the set 
. Given a string 
from the set 
, we want to count the ballot paths with a given number of
occurrences of 
. In order to use finite operator calculus, we must put
some restrictions on the string 
we wish to keep track of. A ballot path
ending on the diagonal can be viewed as a Dyck path, thus all of our results
also apply to the enumeration of Dyck paths with a given number of
occurrences of 
. Finally, we give an example of counting ballot paths
with a given number of occurrences of two patterns.
© 2012 by Walter de Gruyter Berlin Boston
Articles in the same Issue
- Masthead
- Odd Catalan Numbers Modulo
- Variations of the Poincaré Map
- Diophantine Equations of Matching Games I
- Norm Euclidean Quaternionic Orders
- A New Proof of Winquist's Identity
- Counting Depth Zero Patterns in Ballot Paths
- Codes Associated with and Power Moments of Kloosterman Sums
- Subprime Factorization and the Numbers of Binomial Coefficients Exactly Divided by Powers of a Prime
- Generalized Nonaveraging Integer Sequences
- The Robin Inequality for 7-Free Integers
- On 3-adic Valuations of Generalized Harmonic Numbers
Articles in the same Issue
- Masthead
- Odd Catalan Numbers Modulo
- Variations of the Poincaré Map
- Diophantine Equations of Matching Games I
- Norm Euclidean Quaternionic Orders
- A New Proof of Winquist's Identity
- Counting Depth Zero Patterns in Ballot Paths
- Codes Associated with and Power Moments of Kloosterman Sums
- Subprime Factorization and the Numbers of Binomial Coefficients Exactly Divided by Powers of a Prime
- Generalized Nonaveraging Integer Sequences
- The Robin Inequality for 7-Free Integers
- On 3-adic Valuations of Generalized Harmonic Numbers
