Volume 10, Issue 1

We discuss a new restricted m -ary overpartition function , which is the number of hyper m -ary overpartitions of n , such that each power of m is allowed to be used at most m times as a non-overlined part. In this note we use generating function dissections to prove the following family of congruences for all n ≥ 0, m ≥ 4, j ≥ 0, 3 ≤ k ≤ m – 1, and t ≥ 1:

We opt to study the convergence of maximal real roots of certain Fibonacci-type polynomials given by G n = x k G n – 1 + G n – 2 . The special cases k = 1 and k = 2 were done by Moore, and Zeleke and Molina, respectively.

This paper presents a new proof that if k α is irrational then the sequence {⌊ α + log k n ⌋} n ≥1 is not k -regular. Unlike previous proofs, the methods used do not rely on automata or language theoretic concepts. The paper also proves the stronger statement that if k α is irrational then the generating function in k non-commuting variables associated with {⌊ α + log k n ⌋} n ≥1 is not algebraic.

In this work, we state and prove Lerch's theorems for Fermat and Euler quotients over function fields defined analogously to the number fields.

Let α > 1 be an algebraic number and ξ > 0. Denote the fractional parts of ξα n by { ξα n }. In this paper, we estimate a lower bound for the number λ N ( α , ξ ) of integers n with 0 ≤ n < N and Our results show, for example, the following: Let α be an algebraic integer with Mahler measure M( α) and ξ > 0 an algebraic number with ξ ∉ ℚ ( α ). Put [ℚ ( α , ξ ) : ℚ ( α )] = D . Then there exists an absolute constant c satisfying for all large N .

We consider the values of the consecutive minima of the quantities . W. Schmidt, in 1958, calculated the first and second minimum for j = 1 and d = 2. Schweiger, in 1975, considered the case j = 1 for any d ≥ 2. This note is a continuation of these investigations.

The partitions of a positive integer n in which the parts are in arithmetic progression possess interesting combinatorial properties that distinguish them from other classes of partitions. We exhibit the properties by analyzing partitions with respect to a fixed length of the arithmetic progressions. We also address an open question concerning the number of integers k for which there is a k -partition of n with parts in arithmetic progression.

Jacobi's two-square theorem states that the number of representations of a positive integer k as a sum of two squares, counting order and sign, is 4 times the surplus of positive divisors of k congruent to 1 modulo 4 over those congruent to 3 modulo 4. In this paper we give numerous identities, each of which yields an analogue of Jacobi's result. Our identities are drawn from a much larger list in [Melham, Analogues of Jacobi's two-square theorem], and involve polygonal numbers. The formula for the n -th k -gonal number is F k = F k ( n ) = n (( k – 2) n – ( k – 4))/2.

We derive several new convolution identities for the Stirling numbers of the first kind. As a consequence we obtain a new linear recurrence relation which generalizes known relations.

Recently, designs of pseudorandom number generators (PRNGs) using integer-valued variants of logistic maps and their applications to certain cryptographic schemes have been studied, due mostly to their ease of implementation and performance. However, it has been noted that this ease is reduced for some choices of the PRNGs accuracy parameters. In this article, we show that the distribution of such undesirable accuracy parameters is closely related to the occurrence of some patterns in the dyadic expansion of the square root of 2. We prove that for an arbitrary infinite binary word, the asymptotic occurrence rate of these patterns is bounded in terms of the asymptotic occurrence rate of zeroes. As a consequence, a classical conjecture on asymptotic evenness of occurrence of zeroes and ones in the dyadic expansion of the square root of 2 implies that the asymptotic rate of the undesirable accuracy parameters for the PRNGs is at least 1/6.

Motivated by the study of Mahonian statistics, in 2000, Babson and Steingrímsson [Sém. Lothar. Comb] introduced the notion of a “generalized permutation pattern” (GP) which generalizes the concept of “classical” permutation pattern introduced by Knuth in 1969. The invention of GPs led to a large number of publications related to properties of these patterns in permutations and words. Since the work of Babson and Steingrímsson, several further generalizations of permutation patterns have appeared in the literature, each bringing a new set of permutation or word pattern problems and often new connections with other combinatorial objects and disciplines. For example, Bousquet-Mélou et al. [J. Comb. Theory A] introduced a new type of permutation pattern that allowed them to relate permutation patterns theory to the theory of partially ordered sets. In this paper we introduce yet another, more general definition of a pattern, called place-difference-value patterns (PDVP) that covers all of the most common definitions of permutation and/or word patterns that have occurred in the literature. PDVPs provide many new ways to develop the theory of patterns in permutations and words. We shall give several examples of PDVPs in both permutations and words that cannot be described in terms of any other pattern conditions that have been introduced previously. Finally, we discuss several bijective questions linking our patterns to other combinatorial objects.

We investigate properties of a multivariate function E ( m 1 , m 2 , …, m r ), called orbicyclic , that arises in enumerative combinatorics in counting non-isomorphic maps on orientable surfaces. E ( m 1 , m 2 , …, m r ) proves to be multiplicative, and a simple formula for its calculation is provided. It is shown that the necessary and sufficient conditions for this function to vanish are equivalent to familiar Harvey's conditions that characterize possible branching data of finite cyclic automorphism groups of Riemann surfaces.