scholarly journals On Bergeron's Positivity Problem for $q$-Binomial Coefficients

10.37236/7358 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Fabrizio Zanello
Keyword(s):  
Binomial Coefficients ◽  
Main Ingredient ◽  
Technical Application ◽  
Nonnegative Coefficients ◽  
Positive Integers

F. Bergeron recently asked the intriguing question whether $\binom{b+c}{b}_q -\binom{a+d}{d}_q$  has nonnegative coefficients as a polynomial in $q$, whenever $a,b,c,d$ are positive integers, $a$ is the smallest, and $ad=bc$. We conjecture that, in fact, this polynomial is also always unimodal, and combinatorially show our conjecture for $a\le 3$ and any $b,c\ge 4$. The main ingredient will be a novel (and rather technical) application of Zeilberger's KOH theorem.

scholarly journals Statistics on Lattice Walks and q-Lassalle Numbers

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Lenny Tevlin
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Integer Coefficients ◽  
Major Index ◽  
Lattice Walks ◽  
International Audience ◽  
The Difference ◽  
Positive Integers

International audience This paper contains two results. First, I propose a $q$-generalization of a certain sequence of positive integers, related to Catalan numbers, introduced by Zeilberger, see Lassalle (2010). These $q$-integers are palindromic polynomials in $q$ with positive integer coefficients. The positivity depends on the positivity of a certain difference of products of $q$-binomial coefficients.To this end, I introduce a new inversion/major statistics on lattice walks. The difference in $q$-binomial coefficients is then seen as a generating function of weighted walks that remain in the upper half-plan. Cet document contient deux résultats. Tout d’abord, je vous propose un $q$-generalization d’une certaine séquence de nombres entiers positifs, liés à nombres de Catalan, introduites par Zeilberger (Lassalle, 2010). Ces $q$-integers sont des polynômes palindromiques à $q$ à coefficients entiers positifs. La positivité dépend de la positivité d’une certaine différence de produits de $q$-coefficients binomial.Pour ce faire, je vous présente une nouvelle inversion/major index sur les chemins du réseau. La différence de $q$-binomial coefficients est alors considérée comme une fonction de génération de trajets pondérés qui restent dans le demi-plan supérieur.

scholarly journals On the prime factorization of binomial coefficients

1978 ◽  
Vol 26 (3) ◽  
pp. 257-269 ◽  
Author(s):  
E. F. Ecklund ◽  
R. B. Eggleton ◽  
P. Erdös ◽  
J. L. Selfridge
Keyword(s):  
Prime Factor ◽  
Binomial Coefficients ◽  
Prime Factorization ◽  
Positive Integers

AbstractFor positive integers n and k, with n≥2k, let , where each prime factor of u is less than k, and each prime factor of v is at least equal to k. It is shown that u<v holds with just 12 exceptions, which are determined. If , where each prime factor of U is at most equal to k, and each prime factor of V is greater than k, then U<V holds with at most finitely many exceptions, 19 of which are determined. It is conjectured that there are no others.

scholarly journals Is the Sibuya distribution a progeny?

2019 ◽  
Vol 56 (01) ◽  
pp. 52-56
Author(s):  
Gérard Letac
Keyword(s):  
Power Series ◽  
Series Expansion ◽  
Generating Function ◽  
Branching Process ◽  
Power Series Expansion ◽  
Nonnegative Coefficients ◽  
Positive Integers

AbstractFor 0 &lt; a &lt; 1, the Sibuya distribution sa is concentrated on the set ℕ+ of positive integers and is defined by the generating function $$\sum\nolimits_{n = 1}^\infty s_a (n)z^{{\kern 1pt} n} = 1 - (1 - z)^a$$. A distribution q on ℕ+ is called a progeny if there exists a branching process (Zn)n≥0 such that Z0 = 1, such that $$(Z_1 ) \le 1$$, and such that q is the distribution of $$\sum\nolimits_{n = 0}^\infty Z_n$$. this paper we prove that sa is a progeny if and only if $${\textstyle{1 \over 2}} \le a &#x003C; 1$$. The main point is to find the values of b = 1/a such that the power series expansion of u(1 − (1 − u)b)−1 has nonnegative coefficients.

SUMS OF SQUARES AND PARTITION CONGRUENCES

2020 ◽  
pp. 1-19
Author(s):  
SU-PING CUI ◽  
NANCY S. S. GU
Keyword(s):  
Positive Integer ◽  
Sums Of Squares ◽  
Binomial Coefficients ◽  
Partition Congruences ◽  
Arithmetic Properties ◽  
Positive Integers ◽  
Pair Function

For positive integers $n$ and $k$ , let $r_{k}(n)$ denote the number of representations of $n$ as a sum of $k$ squares, where representations with different orders and different signs are counted as distinct. For a given positive integer $m$ , by means of some properties of binomial coefficients, we derive some infinite families of congruences for $r_{k}(n)$ modulo $2^{m}$ . Furthermore, in view of these arithmetic properties of $r_{k}(n)$ , we establish many infinite families of congruences for the overpartition function and the overpartition pair function.

Some Divisibility Properties of Binomial Coefficients and the Converse of Wolstenholme's Theorem

Integers ◽  
2010 ◽  
Vol 10 (5) ◽  
Author(s):  
Kevin A. Broughan ◽  
Florian Luca ◽  
Igor E. Shparlinski
Keyword(s):  
Binomial Coefficients ◽  
Divisibility Properties ◽  
Positive Integers

AbstractWe show that the set of composite positive integersis of cardinality at most

scholarly journals Proof of two Divisibility Properties of Binomial Coefficients Conjectured by Z.-W. Sun

10.37236/4258 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Victor J. W. Guo
Keyword(s):  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Divisibility Properties ◽  
Positive Integers

For all positive integers $n$, we prove the following divisibility properties:\[ (2n+3){2n\choose n}  \left|3{6n\choose 3n}{3n\choose n},\right. \quad\text{and}\quad(10n+3){3n\choose n} \left|21{15n\choose 5n}{5n\choose n}.\right. \]This confirms two recent conjectures of Z.-W. Sun. Some similar divisibility properties are given. Moreover, we show that, for all positive integers $m$ and $n$, the product $am{am+bm-1\choose am}{an+bn\choose an}$ is divisible by $m+n$. In fact, the latter result can be further generalized to the $q$-binomial coefficients and $q$-integers case, which generalizes the positivity of $q$-Catalan numbers. We also propose several related conjectures.

scholarly journals FACTORS OF SUMS AND ALTERNATING SUMS INVOLVING BINOMIAL COEFFICIENTS AND POWERS OF INTEGERS

2011 ◽  
Vol 07 (07) ◽  
pp. 1959-1976 ◽  
Author(s):  
VICTOR J. W. GUO ◽  
JIANG ZENG
Keyword(s):  
Positive Integer ◽  
Nonnegative Integer ◽  
Binomial Coefficients ◽  
Alternating Sums ◽  
Divisibility Properties ◽  
Positive Integers

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers n1,…,nm, nm+1 = n1, and any nonnegative integer r, there holds [Formula: see text] and conjecture that for any nonnegative integer r and positive integer s such that r + s is odd, [Formula: see text] where ε = ±1.

scholarly journals ON COMMON DIVISORS OF MULTINOMIAL COEFFICIENTS

2010 ◽  
Vol 83 (1) ◽  
pp. 138-157
Author(s):  
GEORGE M. BERGMAN
Keyword(s):  
Lower Bound ◽  
Binomial Coefficients ◽  
Analogous Statement ◽  
The Common ◽  
Positive Integers ◽  
Multinomial Coefficients

AbstractErdős and Szekeres [‘Some number theoretic problems on binomial coefficients’, Aust. Math. Soc. Gaz.5 (1978), 97–99] showed that for any four positive integers satisfying m1+m2=n1+n2, the two binomial coefficients (m1+m2)!/m1!m2! and (n1+n2)!/n1!n2! have a common divisor greater than 1. The analogous statement for k-element families of k-nomial coefficients (k>1) was conjectured in 1997 by David Wasserman.Erdős and Szekeres remark that if m1,m2,n1,n2 as above are all greater than 1, there is probably a lower bound on the common divisor in question which goes to infinity as a function of m1 +m2 . Such a bound is obtained in Section 2.The remainder of this paper is devoted to proving results that narrow the class of possible counterexamples to Wasserman’s conjecture.

The Monotonicity and Log-Behaviour of Some Functions Related to the Euler Gamma Function

2016 ◽  
Vol 60 (2) ◽  
pp. 527-543
Author(s):  
Bao-Xuan Zhu
Keyword(s):  
Gamma Function ◽  
Catalan Numbers ◽  
Complete Monotonicity ◽  
Binomial Coefficients ◽  
Euler Gamma Function ◽  
Continuous Analogue ◽  
Riemann Zeta ◽  
The One ◽  
Image Position ◽  
Positive Integers

AbstractThe aim of this paper is to develop analytic techniques to deal with the monotonicity of certain combinatorial sequences. On the one hand, a criterion for the monotonicity of the function is given, which is a continuous analogue of a result of Wang and Zhu. On the other hand, the log-behaviour of the functionsis considered, where ζ(x) and Γ(x) are the Riemann zeta function and the Euler Gamma function, respectively. Consequently, the strict log-concavities of the function θ(x) (a conjecture of Chen et al.) and for some combinatorial sequences (including the Bernoulli numbers, the tangent numbers, the Catalan numbers, the Fuss–Catalan numbers, and the binomial coefficients are demonstrated. In particular, this contains some results of Chen et al., and Luca and Stănică. Finally, by researching the logarithmically complete monotonicity of some functions, the infinite log-monotonicity of the sequenceis proved. This generalizes two results of Chen et al. that both the Catalan numbers and the central binomial coefficients are infinitely log-monotonic, and strengthens one result of Su and Wang that is log-convex in n for positive integers d > δ. In addition, the asymptotically infinite log-monotonicity of derangement numbers is showed. In order to research the stronger properties of the above functions θ(x) and F(x), the logarithmically complete monotonicity of functionsis also obtained, which generalizes the results of Lee and Tepedelenlioǧlu, and Qi and Li.

Tribinomial coefficients and Catalan numbers

2009 ◽  
Vol 93 (528) ◽  
pp. 449-455 ◽  
Author(s):  
Thomas Koshy ◽  
Mohammad Salmassi
Keyword(s):  
Binomial Coefficient ◽  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Triangular Numbers ◽  
Interesting Family ◽  
Positive Integers

The concept of the ordinary binomial coefficientcan be employed to construct an interesting family of positive integers. Such a family was introduced around 1974 by W. Hansell using the triangular numbers where we call them tribinomial coefficients since they are binomial coefficients for triangular numbers. To this end, first we define corresponding to and For example,

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