Mohajan, Haradhan (2014): Generating Function for M(m, n). Published in: Turkish Journal of Analysis and Number Theory , Vol. 2, No. 4 (31 July 2014): pp. 125129.

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Abstract
This paper shows that the coefficient of x in the right hand side of the equation for M(m, n) for all n >1 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 π of n with crank C(π) = m is M(m, n) for all n >1.”
Item Type:  MPRA Paper 

Original Title:  Generating Function for M(m, n) 
English Title:  Generating Function for M(m, n) 
Language:  English 
Keywords:  Crank, jtimes, vector partitions, weight, exponent 
Subjects:  C  Mathematical and Quantitative Methods > C3  Multiple or Simultaneous Equation Models ; Multiple Variables C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C65  Miscellaneous Mathematical Tools 
Item ID:  83594 
Depositing User:  Haradhan Kumar Mohajan 
Date Deposited:  03 Jan 2018 16:31 
Last Modified:  01 Oct 2019 18:18 
References:  [1] Andrews, G.E., The Theory of Partitions, Encyclopedia of Mathematics and its Application, vol. 2 (Gc, Rotaed) AddisonWesley, Reading, mass, 1976 (Reissued, Cambridge University, Press, London and New York 1985). 1985. [2] Andrews, G.E. and Garvan, F.G., Dyson’s Crank of a Partition, Bulletin (New series) of the American Mathematical Society, 18(2): 167–171. 1988. [3] Atkin, A.O.L. and SwinnertonDyer, P., Some Properties of Partitions, Proc. London Math. Soc. 3(4): 84–106. 1954. [4] Garvan, F.G., Ramanujan Revisited, Proceeding of the Centenary Conference, University of Illinois, UrbanChampion. 1988. [5] Garvan, F.G.. Dyson’s Rank Function and Andrews’ sptfunction, University of Florida, Seminar Paper Presented in the University of Newcastle on 20 August 2013. 2013. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/83594 