Qing Zou 1
1Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA
Abstract:

In this paper, we first give a new \( q \)-analogue of the Lah numbers. Then we show the irreducible factors of the \( q \)-Lah numbers over \( \mathbb{Z} \).

Octavio A. Agustín-Aquino 1
1Instituto de Física y Matemáticas, Universidad Tecnológica de la Mixteca, Carretera a Acatlima Km. 2.5, Huajuapan León, Oaxaca, México, C.P. 69000
Abstract:

Let \( A \) and \( B \) be additive sets of \( \mathbb{Z}_{2k} \), where \( A \) has cardinality \( k \) and \( B = v \cdot C A \) with \( v \in \mathbb{Z}_{2k}^\times \). In this note, some bounds for the cardinality of \( A + B \) are obtained using four different approaches. We also prove that in a special case, the bound is not sharp and we can recover the whole group as a sumset.

Guy Louchard 1
1Département d’Informatique, Université Libre de Bruxelles, CP 212, Boulevard du Triomphe, B-1050, Bruxelles, Belgium
Abstract:

In this paper, we analyze the asymptotic number \( I(m,n) \) of involutions of large size \( n \) with \( m \) singletons. We consider a central region and a non-central region. In the range \( m = n – n^\alpha \), \( 0 < \alpha < 1 \), we analyze the dependence of \( I(m,n) \) on \( \alpha \). This paper fits within the framework of Analytic Combinatorics.

Yu-Hong Guo1, Augustine O. Munagi2
1School of Mathematics and Statistics, Hexi University, Zhangye, Gansu, 734000, P.R.China
2The John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand, Johannesburg, South Africa, Augustine.
Abstract:

An inverse-conjugate composition of a positive integer \(m\) is an ordered partition of \(m\) whose conjugate coincides with its reversal. In this paper, we consider inverse-conjugate compositions in which the part sizes do not exceed a given integer \(k\). It is proved that the number of such inverse-conjugate compositions of \(2n – 1\) is equal to \(2F_n^{(k-1)}\), where \(F_n^{(k)}\) is a Fibonacci \(k\)-step number. We also give several connections with other types of compositions, and obtain some analogues of classical combinatorial identities.

Jay Pantone1
1Department of Mathematics, Dartmouth College, Hanover, New Hampshire USA
Abstract:

S. Ekhad and D. Zeilberger recently proved that the multivariate generating function for the number of simple singular vector tuples of a generic \(m_1 \times · · · \times m_d\) tensor has an elegant rational form involving elementary symmetric functions, and provided a partial conjecture for the asymptotic behavior of the cubical case \(m_1 = · · · = m_d\). We prove this conjecture and further identify completely the dominant asymptotic term, including the multiplicative constant. Finally, we use the method of differential approximants to conjecture that the subdominant connective constant effect observed by Ekhad and Zeilberger for a particular case in fact occurs more generally

Istvan Mező1, José Ramírez2
1Department of Mathematics, Nanjing University of Information Science and Technology, Nanjing 210044, P. R. CHINA.
2Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, COLOMBIA
Abstract:

In this paper, we define the q-analogue of the so-called symmetric infinite matrix algorithm. We find an explicit formula for entries in the associated matrix and also for the generating function of the k-th row of this matrix for each fixed k. This helps us to derive analytic and number theoretic identities with respect to the q-harmonic numbers and q-hyperharmonic numbers of Mansour and Shattuck.

Aubrey Blecher1, Charlotte Brennan1, Arnold Knopfmacher 1
1The John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa
Abstract:

Bargraphs are lattice paths in \(\mathbb{N}_0^2\) with three allowed types of steps: up \((0,1)\), down \((0,-1)\), and horizontal \((1,0)\). They start at the origin with an up step and terminate immediately upon return to the \(x\)-axis. A wall of size \(r\) is a maximal sequence of \(r\) adjacent up steps. In this paper, we develop the generating function for the total number of walls of fixed size \(r \geq 1\). We then derive asymptotic estimates for the mean number of such walls.

Sean Prendiville1
1School of Mathematics, University of Manchester, Manchester, UK
Abstract:

We survey four instances of the Fourier analytic ‘transference principle’or ‘dense model lemma’, which allows one to approximate an unbounded function on the integers by a bounded function with similar Fourier transform. Such a result forms a component of a general method pioneered by Green to count solutions to a single linear equation in a sparse subset of integers.

Giorgis Petridis1
1Department of Mathematics, University of Georgia, Athens, GA 30602, USA
Abstract:

Let \( E \subseteq \mathbb{F}_q^2 \) be a set in the 2-dimensional vector space over a finite field with \( q \) elements, which satisfies \(|E| > q\). There exist \( x, y \in E \) such that \(|E \cdot (y – x)| > q/2\). In particular, \( (E + E) \cdot (E – E) = \mathbb{F}_q.\)

Abstract:

Let \((x(n))_{n \geq 1}\) be an \(s\)-dimensional Niederreiter-Xing sequence in base \(b\). Let \(D((x(n))_{n=1}^N)\) be the discrepancy of the sequence \((x(n))_{n=1}^N\). It is known that \(ND((x(n))_{n=1}^N) = O(\ln^s N)\) as \(N \to \infty\). In this paper, we prove that this estimate is exact. Namely, there exists a constant \(K > 0\), such that
\[
\inf_{w \in [0,1]^s} \sup_{1 \leq N \leq b^m} ND((x(n) \oplus w)_{n=1}^N) \geq K \ln^s \quad \text{ for } m = 1, 2, \ldots.
\]

We also get similar results for other explicit constructions of \((t,s)\)-sequences.

Maxie D. Schmidt1
1School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332
Abstract:

We define a new class of generating function transformations related to polylogarithm functions, Dirichlet series, and Euler sums. These transformations are given by an infinite sum over the jth derivatives of a sequence generating function and sets of generalized coefficients satisfying a non-triangular recurrence relation in two variables. The generalized transformation coefficients share a number of analogous properties with the Stirling numbers of the second kind and the known harmonic number expansions of the unsigned Stirling numbers of the first kind.

We prove a number of properties of the generalized coefficients which lead to new recurrence relations and summation identities for the k-order harmonic number sequences. Other  applications of the generating function transformations we define in the article include new series expansions for the polylogarithm function, the alternating zeta function, and the Fourier series for the periodic Bernoulli polynomials. We conclude the article with a discussion of several specific new “almost” linear recurrence relations between the integer-order harmonic numbers and the generalized transformation coefficients, which provide new applications to studying the limiting behavior of the zeta function constants, ζ(k), at integers k ≥ 2.

Ali Boussayoud1
1LMAM Laboratory and Department of Mathematics, Mohamed Seddik Ben Yahia University, Jijel, Algeria.
Abstract:

Generating functions for Pell and Pell-Lucas numbers are obtained. Applications are given for some results recently obtained by Mansour [Mansour12]; by using an alternative approach that considers the action of the operator \(\delta_{e_1 e_2}^k\) to the series \(\sum_{j=0}^\infty a_j (e_1 z)^j\).

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