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 evenand 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 partsof 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 smallestpart 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 of1
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)
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1 s
M m
number of smallest parts of
in M1
n .
1 n
M spt =
n 1 s
M m
1n
M be denote set of all M partitions1
of n.
7 : 1
7 11 1
7 287, 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 p M 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
12 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
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 2k1,kNbe any rM partition1 of n with l distinct parts such that even parts not repeated and smallest parts are odd numbers
Case 1: Letrl t which impliesr t k1. 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
1r t M partition of n tk 1withl- 1 distinct parts and each part is greater than or equal tok11. 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
k11,n tk 1
.Case 2: Letrl t which impliesr t k1
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 tk 1 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: Letrl t 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, 12N1
is equal to thenumber of divisors of 2n1. Since the number of divisors of 2n1 is d
2n1 ,
the numberof partitionsof n with equal parts in set
M1
n k, 12N1
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 k n tk
11
1 1
1 if |
, where
0 otherwise r t
k n
M p k n tk
1 1 1 1 1 1 11 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
2k1,n
M p n1 r
2kr
Proof: Let n( , , ... , ), 1 2 r i 2k i, be any rM partition1 of nsuch that even
numbers not repeated and smallest parts are odd numbers. Subtracting 2k from each part, we get n2kr
12 ,k
22 ,...,k
r 2k
Hence n2kr
12 ,k
22 ,...,k
r 2k
is a rM partition1 of n2kr with even parts not repeated and smallest parts are odd.Therefore the number of rM partitions1 ofnwith parts greater than or equal to 2k1is
1
2 .
r
M p n kr
Hence M p1 r
2k1,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
11 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
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 12 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
11 1
1 2 1 , 2 1
k t
c M spt n M p c k n c k t
1 1 11 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 c12c1,cNProof: Let n( , , ... , ), 1 2 r i 2c k1 i, 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 n2c kr1
12c k1 ,
22c k1 ,...,
r 2c k1
Hence n2c kr1
12c k1 ,
22c k1 ,...,
r2c k1
is a 1rM partitionofn2c 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
Hence M p1 r
2c k1 1,n
M p n1 r
2c kr1
where c1 2c1,cN2.6.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