Generating Functions for Certain Weighted Cranks

Recently, George Beck posed many interesting partition problems considering the number of ones in partitions. In this paper, we first consider crank generating function weighted by and obtain analytic formulas for under conditions being less than or equal to some specific integer. We connect these cumulative point functions partitions with certain sizes Durfee rectangles. then a generalization $k$-colored partitions, which was introduced Fu Tang, investigate corresponding parts subpartition partition. show that are same as unimodal sequences.