MPRA Munich Personal RePEc Archive Generating Function for M(m, n) Haradhan Mohajan Assistant Professor, Premer University, Chittagong, Bangladesh. 4 April 4 Online at https://mpra.ub.uni-muenchen.de/8594/ MPRA Paper No. 8594, posted January 8 6: UTC
Generating Function for M(m, n) Sabuj Das Senior Lecturer, Department of Mathematics Raoan University College, Bangladesh Email: sabujdas.ctg@gmail.com Haradhan Kumar Mohajan Premier University, Chittagong, Bangladesh Email: haradhan_km@yahoo.com Abstract This paper shows that the coefficient of in the right hand side of the equation for M(m, n) for all n > is an algebraic relation in terms of. The eponent of 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 >. Keywords: Crank, j-times, vector partitions, weight, eponent.. Introduction, the crank of partitions,,, First we give definitions of P n ; M m, n. We generate some generating functions related to the crank and show the coefficient of is the algebraic relations in terms of various powers of, the eponent of represent the crank of partitions of n (for all n ). We show the results with the help of eamples when n = 5 and 6 respectively. We introduce the special term weight related to the vector partitions V and show the relations in terms of M m, n, weight and crank. We prove the Theorem The number of partitions of n with crank C m M m, n for all n.. Definitions is Now we give some definitions following ([], [4] and [5]). and
P n: Number of partitions of n, like 4, +, +, ++, +++. Therefore, 4 5 similarly P5 7 etc. Crank of partitions []: For a partition, let l denotes the largest part of, the number of s in, and denote the number of parts of larger than c is given by; c l ; if ; if...... ; 4.... M m, n : The number of partitions of n with crank m.. Notations P and denote, the crank For all integers n and all integers m, the number of n with crank equal to m is M,, like; Partitions Largest Number of s Number of parts larger than Crank of part c l, M. But we see that;, M, M, M. Since, the coefficient of in the right hand side of the equation; m n M m, n m n is i.e., the eponent of being the crank of partition. Therefore,, M, M, M.
. The Generating Function for M m, n The generating function for M m, n is given by []; n m n m, n n n n M m n........................ j j j j, by Andrews [],......... 4............ j... j ; j j- j () 4 4 4 5 5 5
4 4 6 6 6... We see that the eponent of represents the crank of partitions of n (for n ). As for eamples when n = 5 and 6, For n = 5, Partitions of 5 5 4+ + ++ ++ +++ ++++ Largest part l 5 4 Number of s 5 Number of parts larger than Crank c 5 5 For n = 6, Partitions Largest part Numbers of ones Number of parts Crank l larger than c 6 6 6 5+ 5 4+ 4 4 4++ 4 + ++ +++ ++ +++ ++++ 4 4 +++++ 6 6 4. Vector Partitions of n Let, V DPP, where D denotes the set of partitions into distinct parts and P denotes the set of partitions. The set of vector partitions V is defined by the Cartesian product, V DPP.,, where #. For, V # # weight =, the crank 4
We have 4 vector partitions of 4 are given in the following table: Vector partitions of 4 Weight Crank,,4 +,, +,, + 4,, + 5,, + 4 6,, +,, + 7, + 8, + 9,, +,, +,, +,, + 4,, + 5,, + 6,4, + 7,, + 8,, + 9,, +,, + 4,,,,,, 4,, 5,, 6,, 7,, 8,, 9,,,,,, 5
From the above table we have,,,,, 4,, 5,, 6,, 7,, + 8,, 9,, + 4 4,,,, + 4 M,4 6 9 4 + 6 4 4 = +++ + = M,4 4... 9 = + + + =. and M, 4 7... 7 = + + + = M,4 + + = M, 4 + + = M,4 = M, 4 = M 4,4 M 4, 4 6
m,4 M ; i.e., M m 4 m, = 5 V 4 crank m i.e., M m, 4 = 4 m V 4 crank m P. Again we have 8 vector partitions of 5 are given in the following table: Vector partitions of 5 Weight Crank,,5 +,,4 +,, + 4,, +,, + 5,, + 4 6 7,, + 5 8 5,, 9,5, +,4, + 4,, + 4,,,4, 4,4, + 5,,4 + 6,,4 7 4,, 8,, + 9,, +,,,,,, +,, + 7
4,, 5,, 6,, 7,,,, 8 + +,, + 9,, +,, +,,,,,, + 4,, + 5 6,, 7,, +,, 8,, + 9,, + 4 4,, 4,, +,, 4 44,, + 45,, +,, + 46 47,,,, + 4 48,, + 49 5,, +,, 5 5,,,, + 5,, + 54,, + 55,, + 56,, + 57 8
From this table we have; 58,, +,, 59 6,,,, + 5 6,, + 6,, + 6 64,, 4,, 4 65,, + 66,, + 67,, 68,, 69,, 7,, 7,, 7,, 7,, 74 75,, 76,, 77,, 78,, 79,, 8,,,, + 8,, 8,, 8 + M,5 8 4 5 + 8 54 55 + 7 75 76 78 79 79 8 8 9
= ++++++++ + =. M,5 +++++++ = M, 5 +++++++ = M,5 + +++ = M, 5 + +++ = M,5 + += M, 5 + += M4,5 = M 4, 5 = M5,5 M 5, 5 m,5 M ; i.e., M m 5 m, = 7 V 5 crank m i.e., M m, 5 = 5 m V 5 crank m From above discussion we get; m M m, n = n V n crank m P. P.
Theorem: The number of partitions of n with crank c m is m n Proof: The generating function for M m, n is given by; n m n M m, n () n n n m n M, for all n. j... j ; j j- j. Now we distribute the function into two parts where first one represents the crank with l c. c and second one represents the crank with The first function is;... 4 4 5 5... Counts (for n ) the number of partitions with no s and the eponent on being the largest c l, like; part of the partition where Partitions of 4 4 Largest part l 4 Number of s Number of parts larger than Crank c 4 +. Here n = 4, the 5 th term is 4 4 Again second partition is, j j j j, j- 4...... 4 5...... 4 4...
which counts the number of partitions with j c, since i, like; and the eponent on is clearly Partitions + of 4 Largest part l Number of s Number of parts larger than Crank c ++ +++ 4 4 Here n = 4, the 5 th term is 4 4 i.e., 4 4 Thus in the double series epansion of j... j, j j- j., we see that the coefficient of number of partitions of n in which c m () we get the number of partitions of n with c m is m n Theorem. 5. Conclusion m n n is the. Equating the coefficient of m n from both sides in M, for all n. Hence the We have verified that the coefficient of in the right hand side of the generating function for M m, n is an eplanation of, the eponents of represent the crank of partitions, it is already M m, n = shown with eamples for n = 5 and 6. We have satisfied the result n m V n crank m P, it is already shown when n = 4 and 5 respectively. For any positive integer of n we can verify the corresponding Theorem. We have already satisfied the Theorem for n = 4 and 5. Acknowledgment It is a great pleasure to epress our sincerest gratitude to our respected Professor Md. Falee Hossain, Department of Mathematics, University of Chittagong, Bangladesh. We will remain ever grateful to our respected Late Professor Dr. Jamal Narul Islam, RCMPS, University of Chittagong, Bangladesh.
References [] Andrews, G.E., The Theory of Partitions, Encyclopedia of Mathematics and its Application, vol. (G-c, Rotaed) Addison-Wesley, Reading, mass, 976 (Reissued, Cambridge University, Press, London and New York 985). 985. [] Andrews, G.E. and Garvan, F.G., Dyson s Crank of a Partition, Bulletin (New series) of the American Mathematical Society, 8(): 67 7. 988. [] Atkin, A.O.L. and Swinnerton-Dyer, P., Some Properties of Partitions, Proc. London Math. Soc. (4): 84 6. 954. [4] Garvan, F.G., Ramanujan Revisited, Proceeding of the Centenary Conference, University of Illinois, Urban-Champion. 988. [5] Garvan, F.G.. Dyson s Rank Function and Andrews spt-function, University of Florida, Seminar Paper Presented in the University of Newcastle on August..