Enumerations of Rational Non-decreasing Dyck Paths with Integer Slope

Area of Brownian Motion with Generatingfunctionology

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3321 ◽

2003 ◽

Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽

Author(s):

Michel Nguyên Thê

Keyword(s):

Brownian Motion ◽

Random Walks ◽

Generating Functions ◽

Limit Distributions ◽

Dyck Paths ◽

Different Types ◽

International Audience

International audience This paper gives a survey of the limit distributions of the areas of different types of random walks, namely Dyck paths, bilateral Dyck paths, meanders, and Bernoulli random walks, using the technology of generating functions only.

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A General Theory of Wilf-Equivalence for Catalan Structures

The Electronic Journal of Combinatorics ◽

10.37236/5629 ◽

2015 ◽

Vol 22 (4) ◽

Cited By ~ 3

Author(s):

Michael Albert ◽

Mathilde Bouvel

Keyword(s):

General Theory ◽

Equivalence Relation ◽

Generating Functions ◽

Asymptotic Estimate ◽

Equivalence Classes ◽

Catalan Number ◽

Combinatorial Structures ◽

Dyck Paths ◽

Significant Interest ◽

Wilf Equivalence

The existence of apparently coincidental equalities (also called Wilf-equivalences) between the enumeration sequences or generating functions of various hereditary classes of combinatorial structures has attracted significant interest. We investigate such coincidences among non-crossing matchings and a variety of other Catalan structures including Dyck paths, 231-avoiding permutations and plane forests. In particular we consider principal subclasses defined by not containing an occurrence of a single given structure. An easily computed equivalence relation among structures is described such that if two structures are equivalent then the associated principal subclasses have the same enumeration sequence. We give an asymptotic estimate of the number of equivalence classes of this relation among structures of size $n$ and show that it is exponentially smaller than the $n^{th}$ Catalan number. In other words these "coincidental" equalities are in fact very common among principal subclasses. Our results also allow us to prove in a unified and bijective manner several known Wilf-equivalences from the literature.

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A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path

Canadian Journal of Mathematics ◽

10.4153/cjm-2011-078-4 ◽

2012 ◽

Vol 64 (4) ◽

pp. 822-844 ◽

Cited By ~ 20

Author(s):

J. Haglund ◽

J. Morse ◽

M. Zabrocki

Keyword(s):

Building Blocks ◽

Main Diagonal ◽

Combinatorial Theory ◽

Dyck Path ◽

Dyck Paths ◽

Littlewood Polynomials ◽

Diagonal Harmonics ◽

Littlewood Polynomial ◽

Theory Operator ◽

Shuffle Conjecture

Abstract We introduce a q, t-enumeration of Dyck paths that are forced to touch the main diagonal at specific points and forbidden to touch elsewhere and conjecture that it describes the action of the Macdonald theory ∇ operator applied to a Hall–Littlewood polynomial. Our conjecture refines several earlier conjectures concerning the space of diagonal harmonics including the “shuffle conjecture” (Duke J. Math. 126 (2005), pp. 195 − 232) for ∇ en[X]. We bring to light that certain generalized Hall–Littlewood polynomials indexed by compositions are the building blocks for the algebraic combinatorial theory of q, t-Catalan sequences, and we prove a number of identities involving these functions.

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