Andrews-Garvan-Liang’sSpt-crankforMarkedOverpartitions Bhattacharjee,NilandDas,SabujandMohajan,Haradhan MunichPersonalRePEcArchive

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Andrews-Garvan-Liang’s Spt-crank for Marked Overpartitions

Bhattacharjee, Nil and Das, Sabuj and Mohajan, Haradhan

Assistant Professor, Premier University, Chittagong, Bangladesh

13 October 2014

Online at https://mpra.ub.uni-muenchen.de/83047/

MPRA Paper No. 83047, posted 29 Dec 2017 17:37 UTC

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Andrews-Garvan- Liang’s sptcrank for Marked Overpartitions

Nil Ratan Bhattacharjee

Department of Mathematics, University of Chittagong, Bangladesh Sabuj Das

Senior Lecturer, Department of Mathematics Raozan University College, Bangladesh

E-mail: sabujdas.ctg@gmail.com Haradhan Kumar Mohajan

Premier University, Chittagong, Bangladesh E-mail: haradhan1971@gmail.com

ABSTRACT

In 2009, Bingmann, Lovejoy and Osburn have shown the generating function for spt2(n). In 2012, Andrews, Garvan, and Liang have defined the sptcrank in terms of partition pairs. In this article the number of smallest parts in the overpartitions of n with smallest part not overlined and even are discussed, and the vector partitions and S-partitions with 4 components, each a partition with certain restrictions are also discussed. The generating function for spt2(n), and the generating function for MS(m, n) are shown with a result in terms of modulo 3. This paper shows how to prove the Theorem 1, in terms of MS(m, n) with a numerical example, and shows how to prove the Theorem 2, with the help of sptcrank in terms of partition pairs. In 2014, Garvan and Jennings-Shaffer are capable to define the sptcrank for marked overpartitions. This paper also shows another result with the help of 15 SP2-partition pairs of 8 and shows how to prove the Corollary with the help of 15 marked overpartitions of 8.

Key words: crank, non-negative, overpartitions, overlined, sptcrank, weight.

INTRODUCTION

In this paper we give some related definitions of spt₂

 

n , various product notations, vector partitions and S-partitions, ( , )

2 m n

M_S , ( , , )

2 m t n

M_S , S2

 

z,x , marked partition and sptcrank for marked overpartitions. We discuss the generating function for ^spt₂

 

ⁿ ^and

prove the Corollary 1 with the help of generating function to prove the Result 1 with the help of 3 vector partitions from S2 of 4. We prove the Theorem 1 with the help of various generating functions and prove the Corollary 2 with a special series S2

 

z,x , when n =1 and prove the Theorem 2 with the help of sptcrank in terms of partition pairs



₁^,₂



when

).

( ) (

0s₁ s₂ We prove the Result 2 using the crank of partition pairs ^



₁,₂



and

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prove the Corollary 3 and 4 with the help of marked overpartition of 3n and of 3n1 (when n = 2) respectively. Finally we analyze the Corollary 5 with the help of marked overpartitions of 5n3 when n =1.

Some Related Definitions

In this section we have described some definitions related to the article following (Garvan and Shaffer 2014).

 

n

spt₂ (Bringann et al. 2009): The number of smallest parts in the overpartitions of n with smallest part not overlined and even is denoted by spt₂

 

n for example,

n spt₂(n)

1 : 0

2 : 2 1

3 : 0

4 : 4,

.

2 +

.

2 3 5 : 3 2,

.

. 32 2 ... … …

From above we get;

spt2(6) = 6, spt₂(7) = 6, ...

Product Notations

)...

1 ( ) 1 ( ) 1 ( )

(x _  x x² x³ )...

1 ( ) 1 ( )

;

(x² x² _ x² x⁴

) 1 )...(

1 ( ) 1 ( ) 1 ( )

(x _k  x x² x³ x^k )...

1 ( ) 1 ( ) 1 ( )

;

(x⁵ x _ x⁵ x⁶ x⁷

Vector Partitions and S-Partitions

A vector partition can be done with 4 components each partition with certain restrictions (Bringann et al. 2013). Let, V^ DPPD, where D denote the set of all partitions into distinct parts, P denotes the set of all partitions. For a partition , we let, s() denotes the

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smallest part of  (with the convention that the empty partition has smallest part ), #() the number of parts in , and  the sum of the parts of  .

For ( ₁, ₂, ₃, ₄) ,



     V

 we define the weight ( ) ( 1)^#⁽ ¹⁾^¹,

   ^



 the crank c

) (

# ) (

# )

(^  ₂  ₃ , the norm ^  ₁ ₂ ₃ ₄. We say

 is a vector partition of n if ^n. Let S denotes the subset of V and it is given by; S(₁,₂,₃,₄)V^,1s(₁),s(₁)

s⁽2^),s⁽1⁾s⁽3^),s⁽1⁾s⁽4⁾



.

Let S₂ denotes the subset of S with s

 

₁ even.

) ,

2(m n

M_S : The number of vector partitions of n in S2 with crank m are counted according to the weight  is exactly ( , )

2 m n

M_S . )

, ,

2(m t n

M_S : The number of vector partitions of n in S2with crank congruent to m modulo t are counted according to the weight  is exactly ( , , )

2 m t n

M_S .

S2(z, x): The series S2(z, x) is defined by the generating function for ( , )

2 m n

M_S . i.e., S2(z, x)

=







 







₍⁽ 2 _; ^;₎ ⁾₍ ⁽1 2 _; ^;₎ ⁾ 1 2 1

2 2

1 zx x z x x

x x x x x

n n

n

=

 

^







 m

n m S

n

x z n m M ( , )

1 2

.

Marked Partition (Andrews et al. 2013): We define a marked partition as a pair (,k) where

 is a partition and k is an integer identifying one of its smallest parts i.e., k =1, 2, ..., (), where () is the number of smallest parts of .

sptcrank for Marked overpartitions (Chen et al. 2013): We define a marked overpartitions of n as a pair (, j) where  is an overpartition of n in which the smallest part is not overlined and even. It is clear that spt₂(n) = # of marked overpartitions (, j) of n. For example, there are 3 marked overpartitions of 4, like:

(4,1), (2+2,1), and (2+2,2).

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Then, spt₂(4)3.

The Generating Function for spt₂(N)

The generating function (Bringann et al. 2013) for spt₂(n) is given by;

  

 

 



₍₁ 2⁽₎2₍ 2^;1_;⁾ ₎ 1 2 2

1 x x x

x x x

n n

n

) ...

; ( ) 1 (

)

; ( )

; ( ) 1 (

)

; (

5 2 4

5 4 3

2 2

3

2 



 



 



x x x

x x x x

x x

x x x

...

. 6 . 2 . 3 . . 1

.  ² ³ ⁴  ⁵ ⁶

ox x ox x x x



spt₂(1)x spt₂(2)x² spt₂(3)x³ spt₂(4)x⁴spt₂(5)x⁵....

. )

2(

1

n n

x n



^ spt



For convenience, define spt₂(1) = 0.

From above we get spt₂(3)0, spt₂(6)6,...

i.e., spt₂(3.1)00 (mod 3), 0

6 ) 2 . 3

2(  

spt (mod 3), ...

We can conclude that spt₂(3n)0



mod3



. We also get spt₂(4)3, spt₂(7)6,...

i.e., spt₂(31)30



mod3



,



mod3



0 6 ) 1 2 . 3

2(   

spt , …

We can conclude that spt₂⁽³n¹⁾⁰



^mod³



(Bringann 2009). Again from above we get;

, 0 ) 3

2( 

spt spt₂(8)15,...

i.e., spt₂(3)00



mod5



,



mod5



0 15 ) 3 5

2(   

spt , ...

We can conclude that spt₂⁽⁵n³⁾⁰



^mod⁵



.

Corollary 1: spt₂(n)



^



 m

) ,

2(m n M_S . Proof: The generating function for ( , )

2 m n

M_S is given by;

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

^

1

n



^



 m

) ,

2(m n M_S z^mxⁿ

=

 



 







₍⁽ 2 _; ^;₎ ⁾₍ ⁽1 2 _; ^;₎ ⁾ 1 2 1

2 2

1 zx x z x x

x x x x x

n n

n

.

If z = 1, then,



^

1

n



^



 m

) ,

2(m n M_S x ⁿ

=



 







⁽₍ 2 _;^;₎⁾₍⁽2 _; ₎ ^; ⁾ 1 2 1

2 2

1 x x x x

x x x x x

n n

n

=  



 2 2

3 3 2

)

; (

)

; ( )

; (

x x

x x x x x



 

2 4

5 5

4

)

; (

)

; ( )

; (

x x

x x x x

x +...

= 



 _

...

) 1 ( ) 1 (

)...

1 )(

1 ( )

; (

2 3 2 2

4 3 3

2

x x

x x x

...

) 1 ( ) 1 (

)...

1 )(

1 ( )

; (

2 5 2 4

6 5

5 4

x x

x x x

x x



 _ +…

= (1 ) (1 )(1 )...

)

; (

4 3

2 2

3 2

x x

x

x x x



 _

+ ...

)...

1 )(

1 ( ) 1 (

)

; (

6 5

2 4

5

4 



 _

x x

x

x x x

=



^

1

n    



)

; ( ) 1 (

)

; (

1 2 2 2

1 2 2

x x x

n n

=



^

1 n

xn

n spt₂( ) . i.e.,



^

1 n

xn

n

spt₂( ) =



^

1

n



^



 m

) ,

2(m n M_S xⁿ.

Now equating the co-efficient of xⁿfrom both sides we get;

)

2(n

spt



^





m

) ,

2(m n M_S . Hence the Corollary.

Result 1:

) 4 3 ( ) 1 4 , 3 , 2 ( ) 4 , 3 , 1 ( ) 4 , 3 , 0

( ₂

2 2

2 M M spt

M_S  _S  _S  .

Proof: We prove the result with the help of examples. We see the vector partitions from S₂of 4 along with their weights and cranks and are given as follows:

Here we have used  to indicate the empty partition. Thus we have, ,

1 ) 4 , 3 , 0 (

2 

MS (1,3,4) 1

2 

MS ,

1 ) 4 , 3 , 1 ( )

4 , 3 , 2 (

2

2  _S  

S M

M

) 4 , 3 , 1 ( )

4 , 3 , 0 (

2

2 S

S M

M 



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= (3) 3

3 1 3. 1 1 ) 4 , 3 , 2

( ₂

2 spt

M_S    .

Hence the Result.

Table 1

S2-vector partition ( )

 of 4 Weight

 ( )



Crank

) (

 mod 3

) , , , 4

1 (   

^  1 0 0

) , , 2 2 (

2 

^   1 1 1

 

3 (2, , 2,) 1 –1 2

3 )

( 

^

Now from table 1 we get; (^)3, i.e., ( ,3,4) 3

2

0





k M_S

k

.



^



 (4) ( ,3,4) ( ).

2

0

2 M k  

spt _S

k

Now we can define;

) , ( )

, , (

2

2 (mod )

n m M n

t k

M _S

t k m

S







and

 

^









 ¹

0

2( ) ( , ) ( , , ).

2 2

t

k S m

S m n M k t n

M n

spt

Theorem 1: The number of vector partitions of n in S₂ with crank m counted according to the weight  is non-negative, i.e., ( , ) 0.

2 m n 

M_S

Proof: The generating function for ( , )

2 m n

M_S is given by;

n m S

m n

x z n m M ( , )

1



2















 







²₍⁽ ²²_;¹^;₎ ⁾₍ ⁽¹ ²²_;¹^;₎ ⁾

1 zx x z x x

x x x x x

n n

n















 .( ; )

)

; ( )

; (

2 2 4 2

1 2

2

1

x x x

x z x zx

x _n

n n

n

.

[Since ^ _ ^ _







⁽ ² ¹^; ⁾ ⁽ ² ¹^; ⁾

1

x x x

x ⁿ ⁿ

n

...

)

; ( )

;

( ³  ³  ⁵  ⁵ 

 x x _ x x _ x x _ x x _





(1 x³)(1 x⁴)...(1 x³)(1 x⁴)... (1x⁵)(1x⁶)...(1x⁵)......









(1 x⁶)(1 x⁸)... (1 x¹⁰)(1 x¹²)... (1x¹⁴)......

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...

)

; ( )

;

( ⁶ ²  ¹⁰ ² 

 x x _ x x _ ] )

; ( ⁴ ² ²

1



 





 x ⁿ x

n





 



 





 ( ; )

)

; .( )

; ( )

; (

)

; (

4 2 2 4 2

1 2

4 2

1 x x

x x x x z x zx

x x x

n n n

n

 

 



 

 





₍ 2 _; ₎⁽₍ ^;1⁾2 _; ₎ ^.₍₁ 4 ₎₍¹4 1_; 2₎ 4

2

1 zx x z x x x x x

x x x

n n n

n

[Since, ...

)

; (

)

; ( )

; (

)

; ( )

; (

)

; (

8 2 10 4

2 6 4

2 2 4

1







 



 ^x_xⁿⁿ _x^x ^x_x ^x_x ^x_x ^x_x n

 



 

)...

1 )(

1 (

)...

1 )(

1 (

6 5

4

8 6

x x x

x

x ...

)...

1 )(

1 (

)...

1 )(

1 (

11 10

9 8

12

10 



x x

 



 

)...

1 )(

1 (

1

7 5

4 x x

x (1 )(1 )(1 )... ^...

1

11 9

8 



x x x

) ]

; ( . 1 1

1

2 1 4 4

1  



 





_x ⁿ _x ⁿ _x

n









 





 



 

₍ ⁽2 _; ₎⁾₍ ₎ ^.₍₁ 4 ₎₍¹4 1_; 2₎ 2

1

0 1

2

x x x x

x zx

x

x z _n _n

k k

n k n

k n

n

[Since,







 



 ₍ 2 _; ₎⁽ ₍ ^;1 ⁾2 _; ₎ 4

2

1 zx x z x x

x x x

n n

n

) ] ( )

; (

) (

2 2 1

0 1

2

k k

n k n

k n

n

x x x z

x x z





 









 . (by Berkovich and Garvan 2008)

We see that the coefficient of any power x in the right hand side is non-negative so the coefficient ( , )

2 m n

M_S of z^mxⁿ is non-negative, i.e., ( , ) 0.

2 m n 

M_S Hence the Theorem.

Numerical example 1

The vector partitions from S₂ of 5 along with their weights and cranks are given as follows:

Here we have used  to indicate the empty partition. Thus we have;

, 0 1 1 ) 5 , 0 (

2   

MS (1,5) 1,

2 

MS and ( 1,5) 1,

2  

MS i.e., ( ,5) 2

2 



^MS ^m m

, i.e., every term is non-negative, i.e., ( , ) 0.

2 m n 

M_S

So we can conclude that, ( , ) 0.

2 m n 

M_S

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Corollary 2: S2(1,x) = ₂

1

spt

n



^



(n)xⁿ. Proof: We get;







 







₍⁽ _; ^;₎⁾₍⁽ _; ^;₎ ⁾ )

,

( 2 1 2

1 2 1

2 2

1

2 zx x z x x

x x x x x x

z

S _n _n

n n

n

(Andrews et al. 2012).

If z = 1, then we get;







 





⁽₍ _;^;₎⁾₍⁽ _; ₎^; ⁾ )

, 1

( ₂ ₂

1 2 1

2 2

1

2 x x x x

x x x x x x

S _n _n

n n

n

=  



 2 2

3 3

2

)

; (

)

; ( )

; (

x x

x x x x

x ...

)

; (

)

; ( )

; (

2 4

5 5

4  



x x

x x x x x

= 



 _

...

) 1 ( ) 1 (

)...

1 )(

1 ( )

; (

2 3 2 2

4 3 3

2

x x

x ...

...

) 1 ( ) 1 (

)...

1 ( ) 1 ( )

; (

2 5 2 4

6 5 5

4 



 _

x x

x x x

x x

= 



 _

)...

1 ( ) 1 (

)

; (

3 2 2

3 2

x x

x ...

)...

1 ( ) 1 (

)

; (

5 2 4

5

4 



 _

x x

x x x

=



 

 

 



₍₁ ⁽2₎2₍ 2^; 1_;⁾ ₎ 1 2 2

1 x x x

x x x

n n

n



^

 



1 2 n

n spt xⁿ. i.e., S2(1,x) = ₂

1

spt

n



^



(n)xⁿ. Hence the Corollary.

Theorem 2:













n SP

n spt

2 1 2

1 )

2(



Proof: First we define the sptcrank in terms of partition pairs, )

( ) ( 0 : )

, (

{ ₁ ₂ P P s ₁ s ₂

SP      and all parts of ₂ that are 2s(₁)1 are odd}.

Let SP2 be the set of ^(₁,₂)SP with s(₁)even. The generating function for spt₂(n) is given by;



 

 





 



⁽ ⁾



₍₁ ² ²⁽ ₎²²₍ ²¹^;¹⁾_; ₎

1 1

2 x x x

x x x x

n

spt _n _n

n n

n













 





₍₁ ² ₎²₍² ² ¹_; ₎ ⁽ ² ¹^; ⁾

1

x x x

x x

x n

n n

n

 



 



 





₍₁ ₎ ₍ _; ₎ ^.⁽₍ 2 1^;_; ₎⁾ 2 2 4 1

2 2 2

2

1 x x

x x x x x

x

n n n

n n

n

[Since, ( ; ) ( ³; ) ( ⁵; ) ...

1

2     

 _ _



  



^x ^x ^x ^x ^x ^x

n

(10)

=(1x³)(1x⁴)...(1x⁵)(1x⁶)... (1x⁷)(1x⁸)......

)...

1 )(

1 (

)...

1 )(

1 ( )...

1 )(

1 (

)...

1 )(

1 (

6 5

12 10

4 3

8 6

x x



 



  ...

)...

1 (

)...

1 (

7

14 



  x x

) ..

; (

)

; ( )

; (

)

; (

5 2 10 3

2

6  





x x

x x x

x x

x .



 

 





 ( ; )

)

; (

1 2

2 2 4

1 x x

x x

n n

n

]

 



 



 





₍₁ ₎ ₍ _; ₎ ^.⁽₍ 2 1^;_; ₎⁾ 2 2 4 1

2 2 2

2

1 x x

x x x x x

x

n n n

n n

n



 





 





₍ _; ₎ ^.₍₁ ¹ ₎^.⁽₍ 2 1^;_; ₎⁾ 2 2 4 2 2

2

1 x x

x x x x

x x

n n n n

n

[Since,

 



 



₍₁ 2 ₎2₍ 2 1_; ₎ 2

1 x x x

x

n n

n

) ...

; ( ) 1 ( )

; ( ) 1

( ⁴ ² ⁵

4 3

2 2

2 

 

 



 x x x

x x

x

 



 

)...

1 )(

1 ( ) 1

( ² ² ³ ⁴

2

x x x

x ...

)...

1 )(

1 ( ) 1

( ⁴ ² ⁵ ⁶

4 



x x x

x

)] 1 ( . 1 )

;

( ² ²

2

1

n n

n

n x x x

x

 













   





₍ ² _;² ₎ ^.₍₁ ² ₎₍₁ ² ¹_)...(¹₁ ⁴ ₎₍ ⁴ ¹_; ²₎

1 x x x x x x x

x

n n n

n n

n

[Since, ...

)

; (

)

; ( )

; (

)

; ( )

; (

)

; (

5 2 10 3

2 6 1

2 2 2 4

1







 

 



 ^x_xⁿⁿ ^x_x ^x_x ^x_x ^x_x ^x_x n

 



 

)...

1 )(

1 (

)...

1 )(

1 (

4 3

8 6

x x

x

x ...

)...

1 )(

1 )...(

1 )(

1 (

)...

1 )(

1 (

11 10

6 5

12

10 



x x

 



 

)...

1 ( ) 1 ( ) 1 (

1

5 4

3 x x

x ...

)...

1 )(

1 )...(

1 )(

1 (

1

11 9

6

5 



x x x x

=







  



₍₁ 2 1_)....₍₁ ¹ 4 ₎₍ 4 1_; 2₎

1 x ⁿ x ⁿ x ⁿ x

n

]



^



1

n  

1

1 1





x

n S

P



 

 

2

2 2





x

n s

P





all parts in ₂2n1are odd



^













1

2 1 2

n

n SP

x





 .

Equating the co-efficient of xⁿfrom both sides we get;

(11)













n SP

n spt

2 1 2

1 )

2(



. Hence the Theorem.

Numerical Example 2

The overpartitions of 6 with smallest parts not overlined and even are 6, 4+2, 42, and 2+2+2. Consequently, the number of smallest parts in the overpartitions of 6 with smallest part not overlined and even is given by;

•

6 4+

•

2 ,

2•

4 ,

•

2 +

•

2 +

•

2 ,

so that spt₂(6)6 i.e., there are 6SP₂-partition pairs of 6 like:

), , 6

(  (42,),(2,4), (222,),(22,2) and (2, 2+2).

Result 2:

 ) 8 , 5 , 0 (

S2

M (1,5,8,)

S2

M (2,5,8,)

S2

M

 ) 8 , 5 , 3

2( MS

5 3 1 ) 8 , 5 , 4

2(  

MS spt₂(8).

Proof: We prove the result with the help of examples. We can define a crankof partition pairs ^(1,2)SP2.

For ^(₁,₂)SP₂ , we define, k(^) # of pairs j in ₂ such that s(₁) j2s(₁)1, and also define;

 



















 

0 if ; 1 of parts of

# 0 if

; of

parts of

# ) (

1 1 1

k k

k k s crank



 where ( ).

k 

k

We know that spt₂(8)15. There are 15 SP₂ -partition pairs of 8.

Table 2

S2-vector partition

) (

 of 5

Weight Crank

–1 c 0

(2, , ,3) 1 0

1 1

1 –1

 ⁽^^⁾ ⁽^^⁾ )

, , , 2 3

1 ( 

^  





2  

) , , 3 , 2 (

3 

^ 

) , 3 , , 2 (

4  

^ 

2 ) ( 

^

(12)

From the table 2 we get;

 ) 8 , 5 , 0 (

S2

M (1,5,8,)

S2

M (2,5,8,)

S2

M

 ) 8 , 5 , 3

2( MS

5 3 1 ) 8 , 5 , 4

2(  

MS spt₂(8). Hence the Result.

Table 3

SP2 -partition pair of 8 k crank (mod 5)

(3+2, 3) 1 0 0

(4+2, 2) 1 0 0

) , 8

(  0 0 0

(2+2, 4) 0 1 1

(4+4, ) 0 1 1

(6+2, ) 0 1 1

(2, 2+2+2) 3 –3 2

(3+3+2,) 0 2 2

(4+2+2, ) 0 2 2

(2, 3+3) 2 –2 3

(2+2, 2+2) 2 –2 3

(2+2+2+2,) 0 3 3

(2, 4+2) 1 –1 4

(4, 4) 1 –1 4

(2+2+2, 2) 1 –1 4

Now we will describe the sptcrank of a marked overpartition (Chen et al. 2013). To define the sptcrank of a marked overpartition we first need to define a function k

 

m,n for positive integers m, n such that mn1, we write mb2^j, where b is odd and jo. For a given odd integer b and a positive integer n we define j₀  j₀(b,n) to be the smallest non- negative integer j₀ such that b2^j⁰ n1.

We define;

 













 ^

. 2 2 if , 0

2 2 if 2

2 if , 0 ,

0 0 0

n b

n b n

m k

j j j

j

(13)

Table 4 Marked

overpartition (, j) of 6

1 ₂ (₁) k



⁽₂^,s⁽₁⁾



k sptcrank (mod 3)

(6 ,1) 6  1 0 0 0 0

(4+2 ,1) 4+2  1 0 0 1 1

) 1 , 2 4

(  2 4 1 0 0 0 0

(2+2+2, 1) 2+2+2  3 0 2 –2 1

(2+2+2, 2) 2+2+2  3 0 1 –1 2

(2+2+2, 3) 2+2+2  3 0 0 2 2

For a marked overpartitions (, j) we let ₁ be the partition formed by the non-overlined parts of ,₂ be the partition (into distinct parts) formed by the overlined parts of  so that

), ( ) (₂ s ₁

s  we define k(,i)(₁) jk(₂,s(₁)), where (₁) is the number of smallest parts of ₁.

Now we can define;

   



 

























. 0 , if

; 1 of parts of

#

0 , if

, of

parts of

# ) , (

1 1 1

j k k

k s j

sptcrank



Table 5 Marked

1 ₂ (₁) k



(₂,s(₁)



k sptcrank (mod 3)

(5+2, 1) 5+2  1 0 0 1 1

) 1 , 2 5

(  2 5 1 0 0 0 0

(3+2+2, 1) 3+2+2  2 0 1 0 0

(3+2+2, 2) 3+2+2  2 0 0 2 2

( 3 +2+2, 1) 2+2 3 2 1 2 –2 1

( 3 +2+2, 2) 2+2 3 2 1 1 –1 2

(14)

Table 6 Marked

1 ₂ (₁) k



⁽₂^,s⁽₁⁾



k sptcrank (mod 5) )

1 , 2 6

(  2 6 1 2 2 –2 3

) 1 , 2 2 4

(   2+2 4 2 0 1 –1 4

) 2 , 2 2 4

(   2+2 4 2 0 0 1 1

) 1 , 2 3 3

(   3+2 3 1 1 1 0 0

(2+2+2+2, 1) 2+2+2+2  4 0 3 –3 2

(2+2+2+2, 2) 2+2+2+2  4 0 2 –2 3

(2+2+2+2, 3) 2+2+2+2  4 0 1 –1 4

(2+2+2+2, 4) 2+2+2+2  4 0 0 3 3

(3+3+2, 1) 3+3+2  1 0 0 2 2

(4+2+2, 1) 4+2+2  1 0 1 0 0

(4+2+2, 2) 4+2+2  2 0 0 2 2

(6+2, 1) 6+2  1 0 0 1 1

(4+4, 1) 4+4  2 0 1 –1 4

(4+4, 2) 4+4  2 0 0 1 1

(8, 1) 8  1 0 0 0 0

Corollary 3 (Lovejoy and Osburn 2009): The residue of the sptcrank



mod3



divides the marked overpartitions of 3n with the smallest part not overlined and even into 3 equal classes.

Proof: We prove the Corollary with the help of an example when n = 2. There are 6 marked overpartitions of 3n (when n = 2) with the smallest part not overlined and even so that,

6 ) 6

2( 

spt .

We see that the residue of the sptcrank



^mod³



divides the marked overpartitions of 3n (when n = 2) with smallest part not overlined and even into 3 equal classes. Hence the Corollary.

Corollary 4: The residue of the ^sptcrank



^mod³



divides the marked overpartitions of 3n+1 with smallest part not overlined and even into 3 equal classes.

Proof: We prove the Corollary with the help of an example when n = 2. There are 6 marked overpartitions of 7 with the smallest part not overlined and even, so that spt₂(7)6. We see that the residue of the sptcrank(mod 3) divides the marked overpartitions of 3n+1 (when n = 2) with smallest part not overlined and even. Hence the Corollary.

Corollary 5: The residue of the sptcrank



mod5



divides the marked overpartitions of 5n+3 with smallest part not overlined and even into 5 equal classes.

(15)

Proof: We prove the Corollary with the help of example when n = 1. There are 15 marked overpartitions of 5n + 3 (when n =1) with the smallest part not overlined and even so that

. 15 ) 8

2( 

spt We see that the residue of the divides the marked overpartitions of 8 with the smallest part not overlined and even into 5 equal classes. Hence the corollary.

CONCLUSION

In this study we have found the number of smallest parts in the overpartitions of n with the smallest part not overlined and even for n=1, 2, 3, 4 and 5. We have shown various relations



mod3



0 ) 3

2( n 

spt , spt₂(3n1)0



mod3



, spt₂(5n3)0(mod 5), )

4 , 3 , 0 (

S2

M (1,3,4)

S2

M (2,3,4)

S2

M (4)

3 1

spt2

 and (0,5,8)

S2

M (1,5,8)

S2

M (2,5,8)

S2

M

= (3,5,8)

S2

M (4,5,8)

S2

M

5 31

 spt₂(8) with numerical examples respectively. We have verified the Theorem 1 when n = 5 and have verified the Theorem 2 when n = 6. We have verified the Corollary 3 with 6 marked overpartitions of 6 and have verified the Corollary 4 with 6 marked overpartitions of 7 and also have established the Corollary 5 with 15 marked overpartition of 8.

REFERENCES

[1] Andrews, G.; Dyson, F. and Rhoades R., On the distribution of the spt-crank, Mathematics, 1(3): 76–88, 2013.

[2] Andrews, G.E.; Garvan, F.G. and Liang, J., Combinatorial Interpretations of Congruences for the spt-function, The Ramanujan Journal, Springer. 29(1–3): 321–338, 2012.

[3] Berkovich, A. and Garvan, F.G., K. Saito’s Conjecture for Nonnegative eta Products and Analogous Results for other Infinite Products, Journal of Number Theory, 128(6): 1731– 1748, 2008.

[4] Bringann, K.; Lovejoy, J. and Osburn, R., Rank and Crank Moments for Overpartitions, Journal of Number Theory, 129(7):1758–1772, 2009.

[5] Bringann, K.; Lovejoy, J. and Osburn, R., Automorphic Properties of Generating Functions for Generalized Rank Moments and Durfee Symbols, Int. Math. Res. Not. IMRN, (2): 238–260, 2010.

[6] Chen, W.Y.C.; Ji, K.Q. and Zang, W.J.T., The spt-crank for Ordinary Partitions, arXiv e- prints, Aug. 2013.

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[7] Garvan, F.G. and Shaffer, C.J., The spt-crank for Overpartitions, arXiv:1311.3680v2 [Math. NT], 23 Mar 2014.

[8] Lovejoy, J. and Osburn, R., M2-rank Differences for Partitions without Repeated Odd Parts. J. Theor. Nombres Bordeaux, 21(2): 313–334, 2009.