Enumerative combinatorics
A combinatorial interpretation of the recurrence fn+1= 6fn - fn-1
Discrete Mathematics
Bijective Recurrences for Motzkin Paths
Advances in Applied Mathematics
Two bijections for the area of Dyck paths
Discrete Mathematics
Lattice path moments by cut and paste
Advances in Applied Mathematics - Special issue on: Formal power series and algebraic combinatorics in memory of Rodica Simion, 1955-2000
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For fixed positive integer k, let En 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 En 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 En 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.