A decomposition of partitions and numerical sets

Almost symmetric Arf partitions

“…A hook numerical set which is a numerical semigroup is called a hook semigroup. In [11], the authors explained hook set decomposition of a given numerical set S, and they proved that S has hook semigroup decomposition when α −1 β(S) is a strict dominant partition. Recall that a numerical semigroup S is primitive when F (S) < 2m(S).…”

“…In recent years, the authors gave very interesting relations between partitions, numerical sets and Young diagrams, see [7,11,12,16,17]. In [7], the authors studied a correspondence between numerical sets and integer partitions that leaded to a bijection between simultaneous core partitions and the integer points of a certain polytope.…”

“…In [7], the authors studied a correspondence between numerical sets and integer partitions that leaded to a bijection between simultaneous core partitions and the integer points of a certain polytope. In [11], the authors gave an associative operation ⊕ on Young diagrams which was carried to partitions and numerical sets by means of the correspondences obtained in [17]. This operation is used to introduce a decomposition of a partition into the so called hook partitions and a decomposition of a numerical set into the so called hook numerical sets.…”

“…Then a path with n horizontal and g(S) vertical segments will be obtained, the lattice lying above this path and below the horizontal line defines a Young diagram, for details see [6,7,11,16,17].…”

A decomposition of Arf semigroups

“…A connection among partitions, Young diagrams, numerical semigroups was given by [3,5,10,11,14]. We think of a path as lying in N 2 with bottom left corner of Young diagram at the origin.…”