Motzkin Paths with Vertical Edges
نویسندگان
چکیده
This paper considers finite lattice paths formed from the set of step vectors {→, ↗,↘, ↑, and ↓} with the restriction that vertical steps (↑, ↓) can not be consecutive. We provide a recurrence relation for enumerating paths that terminate a horizontal distance n and vertical distance m from the starting point and apply the relation to paths which are restricted to the first quadrant and paths which are not. We provide an explicit algorithmic bijection between these paths and a bisection of the Motzkin triangle paths. Finally, we extend Dyck paths in a similar manner and show that using the same mapping there is a bijection to a subset of Motzkin paths.
منابع مشابه
Enumeration of edges in some lattices of paths
We enumerate the edges in the Hasse diagram of several lattices arising in the combinatorial context of lattice paths. Specifically, we consider the case of Dyck, Grand Dyck, Motzkin, Grand Motzkin, Schröder and Grand Schröder lattices. Finally, we give a general formula for the number of edges in an arbitrary Young lattice (which can be interpreted in a natural way as a lattice of paths). 1 Pa...
متن کاملMotzkin Paths, Motzkin Polynomials and Recurrence Relations
We consider the Motzkin paths which are simple combinatorial objects appearing in many contexts. They are counted by the Motzkin numbers, related to the well known Catalan numbers. Associated with the Motzkin paths, we introduce the Motzkin polynomial, which is a multi-variable polynomial “counting” all Motzkin paths of a certain type. Motzkin polynomials (also called Jacobi-Rogers polynomials)...
متن کاملA Bijection Between 3-Motzkin Paths and Schröder Paths With No Peak at Odd Height
A new bijection between 3-Motzkin paths and Schröder paths with no peak at odd height is presented, together with numerous consequences involving related combinatorial structures such as 2-Motzkin paths, ordinary Motzkin paths and Dyck paths.
متن کاملHilly poor noncrossing partitions and (2, 3)-Motzkin paths
A hilly poor noncrossing partition is a noncrossing partition with the properties : (1) each block has at most two elements, (2) in its linear representation, any isolated vertex is covered by some arc. This paper defines basic pairs as a combinatorial object and gives the number of hilly poor noncrossing partitions with n blocks, which is closely related to Maximal Davenport-Schinzel sequences...
متن کاملNotes on Motzkin and Schr Oder Numbers
The Motzkin numberMn is the cardinality of the set of sequencesMn = f(xi)ni=1 : xi 2 f 1; 0; 1g; all partial sums P k i=1 xi are nonnegative and P n i=1 xi = 0g. These sequences have a pictorial representation as Motzkin paths: lattice paths of upsteps (corresponding to +1), downsteps ( 1) and atsteps (0) that begin and end at, but never dip below, \ground level". For example, the M3 = 4 Motzki...
متن کامل