A Note on M2 Partitions of n

International Journal of Latest Technology in Engineering, Management & Applied Science (IJLTEMAS)
Volume VI, Issue II, February 2017 | ISSN 2278-2540
www.ijltemas.in Page 63
A Note on M2 Partitions of n
Dr. K.Hanumareddy1
, Gudimella V R K Sagar2
1
Department of Mathematics, Hindu College, Acharya Nagarjuna University, Guntur (Dt)
2
L/Mathematics, Govt.Polytechnic, Ponnur, Guntur (Dt).
Abstract: S.Ahlgren, Bringmann and Lovejoy [1] defined M2
spt(n) to be the number of smallest parts in the partitions of n
without repeated odd parts and with smallest part even and
Bringmann, Lovejoy and Osburn [4]derivedthe generating
function for M2 spt(n) .Hanumareddy and Manjusri [7] derived
generating function for the number of smallest parts of
Partitions of n by using r- Partitions of n. Hanumareddy and
Gudimella V R K Sagar[9], [10] defined M2 partitions and Ga
partitions of positive integer n . In this chapter we derived
generating function for M2 spt(n) by using r- M2 partitions of n .
We also derive generating function for sum M2 spt(n).
Keywords: Partition, r- Partition, M2 partitions, Smallest part of
the M2 partitions.
Subject classification: 11P81 Elementary theory of Partitions.
I. INTRODUCTION
et  2 nM  be denote the set of all partitions of n with
odd numbers appears at most one time and smallest
parts are even numbers. Let  2 nM p be the
cardinality of  2 nM  for and  2 0 1M p  . If
, write  2 r nM p for the number of partitions of
n in  2 nM  each consisting of exactly parts, i.e
2r M partitions of n in  2 nM  . If ,
we write  2 0rM p n  . Let  ,2 k nM p represent the
number of partitions of in  2 nM  using natural
numbers atleast as large as only. Let the partitions in
 2 nM  be denoted by 2 .M partitions
Let  2 nM spt be denotes the number of smallest parts
including repetitions in all partitions of in  2 nM  and
 2 nsumM spt be denotes the sum of the smallest parts.
For 1i  ,let us adopt the following notation on the lines of
[3].
 2 sM m  number of smallest parts of  in  2 nM  .
 2 nM spt =  
 
2 s
n
M m
 



Let  nM be denote the set of all partitions of n with odd
numbers appears at most one time and part 1’s not appear and
 nMp the cardinality of  nM . Let  r nMp for the
number of r partitions of n in  nM . Let  ,k nMp
represent the number of partitions of in  nM using
natural numbers atleast as large as only. Let  nMspt
denote the number of smallest parts including repetitions in all
partitions of in  nM . Let  sMm  be number of
smallest parts of  in  nM and  nMspt is
 
 
s
n
Mm
 



We observe that
1.1 The generating function for the number of
2r M partitions of n with odd numbers appears at most
one time and smallest parts are even numbers is
 
 
 
2 2
1
2 2
,
2 (1.2)
,
r
r
r
r
q q q
M p n
q q



1.2 The generating function for the number of
r partitions of n with odd numbers appears at most one
time and 1’s not appear is
n N
1 r n 
r
0 orr r n 
n
k
n
n
k
n
L

 
 
 
2 2
2 2
,
(1.3)
,
r
r
r
r
q q q
Mp n
q q


II. GENERATING FUNCTION FOR  2M spt n
The generating function for the number of smallest
parts of all partitions of positive integer n is derived by
G.E.Andrews. By utilizing 2r M partitions of n, we
propose a formula for finding the number of smallest parts of
2M partitions of n.
2.1Theorem:
Proof:[5] Let
 11 2
1 2 1 2 1 1( , ,..., ) = , ,..., ,l l
r ln k  
      
 ,
   11 2
1 2 1 1, ,..., , 2l l
l k M n  
   
  , 1 2 ,k k k N 
be any 2r M partition of n with l distinct parts such that
odd parts not repeated and smallest parts are even numbers.
Case 1:[6] Let lr t  which implies 1r t k   . Subtract
all 1 'k s , we get  11 2
1 1 2 1, ,..., l
ln tk  
   
  ,
   11 2
1 2 1, ,..., l
l M n 
   
 
Hence  11 2
1 1 2 1, ,..., l
ln tk  
   
  is a
 r t partition  of 1n tk with 1l- distinct parts and
each part is greater than or equal to 1 1k  . Here the number of
2r M partitions with smallest part 1k that occurs
exactly t times among all 2r M partitions of n is
 1 11,r tMp k n tk   .
Case 2: Let lr t  which implies 1r t k  
Omit 1 'k s from last t places, we get
 11 2
1 1 2 1 1, ,..., ,l l t
ln tk k  
    
  ,
   11 2
1 2 1 1, ,..., , 2l l t
l k M n  
    
  . Hence
 11 2
1 1 2 1 1, ,..., ,l l t
ln tk k  
    
  is a
  2r t M partition  of 1n tk with l distinct parts and
the least part is 1k .
Now we get the number of 2r M partitions with smallest
part 1k that occurs more than t times among all
2r M partitions of n is  1 12 ,r tM f k n tk  .
Case 3: Let lr t  which implies all parts in the partition
are equal.
The number of partitions of nwith equal parts in set
  12 , 2M n k N  is equal to the number of divisors of
2
n
. Since the number of divisors of
2
n
is ,
2
n
d
 
 
 
the
number of partitionsof n with equal parts in set
  12 , 2M n k N  is ,
2
n
d
 
 
 
.
1 if 2 |
where
0 otherwise
r n


 

From cases(1), (2) and (3) we get r partitions of n with
smallest part 1k that occurs t times is
   1 1 1 11, 2 ,r t r tMp k n tk M f k n tk      
 1 1,r tMp k n tk   
The number of smallest parts in 2M partitions of n is
   
1
1 1
1 1
2 ,
2k t
n
M spt n Mp k n tk d
 
 
 
    
 

   
1 1
2 2 , 2
2k t
n
M spt n Mp k n tk d
 
 
 
     
 

2.2 Theorem:    2 2, 2r rMp k n Mp n kr  
Proof: Let 1 2 1( , ,..., ), 1r in k i       ,
1 2 ,k k k N  be any 2r M partition of n such that
odd numbers not repeated and smallest parts are even
numbers. Subtracting 1k from each part, we get

 1 1 1 2 1 1, ,..., rn k r k k k      
Hence  1 1 1 2 1 1, ,..., rn k r k k k       is a
2r M partition of 1n k r with odd parts not repeated
and 1’s not appear.
Therefore the number of 2r M partitions of n with parts
greater than or equal to 1 2k  is  1 .rMp n k r
Hence    1 12,r rMp k n Mp n k r  
   2 2, 2r rMp k n Mp n kr   
2.3 Theorem:
 
 
   
2 2 1 2
22 2 2 2
1 1
;
2
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
2 2 , 2
2k t
n
M spt n Mp k n tk d
 
 
 
    
 

 
1 1 1
2 , 2
2
r
k t r
n
Mp k n tk d
  
  
 
    
 

 
  
1 1 1
first replace 2 2 by 2 ,then replace by 2 intheorem 2.2
2 2 2
2
r
t r k
k k n n tk
n
Mp n tk r k d
  
  
 
 
      
 

Where  d n is the number of positive divisors of n .
From (1.5.1) in [8]
 
 
 
2 2 2 2 2 2
22 2
1 1 1 1
,
1,
r tk r k
k
r
k
k t r k
r
q q q q
qq q
  
   
   

 

 
 
 
2 2 2 2
22 2
1 1 1 1
,
1,
tk kr k
r
k
k t r k
r
q q q q
qq q

   
   

 

 
   
 
2 2 2
2
22 2
1 1 1 1
,
1,
rk k
tk r
k
k t r k
r
q q q q
q
qq q
   
   
 
  
  
 
  
   
 
2 22 2
2 22 2
1 1 1
,
1 1,
rkk k
r
k k
k r k
r
q q qq q
q qq q
  
  
 
  
  
 
  
   
 
2 22
2 2 2
1 1
,
1
1 ,
rkk
r
k
k r
r
q q qq
q q q
 
 
 
  
 
 
 
 
 
 
2 2 12
2 2 2
1 0
1
1 1
r kk
k r k
k r
qq
q q
 

 


 
 
   
     
2 2 1 2 3 2 5
2 2 2 2 2 4 2 6
1
1 1 1 ...
1 1 1 1 1 ...
k k k k
k k k k k
k
q q q q
q q q q q
  
  

  

    

 
   
2 2 1 2
22 2 2 2
1
;
1 ;
k k
k k
k
q q q
q q q










 
 
   
2 2 1 2
22 2 2 2
1 1
;
2
1 ;
n n
n
n n
n n
q q q
M spt n q
q q q

 


 




 
2.4 Corollary: The generating function for  2cM spt n , the
number of smallest parts of the 2M partitions of n which
are multiples of c is
 
 
   
2 2 1 2
22 2 2 2
1 1
;
2
1 ;
cn cn
n
cn cn
n n
q q q
cM spt n q
q q q

 


 




 
2.5 Theorem: The generating function for the sum of smallest
parts of the 2M partitions of n is
 
 
   
2 2 1 2
22 2 2 2
1 1
2 ;
2
1 ;
n n
n
n n
n n
nq q q
sumM spt n q
q q q

 


 




 
REFERENCES
[1]. S. Ahlgren, K. Bringmann, J.Lovejoy.l-adic properties of smallest
parts functions. AdvMath.,228(1):629-645, 2011

[2]. G. E. Andrews, The theory of partitions
[3]. G. E. Andrews, The number of smallest parts in the partitions of n.
J.Reine Angew.Math.624:133-142,2008.
[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]. Hanumareddy K (2009), A Note on r-partitions of n in which least
part k, International Journal of Computational Ideas, 2,1, pp. 6-
12.
[6]. Hanumareddy K (2009), A Note on partitions , International
Journal of Mathematical Sciences, 9,3-4, pp. 313-322.
[7]. K.Hanumareddy, A.Manjusri The number of smallest parts of
partitions of n. IJITE.,Vol.03,Issue-03,(March 2015), ISSN:2321-
1776
[8]. Hanumareddy K., Gudimella V R K Sagar "The number of
Smallest parts of overpartitions of n" International Research
Journal of Mathematics, Engineering, and IT Vol.2 Issue 3,
March 2015
[9]. Hanumareddy K., Gudimella V R K Sagar "A Note on M1
Partitions of n" International Research Journal of Mathematics,
Engineering, and IT Vol.3 Issue 12, December 2016
[10]. Hanumareddy K., Gudimella V R K Sagar "A Note on Ga
Partitions of n" International Journal of Latest Technology in
Engineering, Management & Applied Science (IJLTEMAS) ISSN
2278-2540, Vol.V Issue XII, pp.94-98 December 2016

A Note on M2 Partitions of n

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