Counting Ordered Pairs

Combinatorial proofs and generalizations of conjectures related to Euler’s partition theorem

In 1960, Sierpiński proved that there exist infinitely many odd positive integers k such that k · 2n + 1 is composite for all integers n ≥ 0. Variations of this problem, using polynomials with integer coefficients, and considering reducibility over the rationals, have been investigated by several authors. In particular, if S is the set of all positive integers d for which there exists a polynomial f(x) ∈ ℤ[x], with f(1) ≠ -d, such that f(x)xn + d is reducible over the rationals for all integers n ≥ 0, then what are the elements of S? Interest in this problem stems partially from the fact that if S contains an odd integer, then a question of Erdös and Selfridge concerning the existence of an odd covering of the integers would be resolved. Filaseta has shown that S contains all positive integers d ≡ 0 (mod 4), and until now, nothing else was known about the elements of S. In this paper, we show that S contains infinitely many positive integers d ≡ 6 (mod 12). We also consider the corresponding problem over 𝔽p, and in that situation, we show 1 ∈ S for all primes p.

In this article, we investigate the pairs of positive integers for which sum of their factorials divides the factorial of their sum and establish a bound on their difference. We also solve the divisibility question over the set of Fibonacci numbers. We conclude by proving that there are infinitely many such pairs of positive integers with difference 2 and conjecture that for any positive integer k there are infinitely many such pairs (a, b) with

Recently it was shown that the Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. In this paper, we introduce a similar representation for the Padovan numbers. As a corollary, we derive an infinite family of double inequalities.

The integral sum number of complete bipartite graphs Kr, s

Our main results are asymptotic zero-one laws satisfied by the diagrams of unimodal sequences of positive integers. These diagrams consist of columns of squares in the plane; the upper boundary is called the shape. For various types of unimodal sequences, we show that, as the number of squares tends to infinity, 100% of shapes are near a certain curve---that is, there is a single limit shape. Similar phenomena have been well-studied for integer partitions, but several technical difficulties arise in the extension of such asymptotic statistical laws to unimodal sequences. We develop a widely applicable method for obtaining these limit shapes, based in part on a method of Petrov. We also mention a few notable corollaries---for example, we obtain a limit shape for so-called "overpartitions'' by a simple DeSalvo-Pak-type transfer. To aid in the proof of these limit shapes, we prove an asymptotic formula for the number of partitions of the integer n into distinct parts where the largest part is at most t times the square root of n for fixed t. Our method follows a probabilistic approach of Romik, who gave a simpler proof of Szekeres' asymptotic formula for distinct parts partitions when instead the number of parts is bounded by t times the square root of n. The probabilistic approach is equivalent to a circle method/saddle-point method calculation, but it makes some of the steps more intuitive and even predicts the shape of the asymptotic formula, to some degree. Finally, motivated by certain problems concerning Rogers-Ramanujan-type identities, we give combinatorial proofs of three families of inequalities among certain types of integer partitions.

We prove a finite version of the well-known theorem that says that the number of partitions of an integer N into distinct parts is equal to the number of partitions of N into odd parts. Our version says that the number of “lecture hall partitions of length n ” of N equals the number of partitions of N into small odd parts: 1,3,5, ldots, 2n-1 . We give two proofs: one via Bott's formula for the Poincare series of the affine Coxeter group \(\tilde C_n \), and one direct proof.

In this paper, we are going to present sums over partitions. We define a function uₙ(g; q), which can be considered as one generalization of the q-Pochhammer symbol. Using this function uₙ(g; q), many results have been proved for sums of partitions into even distinct parts, odd distinct parts, 1 as the only part, and v = v as a part.

This paper is an extension of a recent work done by the author [4] and here the sums of alternating series of odd powers (up to fifteen) of the reciprocals of odd positive integers are computed. Following this method, the sum of the series of any higher power could be calculated. In the process of computing these sums, the sums of the series of even powers of reciprocals of odd positive integers have been reestablished and enabled the author to compute the values of Riemann’s zeta function for even positive integers.

The number of ways to distribute r identical objects into n identical boxes can usually be obtained by a method of generating function or by a recursive formula. In this paper, for another approach, it is shown that we can obtain this number by using generalization of Polya’s theorem. From this, we can also find the number of partitions of integer n as a sum of k positive integers. Computing for the results is discussed.

Asymptotic behavior in time of the solutions of three nonlinear partial differential equations

In his second notebook, Ramanujan discovered the following identity for the special values of ζ (s) at the odd positive integers where α and β are positive numbers such that αβ = π 2 and m is a positive integer. As shown by Berndt in the viewpoint of general transformation of analytic Eisenstein series, it is a natural companion of Euler’s famous formula for even zeta values. In this note, we prove an analogue of the above Ramanujan’s identity in the functions fields setting, which involves the Bernoulli-Carlitz numbers.

Let $\mathcal A$ denote a set of positive integers, and let $p(\mathcal A,n)$ denote the associated partition function. Let $\beta $ be an odd positive integer, and let $P(z)$ be a polynomial in $ \mathbb F_2[z]$ of order $\beta $ such that $P(0)=1$. J.-L

We discuss the numbers in the title, and in particular whether the minimum period of the Bell numbers modulo a prime p p can be a proper divisor of N p = ( p p − 1 ) / ( p − 1 ) N_p = (p^p-1)/(p-1) . It is known that the period always divides N p N_p . The period is shown to equal N p N_p for most primes p p below 180. The investigation leads to interesting new results about the possible prime factors of N p N_p . For example, we show that if p p is an odd positive integer and m m is a positive integer and q = 4 m 2 p + 1 q=4m^2 p+1 is prime, then q q divides p m 2 p − 1 p^{m^2p}-1 . Then we explain how this theorem influences the probability that q q divides N p N_p .

The aim of this paper is to classify all monogenic ternary semigroups, up to isomorphism. We divide them to two groups: finite and infinite. We show that every infinite monogenic ternary semigroup is isomorphic to the ternary semigroup O, the odd positive integers with ordinary addition. Then we prove that all finite monogenic ternary semigroups with the same index and the same period are isomorphic. We also investigate structure of finite monogenic ternary semigroups and we prove that any finite monogenic ternary semigroup is isomorphic to a quotient ternary semigroup.