Communicated by R. WiegandWe present procedures to calculate the set of Arf numerical semigroups with given genus, given conductor and given genus and conductor. We characterize the Kunz coordinates of an Arf numerical semigroup. We also describe Arf numerical semigroups with fixed Frobenius number and multiplicity up to 7.

The aim of this work is to exhibit some relations between partitions of natural numbers and Arf semigroups. We also give characterizations of Arf semigroups via the hook-sets of Young tableaux of partitions.

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In this work, we give parametrizations in terms of the Kunz coordinates of numerical semigroups with multiplicity up to [Formula: see text]. We also obtain parametrizations of MED semigroups, symmetric and pseudo-symmetric numerical semigroups with multiplicity up to [Formula: see text]. These parametrizations also lead to formulas for the number of numerical semigroups, the number of MED semigroups and the number of symmetric and pseudo-symmetric numerical semigroups with multiplicity up to [Formula: see text] and given conductor.

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