Bounds on the number of numerical semigroups of a given genus

Lower and upper bounds are given for the number ngng of numerical semigroups of genus gg. The lower bound is the first known lower bound while the upper bound significantly improves the only known bound given by the Catalan numbers. In a previous work the sequence ngng is conjectured to behave asymptotically as the Fibonacci numbers. The lower bound proved in this work is related to the Fibonacci numbers and so the result seems to be in the direction to prove the conjecture. The method used is based on an accurate analysis of the tree of numerical semigroups and of the number of descendants of the descendants of each node depending on the number of descendants of the node itself.