Relation Between the Smallest and the Greatest Parts of the Partitions of n

In this paper the formulae for the number of smallest parts of partitions of n ∈ N and relations between the ith smallest parts and the ith greatest parts are obtained.


Introduction
We adopt the common notation on partitions as used in (Andrews, G.E., 1976) and (Andrews, G. E., to appear).A partition of a positive integer n is a finite non-increasing sequence of positive integers λ 1 , λ 2 , ...λ r such that r i = 1 λ i = n and it is denoted by n = (λ 1 , λ 2 , ...λ r ) .The λ i are called the parts of the partition.The number of parts of λ is called the length of λ, and is denoted by l (λ) .λ 1 − I (λ) is called the rank of the partition.Throughout this paper, λ stands for a partition of n, λ = (λ 1 , λ 2 , ...λ r ) , λ 1 ≥ λ 2 ≥ ... ≥ λ r .
Let spt (n) denote the number of smallest parts including repetitions in all partitions of n.For i ≥ 1 let us adopt the following notation n s (λ) = number of smallest parts of λ.
If μ 1 , μ 2 , ..., μ k are the distinct parts in λ where μ 1 > μ 2 > ... > μ k , then μ i = g i = ith greatest part of λ, μ k−i+1 = ith smallest parts of λ, and n s i (λ) is the number of ith smallest parts in λ, that is, the number of μ k−i+1 's in λ.We write s i (λ) = μ k−i+1 where t i = s i − s i−1 if s i and s i−1 both exist, t i = s i if s i exists but s i−1 does not exist, and t i = 0 if s i does not exist but s i−1 exists.Let spt i (n) denote the number of ith smallest parts in all partitions of n and Dually, we define the i th greatest part of λ, gpt (λ) , n g (λ) , ith greatest part g i (λ) of λ and sum (g i ) .
In Table 1, we provide a list of the partitions of 6 with their corresponding s Let ξ (n) denote the set of all partitions of n and p (n) be the cardinality of ξ (n) for n ∈ N and p (0) = 1.If 1 ≤ r ≤ n we write p r (n) for the number of partitions of n each consisting of exactly r parts, that is, r-partitions of n.If r ≤ 0 or r ≥ n, we write p r (n) = 0 and ξ * (n) denotes the set of all conjugate partitions of n.Also let p (k, n) represent the number of partitions of n using natural numbers at least as large as k only, and let G (s, n) denote the number of partitions of n having greatest part n , as in (Atkin, A.O.L et al, 2003) and (Bringmann, K. et al,to appear) Andrews (Andrews, G. E., to appear) proved analytically that In this paper, we give a proof of the theorem for the relation between the ith smallest parts and ith greatest parts of the partitions of the positive integer n.In particular, we show that Proof.The Ferrer diagram for λ consists of r rows of dots, with the ith row having λ i dots.Thus clearly, the columns also have dots in decreasing numbers.Hence the rows in λ * have dots in decreasing numbers.Since λ, λ * have the same number of dots, it follows that Theorem 5.For 1 ≤ i ≤ n, let g i be the ith greatest part of λ.Then Proof.From theorem 1, we have that ξ Theorem 6.For each i, if (ii) the (i + 1) th greatest part does not exist, then n s i (λ) = g * i .
(iii) the ith greatest part does not exist, then n s i (λ) = 0.
The Ferrer diagram of λ and λ * can be partitioned into rows having equal numbers of dots which can expressed in matrix form as shown in Figure 3.We observe that the number of rows in the ith matrix from bottom to top of the diagram for λ is equal to the number for the ith smallest parts of λ.Also, the number of columns in the ith matrix from top to bottom of the diagram for λ * is equal to the ith greatest part of λ * .We also observe that (i) If both the ith and the (i + 1) th greatest parts of λ * exist, then the difference between the ith and the (i + 1) th greatest parts of λ * is equal to the number of the ith smallest parts of λ.
(ii) If the ith greatest part of λ * exists and the (i + 1) th greatest parts of λ * do not exist, then the value of the ith greatest part of λ * is equal to the number of the ith smallest part of λ.
(iii) If the ith greatest part of λ * does not exist and the (i + 1) th greatest part of λ * exists, then the difference between the ith and the (i + 1) th greatest parts of λ * is equal to zero. Hence and from theorem 2, we have that As a consequence of theorem 4, we have the following corollary: Corollary 1.Let g 1 , g 2 be the first and second greatest parts of λ respectively.Then Proof.From theorem 3, the ith matrix from top to bottom in the Ferrer diagram is the same as the (k − i + 1) th matrix from bottom to top.Hence Proof.From theorem 4, we have that As a consequence of theorem 6, we have the following corollary: Corollary 2. Let s 1 be the smallest parts of λ.Then s t−1 (λ) , since when s i (λ) does not exist, then s t (λ)−s t−1 (λ) = 0.
Proof.From (Reddy, K. H., 2010), for every g, we have If the greatest parts g 1 appear k 1 times followed by its successor g 2 , then The sum of these second greatest parts taken over all partitions is The theorem follows by repeating this process ...
Proof.By induction.
Theorem 13.The number of r-partitions of n having k as a smallest part is j where j = 1 if r divides n, otherwise j = 0.
Proof.From (Reddy, K.H., 2010), the number of r-partitions of n with smallest part k is We fix k ∈ {1, 2, ..., n} .For 1 ≤ i ≤ r, the number of r-partitions of n with (r − i) smallest parts each being k is the number of i-partitions of n − (r − i) k.Summing over i = 1 to r, we get the total number of r-partitions of n with smallest parts k.This number is where j = 1 if r divides n, otherwise j = 0.
As k varies from 1 to n, we have the following corollaries: Corollary 4. The total number of r-partitions is where j = 1 if r divides n, otherwise j = 0.
Corollary 5.By taking the sum as r varies, we get where j = 1 if r divides n, otherwise j = 0.
We now independently derive another formula for spt (n) .
Proof.Any partition in ξ (n) has a smallest part which possibly repeats.
(i) If the smallest part d is a divisor of n, then the number of partitions with d as a smallest part is 1 + where 1 corresponds to the partition d, d, ... up to n d times .
(ii) If d is not a divisor of n there is no partition with equal parts.In this case the total partitions with d as smallest part is As a consequence of theorem 11, we have d 1 (n − ts) = number of divisors of n − ts that are greater than s From theorem 10, where j = 1 if r divides n, otherwise j = 0. From equations (1) and (2) we have As a consequence of theorem 12, we have the following t 1 s 1 − t 2 s 2 − ... − t i−1 s i−1 ) Theorem 15.If {a, m, n, r, S } ⊂ N, b ∈ Z, r| n − br a and S = {am + b|m = 1, 2, ..., n} , then spt (S ; n) − 1) r − 1 − i ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦Proof.From (Reddy, K. H., 2010), if a|n − br and n − br > 0, then p r ("S ", n) = p r n