Lattice paths not touching a given boundary

Tamm, Ulrich

doi:10.1016/s0378-3758(01)00273-7

Cited by 15 publications

(11 citation statements)

References 7 publications

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“…This result is also implicit in Tamm [9,Propositions 2,3], where it appears in generating series form. The proof given there follows a probabilistic argument (originally due to Gessel) reliant on Lagrange inversion, whereas our derivation is purely combinatorial.…”

Section: Counting Paths Dominated By Periodic Boundariesmentioning

confidence: 69%

The number of lattice paths below a cyclically shifting boundary

Irving

Rattan

2009

Journal of Combinatorial Theory, Series A

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We count the number of lattice paths lying under a cyclically shifting piecewise linear boundary of varying slope. Our main result can be viewed as an extension of well-known enumerative formulae concerning lattice paths dominated by lines of integer slope (e.g. the generalized ballot theorem). Its proof is bijective, involving a classical "reflection" argument. Moreover, a straightforward refinement of our bijection allows for the counting of paths with a specified number of corners. We also show how the result can be applied to give elegant derivations for the number of lattice walks under certain periodic boundaries. In particular, we recover known expressions concerning paths dominated by a line of half-integer slope, and some new and old formulae for paths lying under special "staircases."

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Section: Counting Paths Dominated By Periodic Boundariesmentioning

confidence: 69%

The number of lattice paths below a cyclically shifting boundary

Irving

Rattan

2009

Journal of Combinatorial Theory, Series A

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“…The method of obtaining generating functions from an Appell relation or functional equation is an old one and dates back, at least, to Pólya. Special cases of results in this paper can be find in [1,2,10,17] and other papers.…”

Section: Paths In the Planementioning

confidence: 77%

Lattice and Schröder paths with periodic boundaries

Kung

Mier

Sun

et al. 2009

Journal of Statistical Planning and Inference

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“…Why does the formula apply to an individual entry, then to sums of pairs of entries from different rows, and then to the negative of an entry?". This question had been answered in our papers [11] (without being aware of the reference [3] at that time) and [13].…”

Section: Berlekamp's Problemmentioning

confidence: 97%