All a b c d e f g h i j k l m n o p q r s t u v w x y z Other

Previous Page 2 Next

Displaying 21 – 40 of 286

Showing per page

A note on representation functions with different weights

Zhenhua Qu (2016)

Colloquium Mathematicae

For any positive integer k and any set A of nonnegative integers, let r 1 , k ( A , n ) denote the number of solutions (a₁,a₂) of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. Let k,l ≥ 2 be two distinct integers. We prove that there exists a set A ⊆ ℕ such that both r 1 , k ( A , n ) = r 1 , k ( ℕ ∖ A , n ) and r 1 , l ( A , n ) = r 1 , l ( ℕ ∖ A , n ) hold for all n ≥ n₀ if and only if log k/log l = a/b for some odd positive integers a,b, disproving a conjecture of Yang. We also show that for any set A ⊆ ℕ satisfying r 1 , k ( A , n ) = r 1 , k ( ℕ ∖ A , n ) for all n ≥ n₀, we have r 1 , k ( A , n ) → ∞ as n → ∞.

A note on the determinant of a Toeplitz-Hessenberg matrix

Mircea Merca (2013)

Special Matrices

The nth-order determinant of a Toeplitz-Hessenberg matrix is expressed as a sum over the integer partitions of n. Many combinatorial identities involving integer partitions and multinomial coefficients can be generated using this formula.

An application of Pólya’s enumeration theorem to partitions of subsets of positive integers

Xiao Jun Wu, Chong-Yun Chao (2005)

Czechoslovak Mathematical Journal

Let S be a non-empty subset of positive integers. A partition of a positive integer n into S is a finite nondecreasing sequence of positive integers a 1 , a 2 , ⋯ , a r in S with repetitions allowed such that ∑ i = 1 r a i = n . Here we apply Pólya’s enumeration theorem to find the number ( n ; S ) of partitions of n into S , and the number D P ( n ; S ) of distinct partitions of n into S . We also present recursive formulas for computing ( n ; S ) and D P ( n ; S ) .

Currently displaying 21 – 40 of 286