Young tableaux and Arf partitions

Abstract

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.

Almost symmetric Arf partitions

“…Since S is Arf, the type sequence of S is identical with Young sequence of S by [17]. Thus, Then the following statements hold:…”

“…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].…”

“…This condition, given in [2], is known as the [17], the first type set of K can be calculated as follows:…”