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A Bijective Approach to the Area of Generalized Motzkin Paths

โœ Scribed by E. Pergola; R. Pinzani; S. Rinaldi; R.A. Sulanke

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
105 KB
Volume
28
Category
Article
ISSN
0196-8858

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โœฆ Synopsis

For fixed positive integer k, let E n denote the set of lattice paths using the steps 1 1 , 1 -1 , and k 0 and running from 0 0 to n 0 while remaining strictly above the x-axis elsewhere. We first prove bijectively that the total area of the regions bounded by the paths of E n and the x-axis satisfies a four-term recurrence depending only on k. We then give both a bijective and a generating function argument proving that the total area under the paths of E n equals the total number of lattice points on the x-axis hit by the unrestricted paths running from 0 0 to n -2 0 and using the same step set as above. ๏ฃฉ 2002 Elsevier Science (USA)

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