A formula for the r -coloured partition function in terms of the sum of divisors function and its inverse
aa r X i v : . [ m a t h . G M ] A ug A FORMULA FOR THE r -COLOURED PARTITION FUNCTION INTERMS OF THE SUM OF DIVISORS FUNCTION AND ITS INVERSE SUMIT KUMAR JHA
Abstract.
Let p − r ( n ) denote the r -coloured partition function, and σ ( n ) = P d | n d denote the sum of positive divisors of n . The aim of this note is to prove the following p − r ( n ) = θ ( n )+ n − X k =1 r k +1 ( k + 1)! n − X α = k α − X α = k − · · · α k − − X α k =1 θ ( n − α ) θ ( α − α ) · · · θ ( α k − − α k ) θ ( α k )where θ ( n ) = n − σ ( n ), and its inverse σ ( n ) = n n X r =1 ( − r − r (cid:18) nr (cid:19) p − r ( n ) . Main results
Definition 1. [1] A partition of a positive integer n is a finite weakly decreasing sequenceof positive integers λ ≥ λ ≥ · · · ≥ λ r > P ri =1 λ i = n . The λ i ’s are called the parts of the partition. Let p ( n ) denote the number of partitions of n .A partition is said to be r -coloured if each part can occur as r colours. Let p − r ( n ) denotethe number of r coloured partitions of n . The generating function for p − r ( n ) is given by ∞ X n =0 p − r ( n ) q n = E ( q ) − r where E ( q ) := ∞ Y j =1 (1 − q j )where | q | < σ ( n ) = P d | n d denote the sum of divisors of a positive integer n .We first prove the following. Mathematics Subject Classification.
Key words and phrases.
Partition function; Divisor function; Bell polynomials; r-coloured Partition.
Theorem 1.
For all positive integers n ≥ we have p − r ( n ) = θ ( n ) + n − X k =1 r k +1 ( k + 1)! n − X α = k α − X α = k − · · · α k − − X α k =1 θ ( n − α ) θ ( α − α ) · · · θ ( α k − − α k ) θ ( α k )(1) where θ ( n ) = n − σ ( n ) . Lemma 1.
We have − log( E ( q )) = ∞ X n =1 θ ( n ) q n . Proof.
It is easy to see that log( E ( q )) = ∞ X j =1 log(1 − q j )= − ∞ X j =1 ∞ X l =1 q lj l = − ∞ X n =1 q n X d | n d . This completes the proof. (cid:3)
Definition 2.
For n and k non-negative integers, the partial ( n, k )th partial Bell polyno-mials in the variables x , x , . . . , x n − k +1 denoted by B n,k ≡ B n,k ( x , x , . . . , x n − k +1 ) [3, p.206] can be defined byB n,k ( x , x , . . . , x n − k +1 ) = X ≤ i ≤ n,ℓ i ∈ N P ni =1 iℓ i = n P ni =1 ℓ i = k n ! Q n − k +1 i =1 ℓ i ! n − k +1 Y i =1 (cid:16) x i i ! (cid:17) ℓ i . Cvijovi´c [2] gives the following formula for calculating these polynomials B n,k +1 = 1( k + 1)! n − X α = k α − X α = k − · · · α k − − X α k =1 | {z } k k z }| {(cid:18) nα (cid:19)(cid:18) α α (cid:19) · · · (cid:18) α k − α k (cid:19) · x n − α x α − α · · · x α k − − α k x α k ( n ≥ k + 1 , k = 1 , , . . . ) (2) Lemma 2.
We have p − r ( n ) = 1 n ! n X k =1 r k B n,k (1! θ (1) , θ (2) , · · · , ( n − k + 1)! θ ( n − k + 1)) , (3) FORMULA FOR THE r -COLOURED PARTITION FUNCTION IN TERMS OF THE SUM OF DIVISORS FUNCTION AND ITS INVERSE3 where θ ( n ) = n − σ ( n ) .Proof. Let f ( q ) = e rq , and g ( q ) = − log( E ( q )). Using Fa`a di Bruno’s formula [3, p. 134]we have d n dq n f ( g ( q )) = n X k =1 f ( k ) ( g ( q )) · B n,k (cid:0) g ′ ( q ) , g ′′ ( q ) , . . . , g ( n − k +1) ( q ) (cid:1) . (4)Since f ( k ) ( q ) = r k e rq and g (0) = 1, letting q → (cid:3) Combining equations (3) and (2) we can conclude (1).Now we prove the following.
Theorem 2.
We have σ ( n ) = n n X r =1 ( − r − r (cid:18) nr (cid:19) p − r ( n ) . (5) Lemma 3.
We have θ ( n ) = 1 n ! n X k =1 ( − k − ( k − B n,k (1! p (1) , p (2) , · · · , ( n − k + 1)! p ( n − k + 1)) (6) where p ( n ) = p − ( n ) is the partition function.Proof. Let f ( q ) = log q , and g ( q ) = 1 /E ( q ). Then using Fa`a di Bruno’s formula (4) wehave d n dq n f ( g ( q )) = n X k =1 f ( k ) ( g ( q )) · B n,k (cid:0) g ′ ( q ) , g ′′ ( q ) , . . . , g ( n − k +1) ( q ) (cid:1) . Since f ( k ) ( q ) = ( − k − ( k − q k and g (0) = 1, letting q → (cid:3) Lemma 4.
We have for positive integers n, kB n,k (1! p (1) , p (2) , · · · , ( n − k + 1)! p ( n − k + 1)) = n ! k ! k X r =1 ( − k − r (cid:18) kr (cid:19) p − r ( n ) (7) Proof.
We start with the generating function for the partial Bell polynomials as follows ∞ X n = k B n,k (1! p (1) , p (2) , · · · , ( n − k + 1)! p ( n − k + 1)) q n n ! = 1 k ! ∞ X j =1 p ( j ) q j ! k = 1 k ! ( E ( q ) − − k = 1 k ! k X r =0 ( − k − r (cid:18) kr (cid:19) E ( q ) − r SUMIT KUMAR JHA = 1 k ! k X r =0 ( − k − r (cid:18) kr (cid:19) ∞ X n =0 p − r ( n ) q n to conclude the equation (7). (cid:3) Proof of Theorem 2.
Combining equations (6) and (7) we have X d | n d = n X k =1 k k X r =1 ( − r − (cid:18) kr (cid:19) p − r ( n )= n X r =1 ( − r − p − r ( n ) n X k = r k (cid:18) kr (cid:19) = n X r =1 ( − r − r (cid:18) nr (cid:19) p − r ( n ) . Now we can conclude our main result equation (5) after the fact that P d | n nd = P j | n j . (cid:3) References [1] S. Chern, S. Fu & D. Tang, Some inequalities for k-colored partition functions,
Ramanujan J (2011),1544–1547.[3] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions , D. Reidel PublishingCo., Dordrecht, 1974.
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