scholarly journals The first ascent of size $d$ or more in compositions

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Charlotte Brennan ◽  
Arnold Knopfmacher
Keyword(s):  
Positive Integer ◽  
Nonnegative Integer ◽  
Finite Sequence ◽  
Initial Height ◽  
International Audience ◽  
Positive Integers ◽  
Average Position

International audience A composition of a positive integer $n$ is a finite sequence of positive integers $a_1, a_2, \ldots, a_k$ such that $a_1+a_2+ \cdots +a_k=n$. Let $d$ be a fixed nonnegative integer. We say that we have an ascent of size $d$ or more at position $i$, if $a_{i+1}\geq a_i+d$. We study the average position, initial height and end height of the first ascent of size $d$ or more in compositions of $n$ as $n \to \infty$.

scholarly journals The distribution of ascents of size d or more in compositions

2009 ◽  
Vol Vol. 11 no. 1 (Combinatorics) ◽  
Author(s):  
Charlotte Brennan ◽  
Arnold Knopfmacher
Keyword(s):  
Positive Integer ◽  
Nonnegative Integer ◽  
Finite Sequence ◽  
Limiting Distribution ◽  
International Audience ◽  
The Mean ◽  
Average Size ◽  
Positive Integers ◽  
Mean Variance

Combinatorics International audience A composition of a positive integer n is a finite sequence of positive integers a(1), a(2), ..., a(k) such that a(1) + a(2) + ... + a(k) = n. Let d be a fixed nonnegative integer. We say that we have an ascent of size d or more if a(i+1) >= a(i) + d. We determine the mean, variance and limiting distribution of the number of ascents of size d or more in the set of compositions of n. We also study the average size of the greatest ascent over all compositions of n.

scholarly journals On the 2-adic order of Stirling numbers of the second kind and their differences

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Tamás Lengyel
Keyword(s):  
Positive Integer ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Binary Representation ◽  
Exact Order ◽  
Special Cases ◽  
International Audience ◽  
The Difference ◽  
Divisibility Properties ◽  
Positive Integers

International audience Let $n$ and $k$ be positive integers, $d(k)$ and $\nu_2(k)$ denote the number of ones in the binary representation of $k$ and the highest power of two dividing $k$, respectively. De Wannemacker recently proved for the Stirling numbers of the second kind that $\nu_2(S(2^n,k))=d(k)-1, 1\leq k \leq 2^n$. Here we prove that $\nu_2(S(c2^n,k))=d(k)-1, 1\leq k \leq 2^n$, for any positive integer $c$. We improve and extend this statement in some special cases. For the difference, we obtain lower bounds on $\nu_2(S(c2^{n+1}+u,k)-S(c2^n+u,k))$ for any nonnegative integer $u$, make a conjecture on the exact order and, for $u=0$, prove part of it when $k \leq 6$, or $k \geq 5$ and $d(k) \leq 2$. The proofs rely on congruential identities for power series and polynomials related to the Stirling numbers and Bell polynomials, and some divisibility properties.

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 Multiplicity of a Proportionally Modular Numerical Semigroup

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Ze Gu
Keyword(s):  
Positive Integer ◽  
Nonnegative Integer ◽  
Numerical Semigroup ◽  
Diophantine Inequality ◽  
Integer Solutions ◽  
Positive Integers

A proportionally modular numerical semigroup is the set S a , b , c of nonnegative integer solutions to a Diophantine inequality of the form a x   mod   b ≤ c x , where a , b , and c are positive integers. A formula for the multiplicity of S a , b , c , that is, m S a , b , c = k b / a for some positive integer k , is given by A. Moscariello. In this paper, we give a new proof of the formula and determine a better bound for k . Furthermore, we obtain k = 1 for various cases and a formula for the number of the triples a , b , c such that k ≠ 1 when the number a − c is fixed.

scholarly journals 3x+1 Minus the +

2002 ◽  
Vol Vol. 5 ◽  
Author(s):  
Kenneth G. Monks
Keyword(s):  
Positive Integer ◽  
Rational Numbers ◽  
General Mechanism ◽  
Arbitrary Positive Integer ◽  
Arbitrary Integer ◽  
Constant Multiple ◽  
International Audience ◽  
Positive Integers

International audience We use Conway's \emphFractran language to derive a function R:\textbfZ^+ → \textbfZ^+ of the form R(n) = r_in if n ≡ i \bmod d where d is a positive integer, 0 ≤ i < d and r_0,r_1, ... r_d-1 are rational numbers, such that the famous 3x+1 conjecture holds if and only if the R-orbit of 2^n contains 2 for all positive integers n. We then show that the R-orbit of an arbitrary positive integer is a constant multiple of an orbit that contains a power of 2. Finally we apply our main result to show that any cycle \ x_0, ... ,x_m-1 \ of positive integers for the 3x+1 function must satisfy \par ∑ _i∈ \textbfE \lfloor x_i/2 \rfloor = ∑ _i∈ \textbfO \lfloor x_i/2 \rfloor +k. \par where \textbfO=\ i : x_i is odd \ , \textbfE=\ i : x_i is even \ , and k=|\textbfO|. \par The method used illustrates a general mechanism for deriving mathematical results about the iterative dynamics of arbitrary integer functions from \emphFractran algorithms.

scholarly journals On $t$-Common List-Colorings

10.37236/6738 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Hojin Choi ◽  
Young Soo Kwon
Keyword(s):  
Positive Integer ◽  
Planar Graph ◽  
Chromatic Number ◽  
Upper Bound ◽  
Nonnegative Integer ◽  
Planar Graphs ◽  
Four Color Theorem ◽  
List Colorings ◽  
List Chromatic Number ◽  
Positive Integers

In this paper, we introduce a new variation of list-colorings. For a graph $G$  and for a given nonnegative integer $t$, a $t$-common list assignment of $G$ is a mapping $L$ which assigns each vertex $v$ a set $L(v)$ of colors such that given set of $t$ colors belong to $L(v)$ for every $v\in V(G)$. The $t$-common list chromatic number of $G$ denoted by $ch_t(G)$ is defined as the minimum positive integer $k$ such that there exists an $L$-coloring of $G$ for every $t$-common list assignment $L$ of $G$, satisfying $|L(v)| \ge k$ for every vertex $v\in V(G)$. We show that for all positive integers $k, \ell$ with $2 \le k \le \ell$ and for any positive integers $i_1 , i_2, \ldots, i_{k-2}$ with $k \le i_{k-2} \le \cdots \le i_1 \le \ell$, there exists a graph $G$ such that $\chi(G)= k$, $ch(G) =  \ell$ and $ch_t(G) = i_t$ for every $t=1, \ldots, k-2$. Moreover, we consider the $t$-common list chromatic number of planar graphs. From the four color theorem and the result of Thomassen (1994), for any $t=1$ or $2$, the sharp upper bound of $t$-common list chromatic number of planar graphs is $4$ or $5$. Our first step on $t$-common list chromatic number of planar graphs is to find such a sharp upper bound. By constructing a planar graph $G$ such that $ch_1(G) =5$, we show that the sharp upper bound for $1$-common list chromatic number of planar graphs is $5$. The sharp upper bound of $2$-common list chromatic number of planar graphs is still open. We also suggest several questions related to $t$-common list chromatic number of planar graphs.

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 The Distribution of Ascents of Size d or More in Partitions of n

2008 ◽  
Vol 17 (4) ◽  
pp. 495-509 ◽  
Author(s):  
CHARLOTTE BRENNAN ◽  
ARNOLD KNOPFMACHER ◽  
STEPHAN WAGNER
Keyword(s):  
Positive Integer ◽  
Finite Sequence ◽  
Limiting Distribution ◽  
Distinct Part ◽  
The Mean ◽  
Fixed Positive Integer ◽  
Positive Integers

A partition of a positive integer n is a finite sequence of positive integers a1, a2, . . ., ak such that a1+a2+ċ ċ ċ+ak=n and ai+1 ≥ ai for all i. Let d be a fixed positive integer. We say that we have an ascent of size d or more if ai+1 ≥ ai+d.We determine the mean, the variance and the limiting distribution of the number of ascents of size d or more (equivalently, the number of distinct part sizes of multiplicity d or more) in the partitions of n.

scholarly journals Statistics for 3-letter patterns with repetitions in compositions

2016 ◽  
Vol Vol. 17 no. 3 (Combinatorics) ◽  
Author(s):  
Armend Shabani ◽  
Rexhep Gjergji
Keyword(s):  
Positive Integer ◽  
Linear Algebra ◽  
Generating Functions ◽  
International Audience ◽  
Positive Integers

International audience A composition $\pi = \pi_1 \pi_2 \cdots \pi_m$ of a positive integer $n$ is an ordered collection of one or more positive integers whose sum is $n$. The number of summands, namely $m$, is called the number of parts of $\pi$. Using linear algebra, we determine formulas for generating functions that count compositions of $n$ with $m$ parts, according to the number of occurrences of the subword pattern $\tau$, and according to the sum, over all occurrences of $\tau$, of the first integers in their respective occurrences, where $\tau$ is any pattern of length three with exactly 2 distinct letters.

scholarly journals An Efficient Multicore Algorithm for Minimal Length Addition Chains

Computers ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 23
Author(s):  
Hazem Bahig ◽  
Yasser Kotb
Keyword(s):  
Positive Integer ◽  
Experimental Studies ◽  
Finite Sequence ◽  
Minimal Length ◽  
Sequential Algorithm ◽  
Binary Representation ◽  
Running Time ◽  
Addition Chain ◽  
Addition Chains ◽  
Positive Integers

A minimal length addition chain for a positive integer m is a finite sequence of positive integers such that (1) the first and last elements in the sequence are 1 and m, respectively, (2) any element greater than 1 in the sequence is the addition of two earlier elements (not necessarily distinct), and (3) the length of the sequence is minimal. Generating the minimal length addition chain for m is challenging due to the running time, which increases with the size of m and particularly with the number of 1s in the binary representation of m. In this paper, we introduce a new parallel algorithm to find the minimal length addition chain for m. The experimental studies on multicore systems show that the running time of the proposed algorithm is faster than the sequential algorithm. Moreover, the maximum speedup obtained by the proposed algorithm is 2.5 times the best known sequential algorithm.

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