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A NOTE ON M1 PARTITIONS OF

n

Dr. K. Hanumareddy Gudimella V R K Sagar

Department of Mathematics L/Mathematics Hindu College Govt.Polytechnic Acharya Nagarjuna University Ponnur

Guntur, India. Guntur, India.

ABSTRACT

S.Ahlgren, Bringmann and Lovejoy [1] defined M spt n to be the number of 2

 

smallest parts in the partitions of n without repeated odd parts and with smallest part even

and Bringmann, Lovejoy and Osburn [4]derived the generating function for M spt n . 2

 

Hanumareddy and Manjusri [5] derived generating function for the number of smallest parts

of partitions of n by using rpartitionsof n. In this chapter we definedM spt n as the 1

 

number of smallest parts in the partitions of n without repeated even parts and with smallest

part odd and also derive its generating function by using r M partitions 1 of n . We also derive generating function for M spt n . 1

 

Keywords: Partition, r-partition, M1Partition, Smallest part of the M1Partition.

Subject classification: 11P81 Elementary theory of Partitions. 1. Introduction:

Let M1

 

n be denote the set of all M partitions1 of n with even numbers not repeated and smallest parts are odd numbers. Let M p1

 

n be the cardinality of M1

 

n , write M p1 r

 

n for the number of r M partitions 1 of nin M1

 

n each consisting of exactly parts, i.e r M partitions 1 of n in M1

 

n . Let M p1

 

k n, represent the number of

1

M partitions of n in M1

 

n using natural numbers at least as large as only. Let the partitions in M1

 

n be denoted by M partitions1 .

Let M spt1

 

n be denotes the number of smallest parts including repetitions in all partitions of in M1

 

n and sumM spt1

 

n be denotes the sum of the smallest parts.

r

k

n

International Research Journal of Mathematics, Engineering and IT Vol. 3, Issue 12, December 2016 Impact Factor- 5.489 ISSN: (2349-0322)

© Associated Asia Research Foundation (AARF)

(2)

 

1 s

M m

number of smallest parts of

in M1

 

n .

 

1 n

M spt =

 

 n 1 s

M m  

 

1

n

M be denote set of all M partitions1

of n.

 

7 : 1

 

7 11 1

 

7 28

7, 6 1, 4 3, 5 1 1 , 4 2 1, 3 3 1, 4 1 1 1 , 3 2 1 1, 3 1 1 1 1 , 2 1 1 1 1 1, 1 1 1 1 1 1 1.

For exampleM1

M pM spt

                 

          

:

We observe that

1.1. The generating function for the number of rpartitionsofnwith even numbers not repeated is

 

2

1

2 2

, ,

r

r r

r

q q q M p n

q q  

1.2. The generating function for the number of rM partitions1 ofnwith even numbers appears at most one time and smallest parts are odd numbers is

 

2

1

2 2

,

1 (1.1)

,

r

r r

r

q q q M p n

q q

 

2. Generating function forM spt n1

 

The generating function for the number of smallest parts of all partitions of positive integer n is derived by G.E. Andrews. By utilizing rM partitions1 of n, we propose a

formula for finding the number of smallest parts of n.

2.1 Theorem:

 

1

1 1

1 if 2 1|

1 2 1, 2 1 where

0 otherwise

k t

k n

M spt n M p k n k t  

 

 

 

      

(3)

Proof: Let

1 2 1

1 2 1 2 1 1

( , , ... , ) = , ,..., l , l

r l

n         k

 ,

1 2 1

 

1 , 2 ,..., 1 , 1 1

l l

l k M n

 

 

    

  , k1 2k1,kNbe any rM partition1 of n with l distinct parts such that even parts not repeated and smallest parts are odd numbers

Case 1: Letrlt which impliesr tk1. Subtract all k s1' , we get

1 2 1

1 1 , 2 ,..., 1

l

l

n tk      

  ,

1 2 1

 

1 , 2 ,..., 1 1

l

l M n

  

    

 

Hence

1 2 1

1 1 , 2 ,..., 1

l

l

n tk      

  is a

1

r t M partition of n tk1withl- 1 distinct parts and each part is greater than or equal tok11. Here we get the number of

1

rM partitionswith smallest part k1 that occurs exactly t times among all

1

rM partitionsof n is M p1 r t

k11,n tk1

.

Case 2: Letrlt which impliesr tk1

Omit k s1' from last t places, we get

1 2 1

1 1 , 2 ,..., 1 , 1

l l t

l

n tk      k

  ,

1 2 1

 

1 , 2 ,..., 1 , 1 1

l l t

l k M n

   

     

  . Hence 1

1 1, 2 2,..., 1 1, 1

l l t

l

n tk      k

  is a

r t

M partition1 of n tk1 with l distinct parts and the least part isk1.

Now we get the number of rM partitions1 with smallest part k1 that occurs more

than t times among allrM partitions1 of n is M f1 r t

k n tk1,  1

Case 3: Letrlt which implies all parts in the partition are equal which are odd.

The number of partitions of n with equal parts in set

M1

 

n k, 12N1

is equal to the

number of divisors of 2n1. Since the number of divisors of 2n1 is d

2n1 ,

the number

of partitionsof n with equal parts in set

M1

 

n k, 12N1

is

1 if 1|

2 1 where

0 otherwise k n d n   

From cases (1), (2) and (3) we get rM partitions1 of n with smallest partk1that occurs t times is

1 1

1, 1 1 1, 1

r t r t

M f k n tk M p kn tk 

1

1

1 1

1 if |

, where

0 otherwise r t

k n

M p k n tk   

    

(4)

 

1 1 1 1 1 1 1

1 if |

1 , where

0 otherwise

k t

k n

M spt n M p k n tk  

         



 

1

1 1

1 if 2 1|

1 2 1, 2 1 where

0 otherwise

k t

k n

M spt n M p k n k t  

             



2.2. Theorem: M p1 r

2k1,n

M p n1 r

2kr

Proof: Let n( , , ... , ), 1 2  r i 2ki, be any rM partition1 of nsuch that even

numbers not repeated and smallest parts are odd numbers. Subtracting 2k from each part, we get n2kr

12 ,k

22 ,...,k

r 2k

Hence n2kr

12 ,k

22 ,...,k

r 2k

is a rM partition1 of n2kr with even parts not repeated and smallest parts are odd.

Therefore the number of rM partitions1 ofnwith parts greater than or equal to 2k1is

1

2 .

r

M p nkr

Hence M p1 r

2k1,n

M p n1 r

2kr

.

2.3. Theorem:

 

 

2 1 2 2

2

2 1 2 1 2

1 1 ; 1 1 ; n n n n n n n

q q q

M spt n q

q q q

           

Proof: From theorem (2.1) we have

 

1

1 1

1 2 1, 2 1

k t

M spt n M p k n k t

    



   

 

1

1 1 1

first replace 2 1 by 2 1, then replace by 2 1 in theorem 2.2.

2 1 2 2

r t r n

k k n n k t

M p n k t r k

  

  

   



    

1 if 2 1| where 0 otherwise k n       

2 1 2 2 2 2 1

2 1

2 2

1 1 1 1

,

1 ,

r k t r k

k r

k

k t r k

r

q q q q

q q q                 



   

2 1 2 1 2 2 1

2 1

2 2

1 1 1 1

,

1 ,

k t r k

k r

k

k t r k

r

q q q q

q q q                



 

 

2 1 2 2 1

2 1

2 1

2 2

1 1 1 1

,

1 ,

r

k k

k t r

k

k t r k

r

q q q q

(5)

 

2 1 2

2 1 2 1

2 1 2 2 2 1

1 1 1

,

1 , 1

r k

k k

r

k k

k r k

r

q q q

q q

q q q q

                      

 

2 1 2

2 1

2 1 2 2

1 1 , 1 1 , r k k r k k r r

q q q

q

q q q

                 

2 1 2

Put tq k ,a q q, q in theorem 2.1 'The Theory of partitions' by G.E.Andrews

2 2 1 1

2 1

2 1 2 2 1

1 0

1

1 1

r k k

k r k

k r q q q q               

2 2 2 1

2 1 2 2 1

1 0

1

1 1

r k k

k r k

k r q q q q             















2 1 2 2 2 2 4 2 6

2 1 2 1 2 1 2 3 2 5

1

1 1 1 1 ...

1 1 1 1 1 ...

k k k k k

k k k k k

k

q q q q q

q q q q q

                    

 

2 1 2 2

2

2 1 2 1 2

1 ; 1 ; k k k k k

q q q

q q q

         

 

 

2 1 2 2

2

2 1 2 1 2

1 1 ; 1 1 ; n n n n n n n

q q q

M spt n q

q q q

           

2.4 .Corollary: 1

 

1

1

1

1

1 1

1 2 1 , 2 1

k t

c M spt n M p c k n c k t

    



   

1 1 1

1 if 2 1 |

where and 2 1, N

0 otherwise

c k n

c c c

     



2.5. Theorem: M p1 r

2c k1 1,n

M p n1 r

2c kr1

where c12c1,cN

Proof: Let n( , , ... , ), 1 2  r i 2c k1i, be any rM partition1 of nsuch that even numbers not repeated and smallest parts are odd numbers and c is a constant. Subtracting

1

2c k from each part, we get n2c kr1 

12c k1 ,

22c k1 ,...,

r 2c k1

Hence n2c kr1

12c k1 ,

22c k1 ,...,

r2c k1

is a 1

rM partitionofn2c kr1 with even parts not repeated and smallest parts are odd.

Therefore the number of rM partitions1

ofnwith parts greater than or equal to 2c k1 1is

1

1

2 .

r

(6)

Hence M p1 r

2c k1 1,n

M p n1 r

2c kr1

where c1 2c1,cN

2.6.Theorem:

 

 

 

 

 

1 1

1 1

2 1 2 1 1 2

1 2 1

2 1 2 1 2 2

1 1

;

1 where 2 1

1 ;

c n c n

n

c n c n

n n

q q q

c M spt n q c c

q q q

  

 

  

 

  

2.7. Theorem:

 

 

2 1 2 2

2

2 1 2 1 2

1 1

2 1 ;

1

1 ;

n n

n

n n

n n

n q q q

sumM spt n q

q q q

 

 

 

 

3.References:

 

1 S.Ahlgren,K.Bringmann,J.Lovejoy. adic properties of smallest parts functions. . ., 228(1) : 629 645, 2011.

l

Adv Math

 

 

2 G.E.Andrews, The theory of partitions.

 

3 G.E.Andrews, The number of smallest parts in the partitions of . J.Reine Angew.Math ., 624 :133 142, 2008.

n

 

4 K.Bringmann,J.Lovejoy and R.Osburn.Automorphic properties of generating functions for generalized rank moments and Durfee symbols.

Int.Math.Res.Not.IMRN, (2) : 238 260, 2010.

 

5 K.Hanumareddy,A.Manjusri The number of smallest parts of Partition of . IJITE.,Vol.03,Issue-03,(March2015) ,ISSN:2321 1776.

n

www.aarf.asia

References

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