In this paper, we show that the results of Morales, Pak and Panova on
the *q*-Euler numbers can be derived from previously known results due
to Prodinger by manipulating continued fractions. These *q*-Euler
numbers are naturally expressed as generating functions for
alternating permutations with certain statistics involving
*maj*. It has been proved by Huber and Yee that these *q*-Euler
numbers are generating functions for alternating permutations with
certain statistics involving *inv*. By modifying Foata's
bijection we construct a bijection on alternating permutations which
sends the statistics involving *maj* to the statistic involving
*inv*. We also prove the aforementioned two conjectures of
Morales, Pak and Panova.

Received: November 14, 2017. Accepted: February 17, 2018. Final version: April 1, 2018.

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