Bijective recurrences for Motzkin paths

Abstract

Consider the lattice paths on ℤ2 with steps (1, 1), (1, -1), and (1, 0). For n ≥ 2, let En denote the set of such paths running from (0, 0) to (n, 0) and remaining strictly above the x-axis except initially and terminally. The cardinalities, fn = En for n ≥ 2, are the Motzkin numbers, 1, 1, 2, 4, 9, 21, 51, 127, .... We define a bijection yielding the recurrence (n + 1)fn+1 = (2n - 1)fn + 3(n - 2)fn-1, for n ≥ 3. A modification of the bijection proves that the sum of the areas under the paths of En, denoted by An, satisfies An+1 = 2An + 3An-1, for n ≥ 3. A second modification yields a recurrence for a second moment for the paths of En which agrees with Euler's recurrence for the central trinomial numbers. © 2001 Academic Press.