Combinatorics and Number Theory of Counting Sequences

I Counting sequences related to set partitions and permutations

Set partitions and permutation cycles.

Generating functions

The Bell polynomials

Unimodality, log concavity and log convexity

The Bernoulli and Cauchy numbers

Ordered partitions

Asymptotics and inequalities

II Generalizations of our counting sequences

Prohibiting elements from being together

Avoidance of big substructures

Prohibiting elements from being together

Avoidance of big substructures

Avoidance of small substructures

III Number theoretical properties

Congurences

Congruences vial finite field methods

Diophantic results

Appendix

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• A matemática discreta
• Teoria dos números
• Teoria combinatória e dos grafos
• Lógica matemática

Bell Polynomials;Stirling Numbers;Combinatorics;Negative Real Line;Number Theory;Exponential Generating Function;Counting Sequences;Meixner Polynomials;Cryptography;Universal Generating Functions;finite set partitions;Bell Number;number theoretical results;Minimal Polynomial;permutation cycles;Log Concave Sequence;two-parameter counting sequences;Hankel Transform;Eulerian Polynomial;Hankel Determinant;Pascal Triangle;Power Sum;Eulerian Numbers;Cauchy Number;Combinatorial Proof;Riemann Zeta Function;Binomial Coefficients;Bernoulli Polynomials;Young Tableau;Characteristic Polynomial;Bernoulli Numbers;Bessel Polynomials;Euler Gamma Function