Mircea Merca

Mircea Merca
Polytechnic University of Bucharest | UPB · Department of Mathematics

Dr. habil.

About

149
Publications
29,659
Reads
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942
Citations
Citations since 2017
102 Research Items
838 Citations
2017201820192020202120222023050100150200
2017201820192020202120222023050100150200
2017201820192020202120222023050100150200
2017201820192020202120222023050100150200
Introduction
My research centers on the theory of partitions, number theory, combinatorics and special functions.
Additional affiliations
July 2016 - May 2017
University of Craiova
Position
  • Research Associate
October 2011 - September 2014
University of Craiova
Position
  • PhD Student
Education
October 2011 - September 2014
University of Craiova
Field of study
  • Mathematics
October 2001 - June 2003
Polytechnic University of Bucharest
Field of study
  • Informatics
October 1987 - June 1991
Babeş-Bolyai University
Field of study
  • Mathematics

Publications

Publications (149)
Article
Full-text available
In this paper, we show that some classical results from q-analysis and partition theory are specializations of the fundamental relationships between complete and elementary symmetric functions.
Preprint
Full-text available
Let $b_3(n)$ be the number of $3$-regular partitions of $n$. Recently, W. J. Keith and F. Zanello discovered infinite families of Ramanujan type congruences modulo $2$ for $b_3(2n)$ involving every prime $p$ with $p \equiv 13, 17, 19, 23 \pmod {24}$, and O. X. M. Yao provided new infinite families of Ramanujan type congruences modulo $2$ for $b_3(2...
Article
Let S1 and S2 be two subsets of the positive integers. G. E. Andrews called (S1,S2) an Euler pair whenever q(S1;n)=p(S2;n) for all positive integers n, where q(S; n) denotes the number of partitions of n into distinct parts taken from S and p(S; n) denotes the number of partitions of n into parts taken from S. A well known example of such pair is d...
Article
The crank is a partition statistic requested by Dyson in 1944 in order to combinatorially prove a Ramanujan congruence for Euler’s partition function p(n). In this paper, we provide connections between Dyson’s crank and unimodal compositions. Somewhat unrelated, we give a combinatorial proof of a new truncated Euler pentagonal number theorem due to...
Article
Full-text available
The partitions in which even parts come in two colours are known as cubic partitions. In this paper, we introduce and investigate the cubic partition function A(n) which is defined as the difference between the number of cubic partitions of n into an even numbers of parts and the number of cubic partitions of n into an odd numbers of parts. We pres...
Article
Full-text available
In this paper, we explore Ramanujan-type congruences modulo 4 for the function σ0(n), counting the positive divisors of n. We consider relations of the form σ08(αn+β)+r≡0(mod4), with (α,β)∈N2 and r∈{1,3,5,7}. In this context, some conjectures are made and some Ramanujan-type congruences involving overpartitions are obtained.
Article
Full-text available
Let \(\overline{ A}_\ell (n)\) be the number of \(\ell \)-regular overpartitions of n, i.e., overpartitions of n into parts not divisible by \(\ell \). Let \(\overline{ B}_\ell (n)\) be the number of almost \(\ell \)-regular overpartitions of n, i.e., overpartitions of n in which none of its overlined parts is divisible by \(\ell \). In this paper,...
Article
The study of Ramanujan-type congruences for functions specific to additive number theory has a long and rich history. Motivated by recent connections between divisor sums and overpartitions via congruences in arithmetic progressions, we investigate the existence and classification of Ramanujan-type congruences for functions in multiplicative number...
Article
Full-text available
Linear inequalities involving Euler’s partition function p(n) have been the subject of recent studies. In this article, we consider the partition function Q(n) counting the partitions of n into distinct parts. Using truncated theta series, we provide four infinite families of linear inequalities for Q(n) and partition theoretic interpretations for...
Preprint
Full-text available
The crank is a partition statistic requested by Dyson in 1944 in order to combinatorially prove a Ramanujan congruence of Euler's partition function $p(n)$. In this paper, we provide connections between Dyson's crank and unimodal compositions. Somewhat unrelated, we give a combinatorial proof of a new truncated Euler pentagonal number theorem due t...
Article
Full-text available
We give combinatorial proofs for the generalizations of Stanley’s Theorem given in Andrews and Merca (Math Stud 89(1–2): 175–180, 2020). These involve the total number bk(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek}...
Article
Full-text available
In this paper, we investigate two methods to express the natural powers of 2 as sums over integer partitions. First we consider a formula by N. J. Fine that allows us to express a binomial coefficient in terms of multinomial coefficients as a sum over partitions. The second method invokes the central binomial coefficients and the logarithmic differentiati...
Preprint
Full-text available
The study of Ramanujan-type congruences for functions specific to additive number theory has a long and rich history. Motivated by recent connections between divisor sums and overpartitions via congruences in arithmetic progressions, we investigate the existence and classification of Ramanujan-type congruences for functions in multiplicative number...
Article
Full-text available
In this paper we show that the reciprocal of the finite product (q;q)n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(q;q)_n$$\end{document} can be expressed as a sum...
Preprint
Full-text available
The partition function $pod(n)$ enumerates the partitions of $n$ wherein odd parts are distinct and even parts are unrestricted. Recently, a number of properties for $pod(n)$ have been established. In this paper, for $k\in\{0,2\}$ we consider the partitions of $n$ into distinct parts not congruent to $k$ modulo $4$ and the $4$-regular partitions of...
Article
Full-text available
In this paper, the author gives two linear homogeneous recurrence relations to compute the values of Euler’s partition function p(n) when n is odd. These recurrences do not involve the generalized pentagonal numbers and provide new connections between the partition function p(n) and the positive divisor function σx(n) for x = 0 and x = 1. Some rela...
Article
Full-text available
In 1963, Peter Hagis, Jr. provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the partition function $Q(n)$ which counts partitions of $n$ into distinct parts. Computing $Q(n)$ by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this...
Chapter
Let \(\alpha \) and \(\beta \) be two nonnegative integers such that \(\beta < \alpha \). For an arbitrary sequence \(\{a_n\}_{n\geqslant 1}\) of complex numbers, we investigate linear combinations of the form \(\sum _{k\geqslant 1} S(\alpha k-\beta ,n) a_k\), where S(k, n) is the total number of k’s in all the partitions of n into parts not congru...
Chapter
Full-text available
We state and prove product formulae for several generating functions for sequences \((a_n)_{n\ge 0}\) that are defined by the property that \(Pa_n+b^2\) is a square, where P and b are given integers. In particular, we prove corresponding conjectures of the second author. We show that, by means of the Jacobi triple product identity, all these genera...
Article
An alignment of a permutation π on n letters is an ordered sequence of the disjoint cycles of π. We consider several counting numbers related to alignments: the number of alignments of n and its mean, the number of cycles in all alignments of n and its mean, as well as the mean number of supernecklaces of type III (cycles of cycles) of n. We presen...
Article
Full-text available
Waring’s formula expresses the power sum symmetric functions in terms of the elementary symmetric functions. In 1996, Konvalina gives a generalization of Waring’s formula and expresses monomial symmetric functions with equal exponents in terms of the elementary symmetric functions. Computing the generalized Waring coefficients by Konvalina’s formul...
Article
Full-text available
In this paper, we show that the difference between the number of parts in the odd partitions of n and the number of parts in the distinct partitions of n satisfies Euler’s recurrence relation for the partition function p(n) when n is odd. A decomposition of this difference in terms of the total number of parts in all the partitions of n is also deriv...
Article
We consider the squares of the Rogers-Ramanujan functions and for each $S\in\{1,2\}$ we obtain a linear recurrence relation for the number of partitions of $n$ into parts congruent to $\pm S\bmod{5}$. In this context, we conjecture that for $1\leqslant S < R$, $k\geqslant 1$, the theta series $$\frac{(-1)^k}{(q^S,q^{R-S};q^R)_\infty} \sum_{j=k}^{\i...
Article
Full-text available
Let denote the number of overpartitions of n into odd parts. In this paper, we provide a complete characterization of Ramanujan-type congruences modulo 8 for the overpartition function considering the number of odd positive divisors of and the relations of the form with and
Article
We investigate the sum of the parts in all the partitions of n into distinct parts and give two infinite families of linear inequalities involving this sum. The results can be seen as new connections between partitions and divisors.
Article
In 1939, H. S. Zuckerman provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the overpartition function p ( n ). Computing p ( n ) by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we provide a formula to compute the va...
Article
Full-text available
The main result of this paper is an identity expressing the r-Stirling number of the first kind as a sum involving binomial coefficients and the Möbius function of the set-partition lattice. We provide three different proofs of this result: analytic, inductive, and combinatorial.
Article
Full-text available
Let \(B_{k,r}(n)\) be the number of partitions of the form \(n=b_1+b_2+\cdots +b_s\), where \(b_i-b_{i+k-1}\hbox {\,\char 062\,}2\) and at most \(r-1\) of the \(b_i\) are equal to 1. In this paper, we prove that the numbers \(B_{k,r}(n)\) can be expressed in terms of Euler’s partition function p(n) considering generalized \((2m+3)\)-polygonal numbe...
Article
The restricted partitions in which the largest part is less than or equal to $N$ and the number of parts is less than or equal to $k$ were investigated by Andrews. These partitions were extended recently by the author to partitions into parts of two kinds. In this paper, we use a new class of restricted partitions into parts of two kinds to provide...
Article
In this paper, we show that the geometric polynomials can be expressed as sums over integer partitions in two different ways. New formulas involving geometric numbers, Bernoulli numbers, and Genocchi numbers are derived in this context.
Article
The minimal excludant of a partition [Formula: see text], [Formula: see text], is the smallest positive integer that is not a part of [Formula: see text]. For a positive integer [Formula: see text], [Formula: see text] denotes the sum of the minimal excludants of all partitions of [Formula: see text]. Recently, Andrews and Newman obtained a new com...
Article
We consider the partitions of n into parts not congruent to 0, \(\pm 3\pmod {12}\) and provide some new results related to the number of these partitions. In this context, we derive a new parity result involving sums of partition numbers p(n) and squares in arithmetic progressions: for \(n\geqslant 0\), $$\begin{aligned} \sum _{8k+1\text { square}}...
Preprint
Full-text available
Let $\alpha$ and $\beta$ be two nonnegative integers such that $\beta < \alpha$. For an arbitrary sequence $\{a_n\}_{n\geqslant 1}$ of complex numbers, we consider the generalized Lambert series in order to investigate linear combinations of the form $\sum_{k\geqslant 1} S(\alpha k-\beta,n) a_k$, where $S(k,n)$ is the total number of non-overlined...
Article
Full-text available
We consider the number of k’s in all the partitions of n, and provide new connections between partitions and functions from multiplicative number theory. In this context, we obtain a new generalization of Euler’s pentagonal number theorem and provide combinatorial interpretations for some special cases of this general result.
Article
Full-text available
Recently, Andrews and Merca considered the number of even parts in all partitions of n into distinct parts and obtained new combinatorial interpretations for this number. Their proofs rely on generating functions. In this paper, we provide purely combinatorial proofs of these results.
Chapter
We examine two truncated series derived from a classical theta identity of Gauss. As a consequence, we obtain two infinite families of inequalities for the overpartition function \( \mathit \overline{p_{o}} \) counting the number of overpartitions into odd parts. We provide partition-theoretic interpretations of these results.
Article
Full-text available
In 1944, Freeman Dyson defined the concept of rank of an integer partition and introduced without definition the term of crank of an integer partition. A definition for the crank satisfying the properties hypothesized for it by Dyson was discovered in 1988 by G.E. Andrews and F.G. Garvan. In this paper, we introduce truncated forms for two theta id...
Article
In this paper, the author considered two specializations of the identity q-Chu Vandermonde and derived two recurrence relations for the number of partitions of n into m parts with the smallest part greater than or equal to k and the minimal difference d.
Article
The Mathematics Student, 89 (1-2) 175-180 (2020)
Preprint
In this paper, we investigate two methods to express the natural powers of $2$ as sums over integer partitions. First we consider a formula by N. J. Fine that allows us to express a binomial coefficient in terms of multinomial coefficients as a sum over partitions. The second method invokes the central binomial coefficients and the logarithmic diff...
Article
In this paper, we investigate the sum of distinct parts that appear at least 2 times in all the partitions of [Formula: see text] providing new combinatorial interpretations for this sum. A connection with subsets of [Formula: see text] is given in this context. We provide two different proofs of our results: analytic and combinatorial. In addition...
Article
Full-text available
The minimal excludant of a partition λ, mex(λ), is the smallest positive integer that is not a part of λ. For a positive integer n, σ mex(n) denotes the sum of the minimal excludants of all partitions of n. Recently, Andrews and Newman obtained a new combinatorial interpretation for σ mex(n). They showed, using generating functions , that σ mex(n)...
Article
For each \(s\in \{2,4\}\), the generating function of \(R_s(n)\), the number of partitions of n into odd parts or congruent to 0, \(\pm \, s\pmod {10}\), arises naturally in regime III of Rodney Baxter’s solution of the hard-hexagon model of statistical mechanics. For each \(s\in \{1,3\}\), the generating function of \(R^*_s(n)\), the number of par...
Preprint
Full-text available
Linear inequalities involving Euler's partition function $p(n)$ have been the subject of recent studies. In this article, we consider the partition function $Q(n)$ counting the partitions of $n$ into distinct parts. Using truncated theta series, we provide four infinite families of linear inequalities for $Q(n)$ and partition theoretic interpretati...
Preprint
Full-text available
In $1944$, Freeman Dyson defined the concept of rank of an integer partition and introduced without definition the term of crank of an integer partition. A definition for the crank satisfying the properties hypothesized for it by Dyson was discovered in 1988 by G. E. Andrews and F. G. Garvan. In this paper, we introduce truncated forms for two thet...
Preprint
In 1939, H. S. Zuckerman provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the overpartition function $\overline{p}(n)$. Computing $\overline{p}(n)$ by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we provide a formu...
Preprint
For each $s\in\{2,4\}$, the generating function of $R_s(n)$, the number of partitions of $n$ into odd parts or congruent to $0$, $\pm s\pmod {10}$, arises naturally in regime III of Rodney Baxter's solution of the hard-hexagon model of statistical mechanics. For each $s\in\{1,3\}$, the generating function of $R^*_s(n)$, the number of partitions of...
Article
Full-text available
The least $r$-gap, $g_r(\lambda)$, of a partition $\lambda$ is the smallest part of $\lambda$ appearing less than $r$ times. In this article we introduce two new partition functions involving least $r$-gaps. We consider a bisection of a classical theta identity and prove new identities relating Euler's partition function $p(n)$, polygonal numbers,...
Preprint
Full-text available
The restricted partitions in which the largest part is less than or equal to $N$ and the number of parts is less than or equal to $k$ were investigated by Andrews in \cite{Andrews76}. These partitions were extended recently by the author to the partitions into parts of two kinds. In this paper, we use a new class of restricted partitions into parts...
Preprint
In this paper, we show that the difference between the number of parts in the odd partitions of $n$ and the number of parts in the distinct partitions of $n$ satisfies Euler's recurrence relation for the partition function $p(n)$ when $n$ is odd. A decomposition of this difference in terms of the total number of parts in all the partitions of $n$ i...
Preprint
Full-text available
In this paper, we consider the generating functions of the complete and elementary symmetric functions and provide a new generalization of these classical symmetric functions. Some classical relationships involving the complete and elementary symmetric functions are reformulated in a more general context. Combinatorial interpretations of these gene...
Article
We introduce new relationships between complete and elementary symmetric functions. As specializations of these relations, we obtain new identities of Guo and Yang type. An open problem involving the Gaussian polynomials is derived in this context: For \(n>0\), the expression $$\begin{aligned} \frac{(-1)^{n-1}}{(q;q)_n} \sum _{k=0}^{\lfloor (n-1)/2...
Article
Full-text available
We prove new variants of the Lambert series factorization theorems studied by Merca and Schmidt (2017) which correspond to a more general class of Lambert series expansions of the form $L_a(\alpha, \beta, q) := \sum_{n \geq 1} a_n q^{\alpha n-\beta} / (1-q^{\alpha n-\beta})$ for integers $\alpha, \beta$ defined such that $\alpha \geq 1$ and $0 \leq...
Article
Full-text available
An asymptotic classification for the linear homogeneous partition inequalities of the form ∑i=1rp(n+xi)⩽∑i=1sp(n+yi) has recently been introduced. In this paper, we investigate partition inequalities of this form when r=s. From an asymptotic point of view, such partition inequalities are considered to be non-trivial because they have the same numbe...
Article
A famous theorem of Euler asserts that there are as many partitions of n into distinct parts as there are partitions into odd parts. The even parts in partitions of n into distinct parts play an important role in the Euler–Glaisher bijective proof of this result. In this paper, we investigate the number of even parts in all partitions of n into dis...
Article
Some of the known properties of the Bernoulli numbers can be derived as specializations of the fundamental relationships between complete and elementary symmetric functions. In this paper, we introduce an infinite family of relationships between complete and elementary symmetric functions. As specializations of this result we derive connections bet...
Article
Full-text available
In this paper, we invoke the bisectional pentagonal number theorem to prove that the number of overpartitions of the positive integer $n$ into odd parts is equal to twice the number of partitions of $n$ into parts not congruent to $0$, $2$, $12$, $14$, $16$, $18$, $20$ or $30 \mod{32}$. This result allows us to experimentally discover new infinite...
Article
Full-text available
We summarize the known useful and interesting results and formulas we have discovered so far in this collaborative article summarizing results from two related articles by Merca and Schmidt arriving at related so-termed Lambert series factorization theorems. We unify the matrix representations that underlie two of our separate papers, and which com...
Preprint
We state and prove product formulae for several generating functions for sequences $(a_n)_{n\ge0}$ that are defined by the property that $Pa_n+b^2$ is a square, where $P$ and $b$ are given integers. In particular, we prove corresponding conjectures of the second author. We show that, by means of the Jacobi triple product identity, all these generat...
Article
Full-text available
We introduce an infinite family of lacunary recurrences for the Fibonacci numbers and give a combinatorial proof. The first entry in the family was proved by Lucas in 1876.
Article
Full-text available
In 2016, X. Xiong provided a complete determination of the overpartition function () pn modulo 16 by relating it to some binary quadratic forms. In this paper, we approach the characterization of () pn modulo 16 considering the relations of the form () () 2 (8) mod16 , p n r +  with 0  and {1,3,5,7}  .
Article
Full-text available
We examine two truncated series derived from a classical theta identity of Gauss. As a consequence, we obtain two infinite families of inequalities for the overpartition function po¯(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepack...
Article
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We prove several identities of the type α(n) = ∞ k=0 β n−k(k+1)/2 2. Here, the functions α(n) and β(n) count partitions with certain restrictions or the number of parts in certain partitions. Since Watson proved the identity for α(n) = Q(n), the number of partitions of n into distinct parts, and β(n) = p(n), Euler's partition function, we refer to...
Article
We investigate two truncated series derived recently by S. H. Chan, T. P. N. Ho, and R. Mao from the Watson quintuple product identity and experimentally discover two stronger results. In this context, for each S∈{1,2}, we obtain two infinite families of linear homogeneous inequalities for the number of partitions of n into parts congruent to ±Smod...
Article
The restricted partitions in which the largest part is less than or equal to N and the number of parts is less than or equal to k were investigated by Andrews in [1]. In this paper, these restricted partitions are extended to the partitions into parts of two kinds. New combinatorial identities are discovered and proved in this way exploring the rel...
Preprint
Full-text available
The maximal excludant of a partition $\lambda$, $\rm{mex}(\lambda)$, is defined to be the least gap of $\lambda$. For each positive integer $n$, the function $ \sigma\, \rm{mex}(n)$ is defined to be the sum of the least gaps in all partitions of $n$. Recently, Andrews and Newman obtained a new combinatorial interpretations for $\sigma\, \rm{mex}(n)...
Article
Full-text available
In this paper, we investigate decompositions of the partition function p(n) from the additive theory of partitions considering the famous Möbius function μ(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\od...
Article
The problem of finding fast computing methods for Bernoulli numbers has a long and interesting history. In this paper, the author provides new proofs for two lacunary recurrence relations with gaps of length four and eight for the Bernoulli numbers. These proofs invoked the fact that the $n$th powers of $\pi^2$, $\pi^4$ and $\pi^8$ can be expressed...
Article
The orthogonality of the (q; t)-version of the Stirling numbers has recently been proved by Cai and Readdy using a bijective argument. In this paper, we introduce new recurrences for the (q; t)-Stirling numbers and provide a (q; t)-analogue for sums of powers. Specializations of these results are given in terms of Stirling numbers or q-Stirling num...
Article
Full-text available
We consider the function \(r_s(n)\) which gives the number of ways to write n as the sum of s squares. Since the generating functions for \(r_4(n)\) and \(r_8(n)\) are Lambert series, we use Merca’s factorization theorem for Lambert series to establish relationships between these functions and partitions into distinct parts. We also obtain convolut...
Article
In this paper, we use the Lambert series generating function for Euler's totient function to introduce a new identity for the number of $1$'s in the partitions of $n$. A new expansion for Euler's partition function $p(n)$ is derived in this context. These surprising new results connect the famous classical totient function from multiplicative numbe...
Article
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Arelationship between the general linear group of degree n over a finite field and the integer partitions of n into parts of k different magnitudes was investigated recently by the author. In this paper, we use a variation of the classical binomial transform to derive a new connection between partitions into parts of k different magnitudes and anot...
Article
Full-text available
We provide generalizations for Euler’s recurrence relation for the partition function p(n) and the recurrence relation for the partial sums of the partition functionp(n). As a corollary, we derive an infinite family of inequalities for the partition function p(n). We present few infinite families of determinant formulas for: the partition function...
Article
Full-text available
In this paper, we give asymptotic formulas that combine the Euler-Riemann zeta function and the Chebyshev-Stirling numbers of the first kind. These results allow us to prove an asymptotic formula related to the $n$th complete homogeneous symmetric function, which was recently conjectured by the second author: $$h_{n}\left(1,\left( \frac{k}{k+1}\rig...
Article
Recently, G.E. Andrews and M. Merca considered specializations of the Rogers-Fine identity and obtained partition-theoretic interpretations of two truncated identities of Gauss solving a problem by V.J.W. Guo and J. Zeng. In this paper, we provide purely combinatorial proofs of these results.
Article
Full-text available
In this paper, we investigate new connections between the seemingly disparate branches of the additive and multiplicative number theory. Two infinite families of identities are proved in this context using an interplay of combinatorial and q-series methods. These general results are illustrated considering relationships between the gcd-sum function...
Article
Let $R_2(n)$ denote the number of partitions of n into parts that are odd or congruent to �$\pm2 \bmod 10$. In 2007, Andrews considered partitions with some negative parts and provided a second combinatorial interpretation for $R_2(n)$. In this paper, we give a collection of linear recurrence relations for the partition function $R_2(n)$. As a coro...
Article
In this paper, we give an infinite sequence of inequalities involving the Riemann zeta function with even arguments ? (2n) and the Chebyshev-Stirling numbers of the first kind. This result is based on a recent connection between the Riemann zeta function and the complete homogeneous symmetric functions [18]. An interesting asymptotic formula relate...
Article
We find accurate approximations for certain finite differences of the Euler zeta function, $\zeta(x)$.
Article
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In this article, we explore the parity of sums of partition numbers at certain places in arithmetic progressions. In particular, we investigate pairs (a,b)∈N2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\od...
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We consider a recent result for expanding augmented monomial symmetric functions in terms of the power sum symmetric functions to illustrate a technique for proving and generating inequalities involving specializations of monomial symmetric functions. © 2017, Academy of Romanian Scientists Publishing House. All rights reserved.
Article
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A factorization for partial sums of Lambert series is introduced in this paper. As corollaries, we derive some connections between partitions and divisors. These results can be easily used to discover and prove new combinatorial identities involving important functions from number theory: the Möbius function μ(n), Euler’s totient φ(n), Jordan’s tot...
Article
In this note, we introduce a q-analogue for sums of powers in terms of the q-Stirling numbers of both kinds.
Article
In this paper, we provide generalizations of two identities of Guo and Yang \cite{GuoYang} for the q-binomial coefficients. This approach allows us to derive new convolution identities for the complete and elementary symmetric functions. New identities involving q-binomial coefficients are obtained as very special cases of these results. A new rela...
Article
We provide partition-theoretic interpretation of two truncated identities of Gauss solving a problem by Guo and Zeng. We also reveal that these results, together with our previous truncation of Euler’s pentagonal number theorem, are essentially corollaries of the Rogers-Fine identity. Finally we examine further positivity questions related to the p...
Article
Full-text available
We prove several new variants of the Lambert series factorization theorem established in the first article "Generating special arithmetic functions by Lambert series factorizations" by Merca and Schmidt (2017). Several characteristic examples of our new results are presented in the article to motivate the formulations of the generalized factorizati...
Article
In this paper, we give an infinite sequence of inequalities involving the Riemann zeta function with even arguments and the Chebyshev-Stirling numbers of the first kind. This result is based on a recent connection between the Riemann zeta function and the complete homogeneous symmetric functions. An interesting asymptotic formula related to the n-t...
Article
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In this note, we build on recent work in [7] to establish formulas for ζ(2n) as sums over all the unrestricted integer partitions of n.
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A relationship between the general linear group GL(n,m) and integer partitions was investigated by Macdonald in order to calculate the number of conjugacy classes in GL(n,m). In this paper, the author introduced two different factorizations for a special case of Lambert series in order to prove that the number of conjugacy classes in the general li...
Article
Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. We use this fact to introduce a family of double inequalities involving the generating function for the number of partitions into odd parts and the generating function for the number of odd divisors.
Article
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In this paper, the author uses the generating function for the Bernoulli numbers in order to obtain a new proof for a known linear recurrence relation of the Riemann zeta function with even integer arguments.
Article
In the paper, the author presents an asymptotic formula for the arithmetic mean of the square roots of the first n positive integers using only tools from undergraduate calculus.

Questions

Question (1)
Question
For a set of n elements, say S_n = {1,2,...,n}, a set partition is a set P = {s_1,s_2,...,s_k} of nonempty subsets s_i of S_n whose intersection is empty and whose union equals S_n.

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