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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 |