The Number of Vector Partitions of n with the Crank m

Munich Personal RePEc Archive

The Number of Vector Partitions of n with the Crank m

Das, Sabuj and Mohajan, Haradhan

Assistant Professor, Premier University, Chittagong, Bangladesh.

10 July 2014

Online at https://mpra.ub.uni-muenchen.de/83697/

MPRA Paper No. 83697, posted 09 Jan 2018 05:11 UTC

1

The Number of Vector Partitions of n with the Crank m

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 article shows how to find all vector partitions of any positive integral values of n, but only all vector partitions of 4, 5, and 6 are shown by algebraically. These must be satisfied by the definitions of crank of vector partitions.

Keywords: Vector partitions, Crank, Congruences, Modulo.

International Journal of Reciprocal Symmetry and Theoretical Physics, Vol. 1, No. 2, 2014: 91-105.

1. Introduction

Here we discuss such a crank which in terms of a weighted count of what we call vector partitions. We give the definitions of , #

 

 , 

 

 , crank of vector partitions, weight of ^, ^NV

 

^m,ⁿ , ^NV



^m,^t,ⁿ



and prove the partitions congruences moduli 5, 7 and 11 with the help of examples by finding all vector partitions of 4, 5 and 6, respectively. We analyze the generating functions for ^NV

 

^m,ⁿ and ^NV



^m,^t,ⁿ



.

2. Definitions

 : A partition.

 



# : The number of parts of .

 



 : The sum of the parts of .

2

Crank of vector partitions: The number of parts of ₂ minus the number of parts of ₃, where ₂ and ₃ are unrestricted partitions in a vector partition 



₁,₂,₃



of n, if the sum of ^ is

       

ⁿ

s^  ₁  ₂  ₃  .

Weight of ^: Weight of vector partition ^ is defined as; 

   

  ^{1# 1} .

 

^{m n}

N_V , : The number of vector partitions of n (counted according to the weight ) with the crank m.



^{m t n}



N_V , , : The number of vector partitions of n (counted according to the weight ) with the crank congruent to m modulo t.

3. The Crank for Vector Partitions

For a partition , let #

 

 be the number of parts of  and 

 

 be the sum of the parts of  with the convention ^#

   

 ^  ^0 for the empty partition  of 0 (Andrews 1985, Andrews and Garvan 1988).

Let, V

 

₁^,₂^,₃



₁

is a partition into unequal parts ₂^, ₃ are unrestricted partitions}. We shall call the elements of V

vector partitions. For 



₁,₂,₃



in V

we define the sum of parts, s, a weight, , and a crank, c, by;

       

  ₁  ₂  ₃

s .

   

 1^{# 1}

   

.

     

 # ₂ # ₃

c .

We say ^ is a vector partition of n, if ^s

 

^{ n}. For example, if ^



^1,11,1



^{, then s}

 

^{ 4,}

 

 1

 ^ , ^c

 

^ 1 and ^ is a vector partition of 4.

The number of vector partitions of n (counted according to the weight ) with the crank m is denoted by ^Nv

 

^m,ⁿ so that;

 

^{m n }



^

 

^

N_V , ; if V, ^s

 

^{ n}, and ^c

 

^{ m}.

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

3 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 

  –1 –1



2, ,1 1



32  

^ –1 –2



^2,1,1



33

^ –1 0

4







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

NV + 

       

^₂₆ ^₃₃ ^₄₀ ^₄₁

= 1+1+1+1–1–1–1–1+1 = 1 (1)

The number of vector partitions of n (counted according to the weight ) with the crank congruent to k modulo t is denoted by ^NV



^k,^t,ⁿ



, so that;

 

^



^



^



^

  







 ^

m V

V k t n N m t k n

N , , , , ; (2)

if V, ^s

 

^{ n}, and ^c

  

^ ^k mod^t



. From the table we get;



1,5,4





NV 

       

^₅ ^₁₁ ^₁₄ ^₁₆ +

         

^₂₇ ^₂₈ ^₃₄ ^₃₈ ^₃₉

= 1+1+1+1–1–1–1–1+1 = 1. (3) By considering the transformation that interchanges ₂ and ₃ we have;

 

^{m n N}



^{m n}



N_V ,  _V  , .

We illustrate with an example;

 

1,4 

NV 

   

^₁₁ ^₁₄ ^...

 

^₃₉

= 1 + 1 + 1–1–1–1–1+1 = 0.

and

5



1,4





NV 

   

^₁ ^₇ ...

 

^₃₇

= 1 + 1 + 1–1–1–1–1+1 = 0

 N_V

 

1,4  N_V



1,4



. Again,

^N_V



51,5,4



^N_V



4,5,4

  

^₂₀ 1

NV



1,5,4



NV



51,5,4



by (3).

Generally we can write,



^{m t n}



^N



^{t m t n}



N_V , ,  _V  , ,

3.1. The Generating Function for ^NV

 

^m,ⁿ The generating function for ^NV

 

^m,ⁿ is;

 



ⁿ



ⁿ



^{m n V}

^{ }

^{n n}

n

n

x z n m N x

z zx

x

 

^







 



 





 

0 1 1

1 , 1

1 (4)

which was proved by Atkin and Swinnerton-Dyer (1954). By putting z = 1 in (4), we get;

 



ⁿ



ⁿ



n

n x x

x





^ 

 1 1

1

1

 

ⁿ

m n

V m n x

 

^ N











0

,

   

ⁿ

m n

V n

n N m n x

x n

P

 



^

















0 0

,



  

^

 







m

V m n N n

P , . (5)

Now we discuss it with an example;

R. H. S. =



^

 



 m

V m n N ,

6



^

 







m

V m N ,4

=…+NV



4,4



+NV



3,4



+NV



2,4



+NV



1,4



+NV

 

0,4 +NV

 

1,4 +NV

 

2,4 +NV

 

3,4 +NV

 

4,4 +…

= 0 + 1 + 0 + 1 + 0 + 1 + 0 + 1 + 0 + 1= 5 = ^P

 

⁴ = L. H. S.

3.2. The Generating Function for ^NV

 

0,ⁿ The generating function for ^NV

 

0,ⁿ is defined as;

  ^  

^{ }





  0

2 2

1

n n

n n

x x x

           

^







 



 



 

 



 ...

1 1 1

1 1 1

1 ₂

2 3 15 2 2

2 8 2

3

x x

x x

x x x

x x

...

. 0

1  ²  ³ ⁴ ⁵  ⁶ 

 x x x x x x

=NV

 

0,0 +^NV

 

0,1^x+N_V

 

0,2 x²+N_V

 

0,3 x³+N_V

 

0,4 x⁴+N_V

 

0,5 x⁵+N_V

 

0,6 x⁶+…

=



^

 

0

, 0

n

n

V n x

N .

3.4 Result

3.4.1. The result is;

   

5 4 4 5

5 , 5

,   P n

n k

N_V ; 0k4.

Proof: We prove the result with an example.

From the table 1 we get;



0,5,4





         

^₆ ^₉ ^₁₂ ^₁₃ ^₂₄

NV + 

       

^₂₆ ^₃₃ ^₄₀ ^₄₁

= 1+1+1+1–1–1–1–1+1 = 1,



1,5,4



NV = 1+1+1+1–1–1–1–1+1 = 1,

7



2,5,4



NV = 1+1+1+1–1–1–1= 1,



^3,5,4



NV = 1+1+1–1–1+1–1= 1,



^4,5,4



NV = 1+1+1–1–1–1–1+1+1 = 1.

 NV



^0,5,4



=NV



^1,5,4



=NV



^2,5,4



=NV



^3,5,4



=NV



^4,5,4



=1=

 

5 4

P , where n = 0.

In general we can write;

   

5 4 4 5

5 , 5

,   P n

n k

N_V ; 0k4.

Hence the Theorem.

3.2.2. The result is;

   

7 4 5 7

7 , 7

,   P n

n k

N_V ; 0k6.

Proof: We prove the result with an example.

The vector partitions of 5 are given in the table below:

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

8







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 

 

+1 0



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

9



 



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    

^{ +1 2}



,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

10



^{1,1,1 2}



83 

^ –1 –1

From this table we have;



0,7,5





NV ^

       

^₈ ^₁₁ ^₁₄ ^₁₅ +^

         

^₁₈ ^₂₂ ^₂₃ ^₅₄ ^₅₅ +

         

^₇₀ ^₇₅ ^₇₆ ^₇₇ ^₇₈

 ^  ^  ^  ^  ^ +^

     

^₇₉ ^₈₀ ^₈₁

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

Similarly,



0,7,5





NV NV



^1,7,5



^…=NV



^6,7,5



^1 =

 

7 5 P .

In general we can write;

   

7 5 5 7

7 , 7

,   P n

n k

N_V ; 0k6.

Hence the result.

3.2.3. The result is;

   

11 6 6 11

11 , 11

,   P n

n k

N_V .

Proof: We prove the result with an example.

The vector partitions of 6 are given in the table below:

Vector partitions of 6 Weight

 



 ^{ Crank}

 

^



, ,6



1 

 

+1 –1



, ,5 1



2   

^{ +1} –2



^{, ,4 2}



3   

^{ +1} –2



^{, ,4 1 1}



4    

^{ +1} –3



, ,3 3



5   

^ +1 –2



, ,3 2 1



6     

^ +1 –3



, ,3 1 1 1



7      

^ +1 –4



, ,2 2 2



8     

^{ +1} –3

11



^{, ,2 2 1 1}



9      

^{ +1} –4



^{, ,2 1 1 1 1}



10      

^ +1 –5



, ,1 1 1 1 1 1



11      

^{ +1} –6



 



₁₂ ,6,

+1 1



 



₁₃ ,51,

+1 2



^{ }



₁₄ ,42,

+1 2



 



₁₅ ,411,

+1 3



 



₁₆ ,33,

+1 2



^{ }



₁₇  ,321,

+1 3



 



₁₈ ,3111,

+1 4



 



₁₉  ,222,

+1 3



 



₂₀ ,2211,

+1 4



^{ }



₂₁ ,21111,

+1 5



^{ }



₂₂  ,111111,

+1 6



 



₂₃ 6, , –1 0



^{ }



₂₄ 51, ,

+1 0



 



₂₅  42, ,

+1 0







₂₆ 321, , –1 0



^,5,1



27 

 

+1 0



^,1,5



28 

 

+1 0



,4,2



29 

 

+1 0



,2,4



30 

 

+1 0



^,4,1



31 

 

+1 1



^{,4,1 1}



32  

^ +1 –1



,1,4 1



33  

^ +1 –1



,1 1,4



34  

^{ +1 1}



^,3,3



35 

 

+1 0



,3 2,1



36  

^ +1 1



,1,3 2



37   

^ +1 –1



,3,2 1



38  

^ +1 –1



^{,2 1,3}



39   

^{ +1 1}



^{,1 3,2}



40   

^ +1 1



,2,1 3



41  

^{ +1} –1



,3,1 1 1



42    

^{ +1} –2



^{,3 1,1 1}



43   

^{ +1 0}

12



5, ,1



44 

  –1 –1







₄₅ 5,1, –1 1



^{4, ,2}



46 

  –1 –1



^



₄₇  4,2, –1 1



,1 1 1,3



48    

^ +1 2



,1 1,3 1



49    

^{ +1 0}



^{,1,3 1 1}



50   

^{ +1} –2



^{,3 1 1,1}



51   

^ +1 2



,2 2,2



52  

^ +1 1



,2,2 2



53  

^{ +1} –1



,2,1 1 1 1



54    

^{ +1} –3



^{,1 1 1 1,2}



55    

^{ +1 3}



,2 1,1 1 1



56    

^ +1 –1



,1 1 1,2 1



57    

^{ +1 1}



^{,2 1 1,1 1}



58    

^{ +1 1}



^{,1 1,2 1 1}



59    

^ +1 –1



,1 1 1 1,1 1



60      

^ +1 2



,1 1,1 1 1 1



61     

^{ +1} –2



^{,1 1 1,1 1 1}



62      

^{ +1 0}



^{,1,1 1 1 1 1}



63     

^ +1 –4



,1 1 1 1 1,1



64      

^ +1 4



3,2,1



65

^ –1 0



3,1,2



66 

^ –1 0



^2,3,1



67 

^ –1 0



2,1,3



68

^ –1 0



1,2,3



69

^ –1 0



^1,3,2



70

^ –1 0



^{3,1,1 1}



71 

^ –1 –1



3,1 1,1



72  

^ –1 1



2,2 1,1



73 

^ –1 1



^{2,1,1 2}



74  

^ –1 –1



^{1,1 1 1,1 1}



75   

^ –1 1



1,1 1,1 1 1



76    

^ –1 –1



1,1,1 1 1 1



77    

^ –1 –3



1,1 1 1 1,1



78   

^ –1 3

13



^{2,1 1,1 1}



79  

^ –1 0



4,1,1



80

^ –1 0



3, ,3



81 

  –1 –1







₈₂ 3,3, –1 1







₈₃ 3,111, –1 3



3, ,1 1 1



84    

^ –1 –3







₈₅ 2,22, –1 2



^{2, ,2 2}



86  

^ –1 –2







₈₇  2,211, –1 3



^{2, ,2 1 1}



88    

^ –1 –3



2, ,2 2



89  

^ –1 –2



2, ,1 1 1 1



90    

^ –1 –4







₉₁ 1,11111, –1 –4



^{1, ,1 1 1 1 1}



92     

^ –1 –5



^



₉₃ 12,3,

+1 1



1 2, ,3



94 

  

+1 –1







₉₅ 31,2,

+1 1



3 1, ,2



96 

  

+1 –1



3 1,1,1



97  

^{ +1 0}







₉₈ 41,1,

+1 1



4 1, ,1



99 

  

+1 –1







₁₀₀ 4,11, –1 2



^{4, ,1 1}



101  

^ –1 –2



^



₁₀₂ 31,11,

+1 2



3 1, ,1 1



103   

^ +1 –2







₁₀₄ 21,111,

+1 3



^{2 1, ,1 1 1}



105    

^{ +1} –3



2 1,1,2



106 

^ +1 0



2 1,2,1



107 

^{ +1 0}



1,2 1,2



108 

^ –1 1



1,2,2 1



109 

^ –1 –1







₁₁₀ 1,23, –1 2



^{1, ,2 3}



111  

^ –1 –2



,4,2



112 

 

+1 1

14



^{2 3, ,1}



113 

  

+1 –1







₁₁₄ 2,13, –1 2



^{2, ,3 1}



115  

^ –1 –2



^



₁₁₆ 1,221, –1 3



1, ,2 2 1



117   

^ –1 –3



2 1,1 1,1



118  

^{ +1 1}



^{2 1,1,1 1}



119  

^{ +1} –1



^{1,1 1,2 1}



120  

^ –1 0



1,2 1,1 1



121  

^ –1 0

From this table we have;



0,11,6





NV 

       

^₂₃ ^₂₄ ^₂₅ ^₂₆ +

         

^₂₇ ^₂₈ ^₂₉ ^₃₀ ^₃₅ +

         

₄₃ ₄₉ ₆₂ ₆₅ ₆₆

 ^  ^  ^  ^  ^ +

     

^₆₇ ^₆₈ ^₆₉ +

     

₇₀ ₇₉ ₈₀

 ^  ^  ^ +

         

^₉₇ ^₁₀₆ ^₁₀₇ ^₁₂₀ ^₁₂₁

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



0,11,6





NV 1 =

 

11 6

P , where n = 0 and k = 0.

Hence the result.

4. Conclusions

We verified that for any positive integral value of n in the relation

  

^

 







m

V m n N n

P , and easily

can find generating function for ^NV

 

^m,ⁿ in terms of various corresponding cranks of vector partitions.

Acknowledgment

It is a great pleasure to express our sincerest gratitude to our respected teacher Professor Md. Fazlee Hossain, Department of Mathematics, University of Chittagong, Bangladesh. We will remain ever grateful to our respected teacher Late Professor Dr. Jamal Nazrul Islam, JNIRCMPS, University of Chittagong, Bangladesh.

15 References

Andrews G.E. (1985), 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).

Andrews G.E. and Garvan F.G. (1988), Dyson’s Crank of a Partition, Bulletin (New series) of the American Mathematical Society, 18(2): 167–171.

Atkin, A.O.L. and Swinnerton-Dyer, P. (1954), Some Properties of Partitions, Proc. London Math.

Soc. 3(4): 84–106.