Sequence Of Nonnegative Integers Research Articles

Diophantine equation with weighted k-Fibonacci numbers

Assume that p, q, and k are integers for which the conditions 1≤p,q≤10\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1\\le p,q\\le 10$$\\end{document} and 2≤k≤10\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\\le k\\le 10$$\\end{document} are satisfied. The initial values G0=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_0=0$$\\end{document}, G1=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_1=1$$\\end{document}, together with the recursive rule Gm=kGm-1+Gm-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G_m=kG_{m-1}+G_{m-2}$$\\end{document} define the non-negative integer sequence {Gm}m=0∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{G_m\\}_{m=0}^\\infty$$\\end{document}. In this paper, we solve completely the diophantine equation G1p+2G2p+⋯+ℓGℓp=Gnq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} G_1^p+2G_2^p+\\cdots +\\ell G_\\ell ^p=G_n^q \\end{aligned}$$\\end{document}in the positive integers k,p,q,ℓ,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k,p,q,\\ell ,n$$\\end{document} unconditionally for ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell$$\\end{document} and n. The method works, at least in theory for arbitrary positive integers p, q, and k.

ON THE DOMAIN OF THE CONVERGENCE OF TAYLOR-DIRECHLET SERIES WITH COMPLEX EXPONENTS

The article deals with Dirichlet series of the form F(z) = Σ+∞ k=0 akezλk , as well as Taylor-Dirichlet series of the form F1(z) =+∞Σk=0 zmkakezλk , where (λk) is some sequence of pairwise different complex numbers, and mk is a sequence of non-negative integers, ak ∈ C. The main statements of the work are Statements 1–3 and relate to the description and relationship between the domains of convergence, absolute convergence, and the domains of existence of the maximum term of the above series F and F1.

Asymptotically deferred statistical equivalent functions of order $ \alpha $ in amenable semigroups

The aim of this paper is to study the asymptotic convergence of the sequences. Concretely, we will give the asymptotically statistical convergence of sequences. The asymptotically statistical convergence is studied in many papers including papers [3], [5], [23], [26], [29], [39], [41]. In this paper, we introduce and investigate the concepts of asymptotically deferred statistical equivalent functions of order $ \alpha $ and strong asymptotically deferred statistical equivalent functions of order $ \alpha $ defined on discrete countable amenable semigroups, which generalizes some of the results known in the literature. This is achieved by introducing the Folner sequences and sequences of non-negative integers. Based on this concept we apply it to the approximation theory, and we have proved the Korovkin type theorem and study the rate of convergence for positive linear operators which are deferred statistically convergent of order $ \alpha $.

On forcibly k-connected and forcibly k-arc-connected digraphic sequences

For nonnegative integer sequences d = { ( d 1 + , d 2 + , … , d n + ) , ( d 1 − , d 2 − , … , d n − ) } and d ′ = { ( d ′ 1 + , d ′ 2 + , … , d ′ n + ) , ( d ′ 1 − , d ′ 2 − , … , d ′ n − ) } , we say that d ′ majorizes d , denoted by d ′ ≥ d , if d j ′ + ≥ d j + and d j ′ − ≥ d j − for 1 ≤ j ≤ n . A digraphic sequence d = { ( d 1 + , d 2 + , … , d n + ) , ( d 1 − , d 2 − , … , d n − ) } is forcibly k -connected (resp., k -arc-connected) if each digraph with degree sequence d is k -connected (resp., k -arc-connected). We give a sufficient condition for forcibly k -connected (resp., k -arc-connected) digraphic sequences and observe that if a digraphic sequence d satisfies the condition (which implies d is forcibly k -connected (resp., k -arc-connected)) then every digraphic sequence d ′ ≥ d also satisfies the condition. If d violates the condition, then we may take a sequence d ′ ≥ d such that there exists a non- k -connected (resp., non- k -arc-connected) d ′ -realization.

Solutions to problems about potentially K s,t -bigraphic pair

Abstract Let S = ( a 1 , … , a m ; b 1 , … , b n ) S=\left({a}_{1},\ldots ,{a}_{m};\hspace{0.33em}{b}_{1},\ldots ,{b}_{n}) , where a 1 , … , a m {a}_{1},\ldots ,{a}_{m} and b 1 , … , b n {b}_{1},\ldots ,{b}_{n} are two nonincreasing sequences of nonnegative integers. The pair S = ( a 1 , … , a m ; b 1 , … , b n ) S=\left({a}_{1},\ldots ,{a}_{m};\hspace{0.33em}{b}_{1},\ldots ,{b}_{n}) is said to be a bigraphic pair if there is a simple bipartite graph G = ( X ∪ Y , E ) G=\left(X\cup Y,E) such that a 1 , … , a m {a}_{1},\ldots ,{a}_{m} and b 1 , … , b n {b}_{1},\ldots ,{b}_{n} are the degrees of the vertices in X X and Y Y , respectively. In this case, G G is referred to as a realization of S S . Given a bigraphic pair S S , and a complete bipartite graph K s , t {K}_{s,t} , we say that S S is a potentially K s , t {K}_{s,t} -bigraphic pair if some realization of S S contains K s , t {K}_{s,t} as a subgraph (with s s vertices in the part of size m m and t t in the part of size n n ). Ferrara et al. (Potentially H-bigraphic sequences, Discuss. Math. Graph Theory 29 (2009), 583–596) defined σ ( K s , t , m , n ) \sigma \left({K}_{s,t},m,n) to be the minimum integer k k such that every bigraphic pair S = ( a 1 , … , a m ; b 1 , … , b n ) S=\left({a}_{1},\ldots ,{a}_{m};{b}_{1},\ldots ,{b}_{n}) with σ ( S ) = a 1 + ⋯ + a m ≥ k \sigma \left(S)={a}_{1}+\cdots +{a}_{m}\ge k is a potentially K s , t {K}_{s,t} -bigraphic pair. This problem can be viewed as a “potential” degree sequence relaxation of the (forcible) Turán problem. Ferrara et al. determined σ ( K s , t , m , n ) \sigma \left({K}_{s,t},m,n) for n ≥ m ≥ 9 s 4 t 4 n\ge m\ge 9{s}^{4}{t}^{4} . In this paper, we further determine σ ( K s , t , m , n ) \sigma \left({K}_{s,t},m,n) for n ≥ m ≥ s n\ge m\ge s and n + m ≥ 2 t 2 + t + s n+m\ge 2{t}^{2}+t+s . As two corollaries, if n ≥ m ≥ t 2 + t + s 2 n\ge m\ge {t}^{2}+\frac{t+s}{2} or if n ≥ m ≥ s n\ge m\ge s and n ≥ 2 t 2 + t n\ge 2{t}^{2}+t , the values σ ( K s , t , m , n ) \sigma \left({K}_{s,t},m,n) are determined completely. These results give a solution to a problem due to Ferrara et al. and a solution to a problem due to Yin and Wang.

Solution to a Forcible Version of a Graphic Sequence Problem

Let \(A_n=(a_1,a_2,\ldots ,a_n)\) and \(B_n=(b_1,b_2,\ldots ,b_n)\) be nonnegative integer sequences with \(A_n\le B_n\). The purpose of this note is to give a good characterization such that every integer sequence \(\pi =(d_1,d_2,\ldots d_n)\) with even sum and \(A_n\le \pi \le B_n\) is graphic. This solves a forcible version of problem posed by Niessen and generalizes the Erdős–Gallai theorem.

On the Reconstruction of 3-Uniform Hypergraphs from Degree Sequences of Span-Two

A nonnegative integer sequence is k-graphic if it is the degree sequence of a k-uniform simple hypergraph. The problem of deciding whether a given sequence pi is 3-graphic has recently been proved to be NP-complete, after years of studies. Thus, it acquires primary relevance to detect classes of degree sequences whose graphicality can be tested in polynomial time in order to restrict the NP-hard core of the problem and design algorithms that can also be useful in different research areas. Several necessary and few sufficient conditions for pi to be k-graphic, with kge 3, appear in the literature. Frosini et al. defined a polynomial time algorithm to reconstruct k-uniform hypergraphs having regular or almost regular degree sequences. Our study fits in this research line providing a combinatorial characterization of span-two sequences, i.e., sequences of the form pi =(d,ldots ,d,d-1,ldots ,d-1,d-2,ldots ,d-2), dge 2, which are degree sequences of some 3-uniform hypergraphs. Then, we define a polynomial time algorithm to reconstruct one of the related 3-uniform hypergraphs. Our results are likely to be easily generalized to k ge 4 and to other families of degree sequences having simple characterization, such as gap-free sequences.

On the SEL Egyptian fraction expansion for real numbers

<abstract><p>In the authors' earlier work, the SEL Egyptian fraction expansion for any real number was constructed and characterizations of rational numbers by using such expansion were established. These results yield a generalized version of the results for the Fibonacci-Sylvester and the Engel series expansions. Under a certain condition, one of such characterizations also states that the SEL Egyptian fraction expansion is finite if and only if it represents a rational number. In this paper, we obtain an upper bound for the length of the SEL Egyptian fraction expansion for rational numbers, and the exact length of this expansion for a certain class of rational numbers is verified. Using such expansion, not only is a large class of transcendental numbers constructed, but also an explicit bijection between the set of positive real numbers and the set of sequences of nonnegative integers is established.</p></abstract>

Expansions in Cantor real bases

We introduce and study series expansions of real numbers with an arbitrary Cantor real base $$\varvec{\beta }=(\beta _n)_{n\in {\mathbb {N}}}$$ , which we call $$\varvec{\beta }$$ -representations. In doing so, we generalize both representations of real numbers in real bases and through Cantor series. We show fundamental properties of $$\varvec{\beta }$$ -representations, each of which extends existing results on representations in a real base. In particular, we prove a generalization of Parry’s theorem characterizing sequences of nonnegative integers that are the greedy $$\varvec{\beta }$$ -representations of some real number in the interval [0, 1). We pay special attention to periodic Cantor real bases, which we call alternate bases. In this case, we show that the $$\varvec{\beta }$$ -shift is sofic if and only if all quasi-greedy $$\varvec{\beta }^{(i)}$$ -expansions of 1 are ultimately periodic, where $$\varvec{\beta }^{(i)}$$ is the i-th shift of the Cantor real base $$\varvec{\beta }$$ .

The sum necessary to ensure that a degree sequence pair has an [formula omitted]-connected realization

Let S=(a1,…,am;b1,…,bn), where a1,…,am and b1,…,bn are two nonincreasing sequences of nonnegative integers. The pair S=(a1,…,am;b1,…,bn) is said to be a bigraphic pair if there is a simple bipartite graph G=(X∪Y,E) such that a1,…,am and b1,…,bn are the degrees of the vertices in X and Y, respectively. Let A be an (additive) Abelian group. We define σ(A,m,n) to be the minimum integer k such that every bigraphic pair S=(a1,…,am;b1,…,bn) with am,bn≥2 and σ(S)=a1+⋯+am≥k has an A-connected realization. In this paper, we determine the values of σ(A,m,n) for |A|=k and m≥n≥2, where k≥6 is an even integer, and the values of σ(A,m,n) for |A|=4 and m≥n≥3. Therefore, together with some previous results [(Yin, 2016); (Yin and Dai, 2017); (Guan and Yin, 2018)], the values of σ(A,m,n) for |A|≥3 are determined completely.

Grossberg–Karshon Twisted Cubes and Hesitant Jumping Walk Avoidance

Let $G$ be a complex simply-laced semisimple algebraic group of rank $r$ and $B$ a Borel subgroup. Let $\mathbf i \in [r]^n$ be a word and let $\boldsymbol{\ell} = (\ell_1,\dots,\ell_n)$ be a sequence of non-negative integers. Grossberg and Karshon introduced a virtual lattice polytope associated to $\mathbf i$ and $\boldsymbol{\ell}$ called a twisted cube, whose lattice points encode the character of a $B$-representation. More precisely, lattice points in the twisted cube, counted with sign according to a certain density function, yields the character of the generalized Demazure module determined by $\mathbf i$ and $\boldsymbol{\ell}$. In recent work, the author and Harada described precisely when the Grossberg–Karshon twisted cube is untwisted, i.e., the twisted cube is a closed convex polytope, in the situation when the integer sequence $\boldsymbol{\ell}$ comes from a weight $\lambda$ of $G$. However, not every integer sequence $\boldsymbol{\ell}$ comes from a weight of $G$. In the present paper, we interpret the untwistedness of Grossberg–Karshon twisted cubes associated with any word $\mathbf i$ and any integer sequence $\boldsymbol{\ell}$ using the combinatorics of $\mathbf i$ and $\boldsymbol{\ell}$. Indeed, we prove that the Grossberg–Karshon twisted cube is untwisted precisely when $\mathbf i$ is hesitant-jumping-$\boldsymbol{\ell}$-walk-avoiding.

Wilf equivalences between vincular patterns in inversion sequences

Inversion sequences are finite sequences of non-negative integers, where the value of each entry is bounded from above by its position. They provide a useful encoding of permutations. Patterns in inversion sequences have been studied by Corteel–Martinez–Savage–Weselcouch and Mansour–Shattuck in the classical case, where patterns can occur in any positions, and by Auli–Elizalde in the consecutive case, where only adjacent entries can form an occurrence of a pattern. These papers classify classical and consecutive patterns of length 3 into Wilf equivalence classes according to the number of inversion sequences avoiding them.In this paper we consider vincular patterns in inversion sequences, which, in analogy to Babson–Steingrímsson patterns in permutations, require only certain entries of an occurrence to be adjacent, and thus generalize both classical and consecutive patterns. Solving three conjectures of Lin and Yan, we provide a complete classification of vincular patterns of length 3 in inversion sequences into Wilf equivalence classes, and into more restrictive classes that consider the number of occurrences of the pattern and the positions of such occurrences. We find the first known instance of patterns in inversion sequences where these two more restrictive classes do not coincide.

Bounded degree complexes of forests

Given an arbitrary sequence of non-negative integers λ→=(λ1,…,λn) and a graph G with vertex set {v1,…,vn}, the bounded degree complex, denoted as BDλ→(G), is a simplicial complex whose faces are the subsets H⊆E(G) such that for each i∈{1,…,n}, the degree of vertex vi in the induced subgraph G[H] is at most λi. When λi=k for all i, the bounded degree complex BDλ→(G) is called the k-matching complex, denoted as Mk(G).In this article, we determine the homotopy type of bounded degree complexes of forests. In particular, we show that, for all k≥1, the k-matching complexes of caterpillar graphs are either contractible or homotopy equivalent to a wedge of spheres, thereby proving a conjecture of Vega (2019, Conjecture 7.3). We also give a closed form formula for the homotopy type of the bounded degree complexes of those caterpillar graphs in which every non-leaf vertex is adjacent to at least one leaf vertex.

Hereditary Semiorders and Enumeration of Semiorders by Dimension

In 2010, Bousquet-Mélou et al. defined sequences of nonnegative integers called ascent sequences and showed that the ascent sequences of length $n$ are in one-to-one correspondence with the interval orders, i.e., the posets not containing the poset $\mathbf{2}+\mathbf{2}$. Through the use of generating functions, this provided an answer to the longstanding open question of enumerating the (unlabeled) interval orders. A semiorder is an interval order having a representation in which all intervals have the same length. In terms of forbidden subposets, the semiorders exclude $\mathbf{2}+\mathbf{2}$ and $\mathbf{1}+\mathbf{3}$. The number of unlabeled semiorders on $n$ points has long been known to be the $n$th Catalan number. However, describing the ascent sequences that correspond to the semiorders under the bijection of Bousquet-Mélou et al. has proved difficult. In this paper, we discuss a major part of the difficulty in this area: the ascent sequence corresponding to a semiorder may have an initial subsequence that corresponds to an interval order that is not a semiorder.
 We define the hereditary semiorders to be those corresponding to an ascent sequence for which every initial subsequence also corresponds to a semiorder. We provide a structural result that characterizes the hereditary semiorders and use this characterization to determine the ordinary generating function for hereditary semiorders. We also use our characterization of hereditary semiorders and the characterization of semiorders of dimension $3$ given by Rabinovitch to provide a structural description of the semiorders of dimension at most $2$. From this description, we are able to determine the ordinary generating function for the semiorders of dimension at most $2$.

Vector parking functions with periodic boundaries and rational parking functions

Vector parking functions are sequences of non-negative integers whose order statistics are bounded by a given integer sequence u=(u0,u1,u2,…). Using the theory of fractional power series and an analog of Newton-Puiseux Theorem, we derive the exponential generating function for the number of u-parking functions when u is periodic. Our method is to convert an Appell relation of Gončarov polynomials to a system of linear equations. Solving the system we obtain an explicit formula of the exponential generating function in terms of Schur functions of certain fractional power series. In particular, we apply our methods to rational parking functions for which the boundary is induced by a linear function with rational slope.

HILBERT FUNCTIONS OF GRADED MODULES OVER AN EXTERIOR ALGEBRA: AN ALGORITHMIC APPROACH

Let $K$ be a field, $E$ the exterior algebra of a finite dimensional $K$-vector space, and $F$ a finitely generated graded free $E$-module with homogeneous basis $g_1, \ldots, g_r$ such that $\deg g_1 \le \deg g_2 \le \cdots \le \deg g_r$. Given the Hilbert function of a graded $E$--module of the type $F/M$, with $M$ graded submodule of $F$, the existence of the unique lexicographic submodule of $F$ with the same Hilbert function as $M$ is proved by a new algorithmic approach. Such an approach allows us to establish a criterion for determining if a sequence of nonnegative integers defines the Hilbert function of a quotient of a free $E$--module only via the combinatorial Kruskal--Katona's theorem.

Matrices in [formula omitted] with minimum t-term ranks

Let R and S be two sequences of nonnegative integers in nonincreasing order which have the same sum, and let A(R,S) be the class of all (0,1)-matrices which have row sums given by R and column sums given by S. For a positive integer t, the t-term rank of a (0,1)-matrix A is defined as the maximum number of 1's in A with at most one 1 in each column and at most t 1's in each row. In this paper, we address conditions for the existence of a matrix in A(R,S) that realizes all the minimum t-term ranks, for t≥1.

Rooted trees with the same plucking polynomial

In this paper we address the following question: When do two rooted trees have the same plucking polynomial? The solution provided in the present paper has an algebraic version (Theorem 2.5) and a geometric version (Theorem 1.2). Furthermore, we give a criterion for a sequence of non-negative integers to be realized as a rooted tree.

A formula for the associated Buchsbaum–Rim multiplicities of a direct sum of cyclic modules II

The associated Buchsbaum–Rim multiplicities of a module are a descending sequence of non-negative integers. These invariants of a module are a generalization of the classical Hilbert–Samuel multiplicity of an ideal. In this article, we compute the associated Buchsbaum–Rim multiplicity of a direct sum of cyclic modules and give a formula for the second to last positive associated Buchsbaum–Rim multiplicity in terms of the ordinary Buchsbaum–Rim and Hilbert–Samuel multiplicities. This is a natural generalization of a formula given by Kirby and Rees.

Arithmetical properties of real numbers related to beta-expansions

The main purpose of this paper is to study the arithmetical properties of values \(\sum_{m=0}^{\infty} \beta^{-w(m)}\), where \(\beta\) is a fixed Pisot or Salem number and \(w(m)\) (\(m=0,1,\ldots\)) are distinct sequences of nonnegative integers with \(w(m+1)>w(m)\) for any sufficiently large \(m\). We first introduce the algebraic independence results of such values. Our results are applicable to certain sequences \(w(m)\) (\(m=0,1,\ldots\)) with \(\lim_{m\to\infty}w(m+1)/w(m)=1.\) For example, we prove that two numbers \[\sum_{m=1}^{\infty}\beta^{-\lfloor \varphi(m)\rfloor}, \quad \sum_{m=3}^{\infty}\beta^{-\lfloor a(m)\rfloor}\] are algebraically independent, where \(\varphi(m)=m^{\log m}\) and \(a(m)=m^{\log\log m}\). Moreover, we also give the linear independence results of real numbers. Our results are applicable to the values \(\sum_{m=0}^{\infty}\beta^{-\lfloor m^\rho\rfloor}\), where \(\beta\) is a Pisot or Salem number and \(\rho\) is a real number greater than 1.