Crossings and Nestings Over Some Motzkin Objects and $q$-Motzkin Numbers

We examine the enumeration of certain Motzkin objects according to numbers crossings and nestings. With respect continued fractions, we compute express distributions statistics nestings over three sets, namely set $4321$-avoiding involutions, $3412$-avoiding $(321,3\bar{1}42)$-avoiding permutations. To get our results, exploit bijection Biane restricted sets $4321$- involutions which was characterized by Barnabei et al. between permutations paths, presented Chen al.. Furthermore, manipulate obtained fractions recursion formulas for polynomial nestings, it follows that results involve two new $q$-Motzkin numbers.