Generating Function for M(m, n)

Munich Personal RePEc Archive

Generating Function for M(m, n)

Mohajan, Haradhan

Assistant Professor, Premer University, Chittagong, Bangladesh.

4 April 2014

Online at

https://mpra.ub.uni-muenchen.de/83594/

Generating Function for

M(m, n)

Sabuj Das

Senior Lecturer, Department of Mathematics Raozan 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 x in the right hand side of the equation for M(m, n) for all n >1is 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.”

Keywords: Crank, j-times, vector partitions, weight, exponent.

1. Introduction

First we give definitions of P

 

n , the crank of partitions,

 

x _,

 

zx _,

 

x2;x _ and M

 

m,n . We generate some generating functions related to the crank and show the coefficient of x is the algebraic relations in terms of various powers of z, the exponent of z represent the crank of partitions of n (for all n1). We show the results with the help of examples 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 is M

 

m,n for all n1.”

2. Definitions

 

n

P : Number of partitions of n, like 4, 3+1, 2+2, 2+1+1, 1+1+1+1. Therefore, P

 

4 5 and similarlyP

 

5 7etc.

Crank of partitions [2]: For a partition



, let l

 

 denotes the largest part of



, 

 

 denote

the number of 1’s in



, and 

 

 denote the number of parts of



larger than 

 

 , the crank

 



c is given by;

 

 

_{   }

_{ }

 

  

 

 

. 0 if

;

0 if

;

  

  

  

 l

c

 

x  



1x





1x2



1x3



...

 

zx _



1zx





1zx2



1zx3



...

 

x2;x  



1x2



1x3



1x4



...

 

mn

M , : The number of partitions of n with crank m.

2.1 Notations

For all integers n0 and all integers m, the number of n with crank equal to m is

 

1,1 1

M , like;

Partitions of 1

 



Largest part

 



l

Number of 1’s

 



 Number of parts larger than _

_{ }

_ 

 

 Crank c

 



1 1 1 0 –1

 

1,1 1

M .

But we see that;

 

1,1 M

 

1,1 M

 

0,1 1

M .

Since, the coefficient of x in the right hand side of the equation;

 

_{ }

 

_{ }

  

 

 









x z zx

x x

z n m

M m n

m n

1 0

,

is z1z1 i.e., 1 1 0

z z

z   the exponent of z being the crank of partition.

3. The Generating Function for M

 

m,n

The generating function for M

 

m,n is given by [2];

 

_



n

_



n

_

n

n n m

m n zx z x

x x z n m M ₁ 1

0 1 1

1 , _            

 













1





1

 

...1



1



... ... 1 1 1 2 1 1 2 3 2 x z x z zx zx x x x   _       













1





1

 

...1



1



... ... 1 1 1 2 1 1 2 3 2 x z x z zx zx x x x   _       





 

_

















_     _{ }

 1 1 ...

... 1 1 1 2 1 1 3 2 x z x z x x zx x





 

 

 

 

_       



    0 1 1 j j j j x xz zx zx x

, by Andrews [1],

 

 

 

 

 

 

 

 

 

 

 

_                ... 1 1 3 3 1 3 2 2 1 2 1 1 1 1 x xz zx x xz zx x xz zx zx x

 

 

_



 





 







 





_            2 2 2 2 1 1 1 1 1 1 1 1 1 x x z x zx zx x xz zx zx x

 











 

 

  

  



     ... 1 1 ... 1 ... 1 1 1 3 2 2 2 2 1 2 zx x z x zx xz zx zx

x













1





1



1



... 1 1 1 3 2 3 3 3 2         x x x z x zx zx zx









... ... 1 1 1 4 3 2 3 3      zx x x z x

 











   

          1 1 1 -j 2 2 ; ... 1 1 1 j j j j zx x x z x zx zx x (1)



 

 

 

 

  



 

 1 2 2 2 3 3 3

1 1

1 z z x z z x z z x



1z2 z2 z4 z4



x4 



  1 3 3 5  5



5 



1zz1z2z2z3z3z4z4z6z6



x6...

We see that the exponent of z represents the crank of partitions of n (for n1). As for examples when n = 5 and 6,

For n = 5,

Partitions of 5

 

 Largest part l

 



Number of 1’s

 



 Number of parts larger than 

 



 





Crank

 



c

5 4+1 3+2 3+1+1 2+2+1 2+1+1+1 1+1+1+1+1

5 4 3 3 2 2 1

0 1 0 2 1 3 5

1 1 2 1 2 0 0

5 0 3

–1 1

–3

–5

For n = 6,

Partitions



Largest part l

 



Numbers of ones

 



 larger than Number of parts 

 



 

 

Crank

 



c

6 6 0 1 6

5+1 5 1 1 0

4+2 4 0 2 4

4+1+1 4 2 1 –1

3+3 3 0 2 3

3+2+1 3 1 2 1

3+1+1+1 3 3 0 –3

2+2+2 2 0 3 2

2+2+1+1 2 2 0 –2

2+1+1+1+1 2 4 0 –4

1+1+1+1+1+1 1 6 0 –6

4. Vector Partitions of n

Let, V DPP, 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 DPP.

For  



₁,₂,₃



V _{, where}

3 2

1  



    weight = 

   

  ₁# 1

, the crank

     

 # 2 # 3



We have 41 vector partitions of 4 are given in the following table:

Vector partitions of 4 Weight

 



  Crank

 





, ,4



1  

  +1 –1



, ,3 1



2  

 +1 –2



, ,2 2



3    

 +1 –2



, ,2 1 1



4    

 +1 –3



, ,1 1 1 1



5      

 +1 –4



,1,3



6 

  +1 0



,1,2 1



7   

 +1 –1



,1 1 1 1



8     

 +1 –2



,2 2



9   

 +1 0



,2,1 1



10   

 +1 –1



,1 1,2



11  

 +1 1



,1 1,1 1



12   

 +1 0



,3,1



13 

  +1 0



,2 1,1



14  

 +1 1



,1 1 1,1



15   

 +1 2



 



₁₆  ,4, +1 1



 



₁₇  ,31, +1 2



 



₁₈ ,22, +1 2



 



₁₉  ,211, +1 3



 



₂₀  ,1111, +1 4



1, ,3



21 

  –1 –1



1, ,2 1



22   

 –1 –2



1, ,1 1 1



23   

 –1 –3

 

1,1,2

24 

 –1 0



1,1,1 1



25  

 –1 –1

 

1,2,1

26 

 –1 0



1 1,1,1



27  

 –1 1







₂₈ 1,3, –1 1







₂₉  1,21, –1 2







₃₀ 1,111, –1 3



2, ,2



31 



2, ,1 1



32   

 –1 –2



2,1,1



33

 –1 0







₃₄ 2,2, –1 1







₃₅ 2,11, –1 2



3, ,1



36 

  –1 –1



2 1, ,1



37 

   +1 –1







₃₈ 3,1, –1 1







₃₉  21,1, +1 1



 



₄₀  4, , –1 0



 



₄₁ 31, , +1 0

From the above table we have,

           

0,4 6 9 12 13 24

 

 



 

 



M

+ 

       

₂₆ ₃₃ ₄₀ ₄₁

= 1+1+1+1–1–1–1–1+1 = 1

 

1,4 

M 

   

₁₁ ₁₄ ...

 

₃₉

= 1 + 1 + 1–1–1–1–1+1 = 0.

and



1,4





M 

   

₁ ₇ ...

 

₃₇

= 1 + 1 + 1–1–1–1–1+1 = 0

 

2,4 

M 1 + 1 + 1–1–1= 1



2,4





M 1 + 1 + 1–1–1= 1

 

3,4 

M 1–1= 0



3,4





M 1–1= 0

 

4,4 

M 1



4,4





 

_

 



M m,4   ;

i.e.,

 

 

 





   

 



m V m

m M

  

 

 



crank 4

4

, = 5

i.e.,

 

 

 





   

 



m V m

m M

  

 

 



crank 4

4

, = P

 

4 .

Again we have 83 vector partitions of 5 are given in the following table:

Vector partitions of 5 Weight

 



  Crank

 





, ,5



1  

  +1 –1



, ,4 1



2   

 +1 –2



, ,3 2



3    

 +1 –2



, ,3 1 1



4    

 +1 –3



, ,2 2 1



5    

 +1 –3



, ,2 1 1 1



6      

 +1 –4



, ,1 1 1 1 1



7       

 +1 –5



 



₈  5, , –1 0



 



₉  ,5, +1 1



 



₁₀ ,41, +1 2



 



₁₁ 41, , +1 0







₁₂ 4,1, –1 1







₁₃ 1,4, –1 1



,4,1



14 

  +1 0



,1,4



15 

  +1 0



1, ,4



16 

  –1 –1



4, ,1



17 

  –1 –1







₁₈ 32, , +1 0



 



₁₉  ,32, +1 2







₂₀ 3,2, –1 1







₂₁ 2,3, –1 1



,3,2



22 

  +1 0



,2,3



23 



3, ,2



24 

  –1 –1



2, ,3



25 

  –1 –1



 



₂₆ ,311, +1 3







₂₇ 31,1, +1 1







₂₈ 1,31, –1 2



,3 1,1



29  

 +1 1



,1,3 1



30  

 +1 –1



3 1, ,1



31 

   +1 –1



1, ,3 1



32  

 –1 –2







₃₃ 3,11, –1 2



,1 1,3



34  

 +1 1



,3,1 1



35  

 +1 –1



3, ,1 1



36  

 –1 –2



 



₃₇ ,221, +1 3







₃₈ 1,22, –1 2



,2 2,1



39  

 +1 1



,1,2 2



40  

 +1 –1



1, ,2 2



41  

 –1 –2







₄₂ 21,2, +1 1







₄₃ 2,21, –1 2



,2,2 1



44   

 +1 1



,2 1,2



45  

 +1 1



2 1, ,2



46 

   +1 –1



2, ,2 1



47   

 –1 –2



 



₄₈ ,221, +1 4



,2 1 1,1



49   

 +1 2



,1,2 1 1



50   

 +1 –2







₅₁ 1,211, –1 3



1, ,2 1 1



52   

 –1 –3







₅₃ 21,11, +1 2



,2 1,1 1



54   

 +1 0



,1 1,2 1



55   

 +1 0



2 1, ,1 1



56   

 +1 –2



,1 1 1,2



57   



,2,1 1 1



58   

 +1 –2







₅₉ 2,111, –1 3



2, ,1 1 1



60   

 –1 –3



 



₆₁ ,11111, +1 5



,1 1 1 1,1



62    

 +1 3



,1,1 1 1 1



63    

 +1 –3



1, ,1 1 1 1



64    

 –1 –4







₆₅ 1,1111, –1 4



,1 1,1 1 1



66    

 +1 –1



,1 1 1,1 1



67    

 +1 1



1,1,1 1 1



68  

 –1 –2



1,1 1 1,1



69  

 –1 2



1,1 1,1 1



70  

 –1 0



1,1 1,2



71 

 –1 1



1,2,1 1



72 

 –1 –1



2,1 1,1



73 

 –1 1



2,1,1 1



74 

 –1 –1



2,2,1



75

 –1 0



2,1,2



76

 –1 0



1,2,2



77

 –1 0

 

3,1,1

78

 –1 0

 

1,3,1

79

 –1 0

 

1,1,3

80 

 –1 0



1 2,1,1



81 

 +1 0



1,1 2,1



82  

 –1 1



1,1,1 2



83 

 –1 –1

From this table we have;

 

0,5 

M 

       

₈ ₁₁ ₁₄ ₁₅ +

         

18 22 23 54 55

          +

         

70 75 76 78 79

          +

     

79 80 81

= –1+1+1+1+1+1+1+1+1–1–1–1–1–1–1–1–1+1 = 1.

 

1,5 

M 1–1–1–1–1+1+1+1+1+1+1+1–1–1–1 =1



1,5





M 1–1–1–1–1+1+1+1+1+1+1+1–1–1–1 =1

 

2,5 

M 1+1–1–1–1–1+1+1+1–1= 0



2,5





M 1+1–1–1–1–1+1+1+1–1= 0

 

3,5 

M 1+1–1–1+1= 1



3,5





M 1+1–1–1+1= 1

 

4,5 

M 1–1= 0



4,5





M 1–1= 0

 

5,5 

M 1



5,5





M 1

 

_

 



M m,5   ;

i.e.,

 

 

 





   

 



m V m

m M

  

 

 



crank 5

5

, = 7

i.e.,

 

 

 





   

 



m V m

m M

  

 

 



crank 5

5

, = P

 

5 .

From above discussion we get;

 

 

 





   

 



m n V m

n m M

  

 

 



crank

Theorem: The number of partitions



of n with crank c

 

 m is M

 

m,n for all n1.

Proof: The generating function for M

 

m,n is given by;

 

_



_n

_

n



_n

_

n n m

m n zx z x

x x z n m M ₁ 1

0 1 1

1 , _            

 

(2)















   

          1 1 1 -j 2 2 ; ... 1 1 1 j j j j zx x x z x zx zx x .

Now we distribute the function into two parts where first one represents the crank with

   

 l

c  and second one represents the crank with c

     

   .

The first function is;







1





1



1



...

1 3 2 zx zx zx x    







  







 

 2 2 3 3 2 4 4

1

1 z x z x z x z z x



z3z5



x5 ...

Counts (for n1) the number of partitions with no 1’s and the exponent on z being the largest part of the partition where c

   

 l  , like;

Partitions of 4

 

 Largest part l

 



Number of 1’s

 



 Number of parts larger than 

 



 

  Crank

 

 c 4 2+2 4 2 0 0 1 2 4 2

Here n = 4, the 5th term is



z2z4



x4.

Again second partition is,

   



     1 1 1 -j 2 , j j j j zx x x z x









  

  







    ... 1 1 1 ... 1

1 2 3 4

2 2 3 2 1 zx zx x z x zx zx xz











... ... 1 1 1 1 5 4 3 2 3 3       zx zx x x z x



1 3

 

3 1 2 4



4 ...

2 2

1       

which counts the number of partitions with 

 

  j _{and the exponent on} z is clearly

     

  

c , since i0, like;

Partitions of 4

 



Largest part

 



l

Number

of 1’s

 

 

Number of parts larger than 

 



 

 

Crank

 



c

3+1

2+1+1

1+1+1+1

3

2

1

1

2

4

1

0

0

0

–2

–4

Here n = 4, the 5th term is



1z2 z4



x4 _i.e.,



z0z2z4



x4.

Thus in the double series expansion of















   



  



 

  

1

1 1 -j 2 2

, ...

1 1

1

j

j j j

zx x x

z x zx

zx x

, we see that the coefficient of m n

x

z



n1



is the

number of partitions of n in which c

 

 m. Equating the coefficient of zmxn from both sides in (2) we get the number of partitions of n with c

 

 m _is M

 

m,n for all n1. Hence the Theorem.

5. Conclusion

We have verified that the coefficient of x in the right hand side of the generating function for

 

mn

M , is an explanation of z, the exponents of z represent the crank of partitions, it is already

shown with examples for n = 5 and 6. We have satisfied the result

 

 

 





   

 



m n V m

n m M

  

 

 



crank

, =

 

n

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

References

[1] Andrews, G.E., The Theory of Partitions, Encyclopedia of Mathematics and its Application, vol. 2 (G-c, Rotaed) Addison-Wesley, 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 Swinnerton-Dyer, 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, Urban-Champion. 1988.