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

Bell Polynomials;Stirling Numbers;Negative Real Line;Exponential Generating Function;Meixner Polynomials;Universal Generating Functions;Bell Number;Minimal Polynomial;Log Concave Sequence;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