eng Science and Education Publishing Turkish Journal of Analysis and Number Theory 2014-07-31 2 4 125 129 10.12691/tjant-2-4-4 TJANT2014244 article Sabuj Das 1 Haradhan Kumar Mohajan 2 Senior Lecturer, Department of Mathematics, Raozan University College, Bangladesh Premier University, Chittagong, Bangladesh This paper shows that the coefficient of x in the right hand side of the equation is an algebraic relation in terms of z. The exponent of z represents the crank of partitions of a positive integral value of n and also shows that the sum of weights of corresponding partitions of n is the sum of ordinary partitions of n and it is equal to the number of partitions of n with crank m. This paper shows how to prove the Theorem "The number of partitions of n with crank C()=m is M(m,n) for all n>1." http://pubs.sciepub.com/tjant/2/4/4/tjant-2-4-4.pdf crank j-times vector partitions weight exponent