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Recently, Naruse discovered a hook length formula for the number of standardYoung tableaux of a skew shape. Morales, Pak and Panova found two $q$-analogsof Naruse's hook length formula over semistandard Young tableaux (SSYTs) andreverse plane partitions (RPPs). As an application of their formula, theyexpressed certain $q$-Euler numbers, which are generating functions for SSYTsand RPPs of a zigzag border strip, in terms of weighted Dyck paths. They founda determinantal formula for the generating function for SSYTs of a skewstaircase shape and proposed two conjectures related to RPPs of the same shape.One conjecture is a determinantal formula for the number of \emph{pleasantdiagrams} in terms of Schr\"oder paths and the other conjecture is adeterminantal formula for the generating function for RPPs of a skew staircaseshape in terms of $q$-Euler numbers.  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 Prodingerby manipulating continued fractions. These $q$-Euler numbers are naturallyexpressed as generating functions for alternating permutations with certainstatistics involving \emph{maj}. It has been proved by Huber and Yee that these$q$-Euler numbers are generating functions for alternating permutations withcertain statistics involving \emph{inv}. By modifying Foata's bijection weconstruct a bijection on alternating permutations which sends the statisticsinvolving \emph{maj} to the statistic involving \emph{inv}. We also prove theaforementioned two conjectures of Morales, Pak and Panova.

Reverse plane partitions of skew staircase shapes and $q$-Euler numbers