scholarly journals ON THE DISTRIBUTION OF THE RANK STATISTIC FOR STRONGLY CONCAVE COMPOSITIONS

2019 ◽  
Vol 100 (2) ◽  
pp. 230-238 ◽  
Author(s):  
NIAN HONG ZHOU
Keyword(s):  
Number Theory ◽  
Asymptotic Formula ◽  
Generating Function ◽  
Rank Statistic ◽  
Integer Partition

A strongly concave composition of $n$ is an integer partition with strictly decreasing and then increasing parts. In this paper we give a uniform asymptotic formula for the rank statistic of a strongly concave composition introduced by Andrews et al. [‘Modularity of the concave composition generating function’, Algebra Number Theory7(9) (2013), 2103–2139].

An asymptotic formula in additive number theory

1976 ◽  
Vol 28 (4) ◽  
pp. 405-412 ◽  
Author(s):  
P. Erdös ◽  
G. Jogesh Babu ◽  
K. Ramachandra
Keyword(s):  
Number Theory ◽  
Asymptotic Formula ◽  
Additive Number Theory ◽  
Additive Number

scholarly journals Partitions and the Maximal Excludant

10.37236/8736 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Shane Chern
Keyword(s):  
Generating Function ◽  
Theta Function ◽  
Nonnegative Integer ◽  
Mock Theta Function ◽  
Integer Partition ◽  
The Difference

For each nonempty integer partition $\pi$, we define the maximal excludant of $\pi$ as the largest nonnegative integer smaller than the largest part of $\pi$ that is not itself a part. Let $\sigma\!\operatorname{maex}(n)$ be the sum of maximal excludants over all partitions of $n$. We show that the generating function of $\sigma\!\operatorname{maex}(n)$ is closely related to a mock theta function studied by Andrews, Dyson and Hickerson, and Cohen, respectively. Further, we show that, as $n\to \infty$, $\sigma\!\operatorname{maex}(n)$ is asymptotic to the sum of largest parts over all partitions of $n$. Finally, the expectation of the difference of the largest part and the maximal excludant over all partitions of $n$ is shown to converge to $1$ as $n\to \infty$.

Asymptotic Expansions II

1957 ◽  
Vol 9 ◽  
pp. 194-209 ◽  
Author(s):  
Leo Moser ◽  
Max Wyman
Keyword(s):  
Asymptotic Formula ◽  
Generating Function ◽  
Asymptotic Expansions ◽  
Image Position

In a previous paper (1) the authors considered the problem of finding an asymptotic formula for numbers or functions Bn,m whose generating function is of the form(1.1),where Pm(x) is a polynomial of degree m in x given by(1.2), am≠0.The above-mentioned paper contained the restriction that ak ≥ 0.

scholarly journals NOTE ON THE NUMBER OF DIVISORS OF REDUCIBLE QUADRATIC POLYNOMIALS

2018 ◽  
Vol 99 (1) ◽  
pp. 1-9
Author(s):  
ADRIAN W. DUDEK ◽  
ŁUKASZ PAŃKOWSKI ◽  
VICTOR SCHARASCHKIN
Keyword(s):  
Number Theory ◽  
Asymptotic Formula ◽  
Tauberian Theorem ◽  
Fixed Integer ◽  
Quadratic Polynomials ◽  
Number Of Divisors ◽  
The Relationship

Lapkova [‘On the average number of divisors of reducible quadratic polynomials’, J. Number Theory 180 (2017), 710–729] uses a Tauberian theorem to derive an asymptotic formula for the divisor sum $\sum _{n\leq x}d(n(n+v))$ where $v$ is a fixed integer and $d(n)$ denotes the number of divisors of $n$. We reprove this result with additional terms in the asymptotic formula, by investigating the relationship between this divisor sum and the well-known sum $\sum _{n\leq x}d(n)d(n+v)$.

STURMIAN WORDS AND AMBIGUOUS CONTEXT-FREE LANGUAGES

1990 ◽  
Vol 01 (03) ◽  
pp. 309-323 ◽  
Author(s):  
FILIPPO MIGNOSI
Keyword(s):  
Number Theory ◽  
Generating Function ◽  
Rational Number ◽  
Asymptotic Density ◽  
Gap Theorem ◽  
Context Free Language ◽  
Sturmian Words ◽  
Context Free ◽  
Free Language

If x is a rational number, 0<x≤1, then A(x)c is a context-free language, where A(x) is the set of factors of the infinite Sturmian words with asymptotic density of 1’s smaller than or equal to x. We also prove a “gap” theorem i.e. A(x) can never be an unambiguous co-context-free language. The “gap” theorem is established by proving that the counting generating function of A(x) is transcendental. We show some links between Sturmian words, combinatorics and number theory.

scholarly journals Modular, $k$-Noncrossing Diagrams

10.37236/348 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Christian M. Reidys ◽  
Rita R. Wang ◽  
Albus Y. Y. Zhao
Keyword(s):  
Asymptotic Formula ◽  
Generating Function ◽  
Exact Enumeration ◽  
Tertiary Structures ◽  
Basic Features

In this paper we compute the generating function of modular, $k$-noncrossing diagrams. A $k$-noncrossing diagram is called modular if it does not contain any isolated arcs and any arc has length at least four. Modular diagrams represent the deformation retracts of RNA tertiary structures and their properties reflect basic features of these bio-molecules. The particular case of modular noncrossing diagrams has been extensively studied. Let ${Q}_k(n)$ denote the number of modular $k$-noncrossing diagrams over $n$ vertices. We derive exact enumeration results as well as the asymptotic formula ${Q}_k(n)\sim c_k n^{-(k-1)^2-{k-1\over2}}\gamma_{k}^{-n}$ for $k=3, \ldots, 9$ and derive a new proof of the formula ${Q}_2(n)\sim 1.4848\, n^{-3/2}\,1.8489^{n}$ (Hofacker et al. 1998).

scholarly journals Distinct Parts Partitions without Sequences

10.37236/4970 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Kathrin Bringmann ◽  
Karl Mahlburg ◽  
Karthik Nataraj
Keyword(s):  
Asymptotic Formula ◽  
Generating Function ◽  
Generating Functions ◽  
Series Representation ◽  
Double Series ◽  
Asymptotic Formulas ◽  
Similar Investigation ◽  
Combinatorial Properties ◽  
Consecutive Integers

Partitions without sequences of consecutive integers as parts have been studied recently by many authors, including Andrews, Holroyd, Liggett, and Romik, among others. Their results include a description of combinatorial properties, hypergeometric representations for the generating functions, and asymptotic formulas for the enumeration functions. We complete a similar investigation of partitions into distinct parts without sequences, which are of particular interest due to their relationship with the Rogers-Ramanujan identities. Our main results include a double series representation for the generating function, an asymptotic formula for the enumeration function, and several combinatorial inequalities.

scholarly journals On sums of coefficients of polynomials related to the Borwein conjectures

2021 ◽  
Author(s):  
Ankush Goswami ◽  
Venkata Raghu Tej Pantangi
Keyword(s):  
Number Theory ◽  
Lower Bound ◽  
Asymptotic Formula ◽  
Recent Result ◽  
Asymptotic Estimate ◽  
Partial Sums ◽  
Absolute Value ◽  
Partial Sum ◽  
Special Cases ◽  
Error Terms

AbstractRecently, Li (Int J Number Theory, 2020) obtained an asymptotic formula for a certain partial sum involving coefficients for the polynomial in the First Borwein conjecture. As a consequence, he showed the positivity of this sum. His result was based on a sieving principle discovered by himself and Wan (Sci China Math, 2010). In fact, Li points out in his paper that his method can be generalized to prove an asymptotic formula for a general partial sum involving coefficients for any prime $$p>3$$ p > 3 . In this work, we extend Li’s method to obtain asymptotic formula for several partial sums of coefficients of a very general polynomial. We find that in the special cases $$p=3, 5$$ p = 3 , 5 , the signs of these sums are consistent with the three famous Borwein conjectures. Similar sums have been studied earlier by Zaharescu (Ramanujan J, 2006) using a completely different method. We also improve on the error terms in the asymptotic formula for Li and Zaharescu. Using a recent result of Borwein (JNT 1993), we also obtain an asymptotic estimate for the maximum of the absolute value of these coefficients for primes $$p=2, 3, 5, 7, 11, 13$$ p = 2 , 3 , 5 , 7 , 11 , 13 and for $$p>15$$ p > 15 , we obtain a lower bound on the maximum absolute value of these coefficients for sufficiently large n.

Some generating functions and inequalities for the andrews–stanley partition functions

2021 ◽  
pp. 1-16
Author(s):  
Na Chen ◽  
Shane Chern ◽  
Yan Fan ◽  
Ernest X. W. Xia
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Partition Functions ◽  
Integer Partition

Abstract Let $\mathcal {O}(\pi )$ denote the number of odd parts in an integer partition $\pi$ . In 2005, Stanley introduced a new statistic $\operatorname {srank}(\pi )=\mathcal {O}(\pi )-\mathcal {O}(\pi ')$ , where $\pi '$ is the conjugate of $\pi$ . Let $p(r,\,m;n)$ denote the number of partitions of $n$ with srank congruent to $r$ modulo $m$ . Generating function identities, congruences and inequalities for $p(0,\,4;n)$ and $p(2,\,4;n)$ were then established by a number of mathematicians, including Stanley, Andrews, Swisher, Berkovich and Garvan. Motivated by these works, we deduce some generating functions and inequalities for $p(r,\,m;n)$ with $m=16$ and $24$ . These results are refinements of some inequalities due to Swisher.

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