Journal of Inequalities and Applications, ISSN 1025-5834, 12/2017, Volume 2017, Issue 1, pp. 1 - 8

In the paper, by the Faà di Bruno formula, the authors establish two explicit formulas for the Motzkin numbers, the generalized Motzkin numbers, and the...

Faà di Bruno formula | 05A20 | Motzkin number | generating function | Mathematics | restricted hexagonal number | 11B37 | explicit formula | 11B83 | Analysis | generalized Motzkin number | Catalan number | Bell polynomial of the second kind | Mathematics, general | 05A19 | Applications of Mathematics | 05A15 | MATHEMATICS | MATHEMATICS, APPLIED | Faa di Bruno formula | CATALAN NUMBERS | Inequalities | Research

Faà di Bruno formula | 05A20 | Motzkin number | generating function | Mathematics | restricted hexagonal number | 11B37 | explicit formula | 11B83 | Analysis | generalized Motzkin number | Catalan number | Bell polynomial of the second kind | Mathematics, general | 05A19 | Applications of Mathematics | 05A15 | MATHEMATICS | MATHEMATICS, APPLIED | Faa di Bruno formula | CATALAN NUMBERS | Inequalities | Research

Journal Article

Linear Algebra and Its Applications, ISSN 0024-3795, 2011, Volume 434, Issue 3, pp. 712 - 722

We use combinatorial methods to evaluate Hankel determinants for the sequence of sums of consecutive t-Motzkin numbers. More specifically, we consider the...

Combinatorial proofs | Nonintersecting lattice paths | Chebychev polynomials | Motzkin numbers | Hankel determinants | MATHEMATICS, APPLIED | PATHS | GENERALIZED CATALAN NUMBERS

Combinatorial proofs | Nonintersecting lattice paths | Chebychev polynomials | Motzkin numbers | Hankel determinants | MATHEMATICS, APPLIED | PATHS | GENERALIZED CATALAN NUMBERS

Journal Article

Discrete Mathematics, ISSN 0012-365X, 2005, Volume 305, Issue 1, pp. 170 - 189

We say that a permutation π is a Motzkin permutation if it avoids 132 and there do not exist a < b such that π a < π b < π b + 1 . We study the distribution of...

Motzkin path | Restricted permutation | Motzkin permutation | Generalized pattern | Permutation statistic | Chebyshev polynomial | generalized pattern | MATHEMATICS | NUMBER | restricted permutation | AVOIDANCE | GENERALIZED PATTERNS | FORBIDDEN SUBSEQUENCES | permutation statistic

Motzkin path | Restricted permutation | Motzkin permutation | Generalized pattern | Permutation statistic | Chebyshev polynomial | generalized pattern | MATHEMATICS | NUMBER | restricted permutation | AVOIDANCE | GENERALIZED PATTERNS | FORBIDDEN SUBSEQUENCES | permutation statistic

Journal Article

AIMS MATHEMATICS, ISSN 2473-6988, 2020, Volume 5, Issue 2, pp. 1333 - 1345

In the paper, the authors find two explicit formulas and recover a recursive formula for generalized Motzkin numbers. Consequently, the authors deduce two...

MATHEMATICS, APPLIED | recursive formula | SERIES | Motzkin number | generating function | restricted hexagonal number | MATHEMATICS | EXPRESSIONS | explicit formula | generalized Motzkin number | CATALAN NUMBERS | Catalan number | DERIVATIVES | MULTIVARIATE BELL POLYNOMIALS | MOMENTS | generalized motzkin number | motzkin number | catalan number

MATHEMATICS, APPLIED | recursive formula | SERIES | Motzkin number | generating function | restricted hexagonal number | MATHEMATICS | EXPRESSIONS | explicit formula | generalized Motzkin number | CATALAN NUMBERS | Catalan number | DERIVATIVES | MULTIVARIATE BELL POLYNOMIALS | MOMENTS | generalized motzkin number | motzkin number | catalan number

Journal Article

Journal of Number Theory, ISSN 0022-314X, 06/2019, Volume 199, pp. 389 - 402

We study the higher-order Euler polynomials and give the corresponding monic orthogonal polynomials, which are the Meixner–Pollaczek polynomials with certain...

Generalized Motzkin number | Higher-order Euler polynomials | Orthogonal polynomial | Meixner–Pollaczek polynomial | MATHEMATICS | BERNOULLI | Meixner-Pollaczek polynomial

Generalized Motzkin number | Higher-order Euler polynomials | Orthogonal polynomial | Meixner–Pollaczek polynomial | MATHEMATICS | BERNOULLI | Meixner-Pollaczek polynomial

Journal Article

Advances in Applied Mathematics, ISSN 0196-8858, 05/2019, Volume 106, pp. 1 - 19

In this paper, we study pattern avoidances of generalized permutations and show that the number of all generalized permutations avoiding π is independent of...

Pattern avoidances | RSK correspondence | Generalized permutations | Motzkin numbers | Riordan numbers | Young tableaux | MATHEMATICS, APPLIED

Pattern avoidances | RSK correspondence | Generalized permutations | Motzkin numbers | Riordan numbers | Young tableaux | MATHEMATICS, APPLIED

Journal Article

Journal of Integer Sequences, 01/2015, Volume 18, Issue 2

Journal Article

Journal of Integer Sequences, 2018, Volume 21, Issue 8, pp. 1 - 31

Journal Article

Acta Mathematica Sinica, English Series, ISSN 1439-8516, 7/2018, Volume 34, Issue 7, pp. 1101 - 1109

We show that many well-known counting coefficients in combinatorics are Hamburger moment sequences in certain unified approaches and that Hamburger moment...

Hankel matrix | 05A99 | total positivity | 15B99 | generalized Motzkin number | infinite convexity | Mathematics, general | Hamburger moment sequence | Mathematics | 44A60 | Stieltjes moment sequence | MATHEMATICS | MATHEMATICS, APPLIED | CATALAN-LIKE NUMBERS | Combinatorics | Polynomials | Combinatorial analysis

Hankel matrix | 05A99 | total positivity | 15B99 | generalized Motzkin number | infinite convexity | Mathematics, general | Hamburger moment sequence | Mathematics | 44A60 | Stieltjes moment sequence | MATHEMATICS | MATHEMATICS, APPLIED | CATALAN-LIKE NUMBERS | Combinatorics | Polynomials | Combinatorial analysis

Journal Article

Linear Algebra and Its Applications, ISSN 0024-3795, 11/2012, Volume 437, Issue 9, pp. 2285 - 2299

We consider weighted large and small Schröder paths with up steps (1,1), down steps (1,-1) assigned the weight of 1 and with level steps (2,0) assigned the...

Non-intersecting lattice paths | Schröder numbers | Combinatorial methods | Hankel determinants | MATHEMATICS | MATHEMATICS, APPLIED | MOTZKIN NUMBERS | Lattice paths | PATHS | Schroder numbers | GENERALIZED CATALAN NUMBERS

Non-intersecting lattice paths | Schröder numbers | Combinatorial methods | Hankel determinants | MATHEMATICS | MATHEMATICS, APPLIED | MOTZKIN NUMBERS | Lattice paths | PATHS | Schroder numbers | GENERALIZED CATALAN NUMBERS

Journal Article

Discrete Mathematics, ISSN 0012-365X, 2009, Volume 309, Issue 12, pp. 3936 - 3953

We consider sequences of polynomials which count lattice paths by area. In some cases the reversed polynomials approach a formal power series as the length of...

Motzkin paths | Schröder paths | Generalized Frobenius partitions | Schroder paths | MATHEMATICS | NUMBERS

Motzkin paths | Schröder paths | Generalized Frobenius partitions | Schroder paths | MATHEMATICS | NUMBERS

Journal Article

Journal of Optimization Theory and Applications, ISSN 0022-3239, 11/2006, Volume 131, Issue 2, pp. 281 - 299

In this paper, we introduce a definition of generalized convexlike functions (preconvexlike functions). Then, under the weakened convexity, we study vector...

Calculus of Variations and Optimal Control; Optimization | Operations Research/Decision Theory | Mathematics | Theory of Computation | Hausdorff topological linear spaces | generalized Motzkin theorems of alternative | Engineering, general | Applications of Mathematics | vector optimization problems | generalized convexlike functions | Optimization | Generalized convexlike functions | Generalized Motzkin theorems of alternative | Vector optimization problems | MATHEMATICS, APPLIED | MINIMAX | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | DUALITY | OPTIMALITY CONDITIONS | Studies | Theorems | Topological manifolds | Multipliers | Mathematical analysis | Convexity | Topology | Vectors (mathematics)

Calculus of Variations and Optimal Control; Optimization | Operations Research/Decision Theory | Mathematics | Theory of Computation | Hausdorff topological linear spaces | generalized Motzkin theorems of alternative | Engineering, general | Applications of Mathematics | vector optimization problems | generalized convexlike functions | Optimization | Generalized convexlike functions | Generalized Motzkin theorems of alternative | Vector optimization problems | MATHEMATICS, APPLIED | MINIMAX | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | DUALITY | OPTIMALITY CONDITIONS | Studies | Theorems | Topological manifolds | Multipliers | Mathematical analysis | Convexity | Topology | Vectors (mathematics)

Journal Article

Annals of Combinatorics, ISSN 0218-0006, 7/2005, Volume 9, Issue 2, pp. 137 - 162

We settle some conjectures formulated by A. Claesson and T. Mansour concerning generalized pattern avoidance of permutations. In particular, we solve the...

generalized patterns avoidance | succession rules | Fibonacci and Motzkin numbers | Mathematics | Combinatorics | 05A05 | permutations | 05A15 | Generalized patterns avoidance | Permutations | Succession rules | MATHEMATICS, APPLIED

generalized patterns avoidance | succession rules | Fibonacci and Motzkin numbers | Mathematics | Combinatorics | 05A05 | permutations | 05A15 | Generalized patterns avoidance | Permutations | Succession rules | MATHEMATICS, APPLIED

Journal Article

Ars Combinatoria, ISSN 0381-7032, 01/2016, Volume 125, pp. 225 - 246

Bizley [J. Inst. Actuar. 80 (1954), 55-62] studied a generalization of Dyck paths from (0, 0) to (pn, qn) (gcd(p, q) = 1), which never go below the line py =...

Lattice paths | 05A19 | Generating functions. 2010 Mathematics Subject Classification: 05A15 | Generalized Dyck paths | 11B83 | Motzkin numbers | MATHEMATICS | lattice paths | generating functions | generalized Dyck paths

Lattice paths | 05A19 | Generating functions. 2010 Mathematics Subject Classification: 05A15 | Generalized Dyck paths | 11B83 | Motzkin numbers | MATHEMATICS | lattice paths | generating functions | generalized Dyck paths

Journal Article

Journal of Neuroscience, ISSN 0270-6474, 2014, Volume 34, Issue 31, pp. 10430 - 10437

Uncertainty is a ubiquitous feature of our daily lives. Although previous studies have identified a number of neural and peripheral physiological changes...

POSTTRAUMATIC-STRESS-DISORDER | GENERALIZED ANXIETY DISORDER | ACTIVATION | HUMAN BRAIN | DECISION-MAKING | HEART-RATE-VARIABILITY | ALZHEIMERS-DISEASE | ANTERIOR CINGULATE | ORBITOFRONTAL CORTEX | NEUROSCIENCES | AMYGDALA | Cues | Oxygen | Uncertainty | Cerebrovascular Circulation - physiology | Humans | Middle Aged | Prefrontal Cortex - physiopathology | Male | Prefrontal Cortex - pathology | Neuropsychological Tests | Magnetic Resonance Imaging | Prefrontal Cortex - blood supply | Image Processing, Computer-Assisted | Neurons - physiology | Heart Rate - physiology | Adult | Female | Photic Stimulation | Brain Injuries - pathology

POSTTRAUMATIC-STRESS-DISORDER | GENERALIZED ANXIETY DISORDER | ACTIVATION | HUMAN BRAIN | DECISION-MAKING | HEART-RATE-VARIABILITY | ALZHEIMERS-DISEASE | ANTERIOR CINGULATE | ORBITOFRONTAL CORTEX | NEUROSCIENCES | AMYGDALA | Cues | Oxygen | Uncertainty | Cerebrovascular Circulation - physiology | Humans | Middle Aged | Prefrontal Cortex - physiopathology | Male | Prefrontal Cortex - pathology | Neuropsychological Tests | Magnetic Resonance Imaging | Prefrontal Cortex - blood supply | Image Processing, Computer-Assisted | Neurons - physiology | Heart Rate - physiology | Adult | Female | Photic Stimulation | Brain Injuries - pathology

Journal Article

Acta Informatica, ISSN 0001-5903, 4/2006, Volume 42, Issue 8, pp. 603 - 616

We consider in this paper the class M k n of generalized Motzkin paths of length n, that is, lattice paths using steps (1,1), (1,−1), (k,0), where k is a fixed...

Rejection | Generalized Motzkin path | Random generation | COMPUTER SCIENCE, INFORMATION SYSTEMS | generalized Motzkin path | random generation | AREA | rejection | Studies | Algorithms | Integers | Origins | Lattices | Running | Complexity

Rejection | Generalized Motzkin path | Random generation | COMPUTER SCIENCE, INFORMATION SYSTEMS | generalized Motzkin path | random generation | AREA | rejection | Studies | Algorithms | Integers | Origins | Lattices | Running | Complexity

Journal Article

Journal of Optimization Theory and Applications, ISSN 0022-3239, 4/2017, Volume 173, Issue 1, pp. 131 - 154

This paper deals with linear systems containing finitely many weak and/or strict inequalities, whose solution sets are referred to as evenly convex polyhedral...

Linear systems | Polyhedra | 52B99 | 15A39 | Mathematics | Theory of Computation | Duality | Optimization | Calculus of Variations and Optimal Control; Optimization | 49N15 | Applications of Mathematics | Engineering, general | Operation Research/Decision Theory | Strict inequalities | Even convexity | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | SET | SEMICLOSED POLYHEDRA | PROGRAMS | CONVEX POLYHEDRA | OPTIMIZATION | Studies | Linear programming | Generalized linear models | Theorems | Mathematical analysis | Inequalities | Polyhedrons | Hulls (structures)

Linear systems | Polyhedra | 52B99 | 15A39 | Mathematics | Theory of Computation | Duality | Optimization | Calculus of Variations and Optimal Control; Optimization | 49N15 | Applications of Mathematics | Engineering, general | Operation Research/Decision Theory | Strict inequalities | Even convexity | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | SET | SEMICLOSED POLYHEDRA | PROGRAMS | CONVEX POLYHEDRA | OPTIMIZATION | Studies | Linear programming | Generalized linear models | Theorems | Mathematical analysis | Inequalities | Polyhedrons | Hulls (structures)

Journal Article

Journal of the Operational Research Society, ISSN 0160-5682, 07/2017, Volume 68, Issue 7, pp. 829 - 833

The Dependency Diagram of a Linear Programme (LP) shows how the successive inequalities of an LP depend on former inequalities, when variables are projected...

generalised Chinese remainder theorem | elimination of variables | linear congruences | projection | integer programming | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MANAGEMENT | Integer programming | Analysis | Polytopes | Mixed integer | Linear programming | Congruences | Inequalities

generalised Chinese remainder theorem | elimination of variables | linear congruences | projection | integer programming | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | MANAGEMENT | Integer programming | Analysis | Polytopes | Mixed integer | Linear programming | Congruences | Inequalities

Journal Article

Annals of Combinatorics, ISSN 0218-0006, 08/2002, Volume 6, Issue 1, pp. 65 - 76

Recently, Babson and Steingrimsson (see [2]) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern...

restricted permutations, generalized patterns, Chebyshev polynomials

restricted permutations, generalized patterns, Chebyshev polynomials

Journal Article

Journal of Combinatorial Theory, Series A, ISSN 0097-3165, 02/2017, Volume 146, pp. 312 - 343

The family Σr consists of all r-graphs with three edges D1,D2,D3 such that |D1∩D2|=r−1 and D1△D2⊆D3. A generalized triangle, Tr∈Σr is an r-graph on...

Turán number | Symmetrization | Stability | Weighted hypergraphs | Steiner systems | Generalized triangle | Blowups | Lagrangian function | THEOREM | HYPERGRAPHS | GRAPHS | MATHEMATICS | REGULARITY | Turan number

Turán number | Symmetrization | Stability | Weighted hypergraphs | Steiner systems | Generalized triangle | Blowups | Lagrangian function | THEOREM | HYPERGRAPHS | GRAPHS | MATHEMATICS | REGULARITY | Turan number

Journal Article

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