On Partitions and Arf Semigroups

Abstract In this study we examine some combinatorial properties of the Arf semigroup. In previous work, the author and Karakaş, Gümüşbaş defined an Arf partition of a positive integer n. Here, we continue this work and give new results on Arf partitions. In particular, we analyze the relation among an Arf partition, its Young dual diagram, and the corresponding rational Young diagram. Additionally, this study contains some results that present the relations between partitions and Arf semigroup polynomials.

Partitions occur in several branches of physics and mathematics such as representation theory and coding theory, see [12,13]. Partitions can be visualized with Young diagrams, see [9,17]. The Young diagram of a partition λ consists of a left-justi ed shape of r columns of boxes with lengths λ , λ , . . . , λr. Flipping a Young diagram over its main diagonal (from upper left to lower right) gives the conjugate diagram. The conjugate partition of λ is the partition corresponding to the conjugate diagram of the Young diagram of λ.
For example, we consider the Young diagram of the partition λ = [ , , ]. In the Young diagram of λ, we have boxes in the rst column, we have boxes in the second column and one box in the third column. Hence, we obtain Young diagrams of λ and the conjugate partition of λ, respectively, as follows: λ= [ , , ] [ , , , ] In a Young diagram, the number of boxes in a column (or a row) is called the length of that column (or, respectively, that row). The length of a row is at most the number of columns of the diagram, and there may be more than one row with the same length.
Assume that there are r columns in a Young diagram and there are u i rows of length i, for each i = , , . . . , r, u i ≥ . Then we denote such a Young diagram of the form (shape) Y = u u u · · · r ur and we have n = r j= ju j . If there is no row of length j, ≤ j ≤ r, then u j = and we omit j in the presentation of a Young diagram Y.
If λ = [λ , λ , . . . , λr] is the partition corresponding to Y = u u · · · r ur , then Note that λ j − λ j+ = u j for each j = , . . . , r − and λr = ur .  The correspondence λ → Y λ is a bijection between the set of partitions of positive integers and the set of Young diagrams.
The Young tableau (plural, "tableaux") of a Young diagram is obtained by placing the numbers , . . . , m in the m boxes of the diagram. A standard Young tableau is a Young tableau in which the numbers form an increasing sequence along each line and along each column.
Given a box of a diagram, the shape formed by the boxes directly to the right of it, the boxes directly below it and the box itself is called the hook of that box. The number of boxes in the hook of a box is called the hook length of that box. Thus a Young diagram with the hook lengths of boxes in it is a Young tableau. Thus we can identify the Young diagram of a partition with a Young tableau.
The hook set of a partition λ is the set of hook lengths of the Young diagram of λ. The hook set of a partition encodes information about the other combinatorial objects related to that partition. The most famous is the hook-length formula which gives the degree of the corresponding irreducible representation of the symmetric group and also counts the number of standard Young tableaux that have the shape of that partition, see [9,13].
We denote the set of positive integers by N and we put N = N ∪ { }. The cardinality of any set K will be denoted by |K|. For two subsets A, B of N , we set A numerical semigroup S is a monoid of N , and it has a nite complement G(S) := N \S. The elements of G(S) are called gaps of S and g := |G(S)| is called the genus of S. The largest element of G(S) is called the Frobenius number and denoted by F(S). The conductor of S is c := F(S) + . We say that S is generated by X ⊆ S, if In this case, X is a system of generators of S and we denote S by X . If X = {x , . . . , x k }, then we write S = x , . . . , x k . Note that a system of generators of a numerical semigroup is a minimal system of generators if none of its proper subsets generates the numerical semigroup. Let {n < n < · · · < ne} be the minimal system of generators of S. Then n is known as multiplicity, and e is called the embedding dimension of S.
If S is a numerical semigroup, then we assume S = { = s , s , . . . , sr , −→}, where "−→" means that all subsequent natural numbers which are bigger then sr belong to S and r denotes the number of small elements of S.
Numerical semigroups have several applications to branches of mathematics such as algebraic geometry, number theory, coding theory. For example, the computation of the minimum distance of algebraic geometric codes involves computations in the Weierstrass semigroup, see [5].
A connection between Young diagrams and numerical semigroups was extended by [7,15]. A partition λ with no hook lengths divisible by a is called an a-core partition. The set of a-cores is in nite but the number of partitions that are both a-cores and b-cores, simultaneous (a, b)-cores, is nite. In [7], the authors studied correspondences between numerical sets (subsets of N which have nite complement and contain zero) and partitions of a positive integer. They count the set of simultaneous (a, b)-cores that come from semigroups for a certain pair (a, b). Moreover, some formulas for the number of partitions with a given hook set and some asymptotic results for the number of semigroups are given in [7]. In [4], the authors proved that a numerical semigroup is presented by a unique Dyck path of order given by its genus, and analyzed some properties such as weight, symmetry by means of a square diagram.
Given a numerical semigroup S ≠ N , we construct a uniquely determined Young diagram and thus a uniquely determined partition as follows. We use the rst quadrant of the cartesian xy-plane for the construction by drawing a continuous polygonal path which starts from the origin. Starting with x = .
• If x ∈ S, then we draw a line segment of unit length to the right.
• If x ∉ S, then we draw a line segment of unit length up. • Repeat for x + .
For any x greater than the Frobenius number of S we draw a line to the right. The lattice lying above the path and below the horizontal line de nes the Young diagram of S. If the Young diagram of a partition λ and a numerical semigroup S are the same, we say that λ is the partition of S. For S = { , , , →} and G(S) = { , , , }, we obtain λ = [ , ] and we have the following path.
x axis y axis The association of a Dyck path to a numerical semigroup follows from the association of numerical semigroups with sequences of s and s and then assigned either up-or-right moves to each, for detail see [17]. There are other papers associating paths in the plane to numerical semigroups and vise versa, for instance [11].
Unless otherwise stated we will make the following assumptions and notations: ].
• Y S : The Young diagram corresponding to a numerical semigroup S.
• The jth column of Y S is denoted by G j , for each j ≥ . The set of hook lengths of boxes which are in the jth column is identi ed with G j . The construction of Y S implies that the jth column G j corresponds to s j , j = , , . . . r − . For j ≥ r, s j is greater than or equal to the Frobenius number of S, then there is no box for sr at the diagram. We know that G = G (S). Given a box in the G , the number of the boxes below shows the number of gaps before the hook length in a given box.
Here note that λ = [ , ] can not represent a numerical semigroup. Otherwise, ∈ S and S must be N , but , are not in S = { , , , →}. We have the following tableau Thus the correspondence λ −→ S is not a bijection between the set of partitions and the set of numerical semigroups. However, the correspondence S −→ Y S is a bijection between the set of numerical sets and the set of Young diagrams.
Numerical semigroups have become important because of their applications in algebraic geometry. Valuations of analytically unrami ed one-dimensional local Noetherian domains are numerical semigroups under certain conditions, and many properties of these rings can be characterized in terms of their associated numerical semigroups, see [2,3,18]. Du Val showed geometrically how multiplicity sequences of the blowups of a curve can be used to classify singularities. Arf showed the algebraic counterpart of Du Val's results. Arf's aim was to calculate the Arf ring closure of the coordinate ring of a curve and then its value semigroup (which is an Arf numerical semigroup), see [2,10].
This property is equivalent to x−y ∈ S, for all x, y ∈ S with y ≤ x. For example, N and S = { , , , , , → } are Arf numerical semigroups. The Arf closure of a numerical semigroup S is the smallest (with respect to set inclusion) Arf semigroup containing S. There are several equivalent conditions on Arf semigroups, see [3,8,10,14,18]. In [10,14], the authors give parametrizations of numerical semigroups with multiplicity up to . In [19], an algorithm is given for nding the Arf closure of a numerical set.
Here, we investigate the properties of the Arf partitions of a positive integer using the set of gaps of an Arf numerical semigroup. In Section 2, rstly, we explain the connection between Arf semigroup and Young tableau. Also, we determine partitions of some numerical semigroup families; Proposition 2.7 is about the partitions of Lipman semigroups (see page 347). Let S (k) be a numerical semigroup with the minimal system of generators < , k, k + t, k + t + >, where k ≡ (mod ) and t is an odd integer with t ≥ . Then, we obtain the Arf partition of S (k) , more precisely, in Proposition 2.8 we show that this partition is The intersection of two semigroups gives a binary operation over a subset of the set of partitions of positive integers. We denote this operation by and it is detailed in the proof of Theorem 2.9, and we prove that the set of partitions obtained from the sets of gaps of all numerical semigroups is a semigroup with the operation . In particular, we obtain that the set of Arf partitions is a semigroup with the operation .
Let λ = [λ , . . . , λr] ∈ N r be a partition of length r. If β i = λ − λ r+ −i , ≤ i ≤ r, then β is called the dual partition of λ. For more details on the concept of the duality, see [9]. In Section 3, we de ne the Young dual of a numerical semigroup using the concept of the dual partition. Given a numerical semigroup S, we determine the elements of the numerical set D which is the Young dual of S, and we give conditions for D to be a numerical semigroup and an Arf semigroup, see Propositions 3.4 and 3.5. Let λ be an Arf partition of a natural number n, and r be the length of λ. Then we show that the dual of λ is also a partition of n with the same length, but it may not be an Arf partition. Furthermore, for the rational diagram of a partition λ, de ned in De nition 2, we analyze the behavior of the numerical semigroup corresponding to λ. Corollary 3.7 states that for any Arf partition λ, there exists a partition β such that the rational diagram of β can be represented with λ as denominator and another Arf partition.
In Section 4, we give some relations between semigroup polynomials and Arf partitions (Lemma 4.1 and Theorem 4.2), and we achieve the generating functions of semigroups given in Proposition 2.7 and Proposition 2.8.

Arf Semigroup and Young Tableau
Let S be a numerical semigroup of genus g and G(S) = {b , . . . , bg}. We set α(S) = (α , . . . αg) with α i = b i − i, for all i ≤ g, which is called the Schubert index of S. The sum w(S) = g i= α i is said to be the weight of S. The notion of the weight w(S) indicates the di erence between the semigroup < g + , . . . , g + > and S.
It is obvious that every S(i) is a numerical semigroup, and we obtain a semigroup chain: . We have another nite chain of semigroups obtained by Lipman semigroup of S: Theorem 2.2 explains that the behavior of the semigroup S over the Young diagram.

Proposition 2.5. If S is an Arf semigroup, then the following statements hold:
(1) If Y S = u u · · · r ur , then u i ≠ , for ≤ i ≤ r.
where c is the conductor of S, s i , s i− ∈ S). In fact, if g j − g j− > , for some g j < F(S), then g j − , g j − ∈ S and (g j − ) − (g j − ) = g j ∈ G . But this is a contradiction. Since u i = s i − s i− − , λ i = r j=i u j , we obtain u i ≥ , ≤ i ≤ r and λ i ≠ λ i+ . Proof. Using Theorem 2.2, we obtain that the hook set G i is a subset of the complement of the semigroup S(i), for ≤ i ≤ r. For an Arf semigroup S, we have the following equivalent conditions: Hence, the related semigroups with an Arf semigroup S can be obtained over the Young diagram Y S . Proof. We prove the proposition by induction on i.
Proposition 2.8. Let S (k) be a numerical semigroup with the minimal system of generators < , k, k+t, k+t+ >, where k ≡ (mod ) and t is an odd integer with t ≥ . Then the corresponding Arf partition to S (k) is Proof. Using induction method, we prove that the set of gaps of S (k) is G(S (k) ) = { , , , , , , , . . . , k − , k − , k − , k − , k − , k + , k + , . . . , k + t − }, and the conductor is k + t − . Hence, the jth part of the partition of S (k) is Using Corollary 3.19 in [18] and induction method, we obtain that S (k) is an Arf semigroup. Hence, λ is an Arf partition.
We remark that the intersection of two numerical semigroups is again a numerical semigroup. A consequence of the closure of this operation can be seen in Theorem 2.9.
Theorem 2.9. Let P denote the set of partitions obtained from the set of numerical semigroups. Then P is a semigroup.
Proof. Let S and T be numerical semigroups corresponding to partitions λ and β, respectively. Since S T = T S , we may assume that F(S) ≥ F(T). Now, we use the following notations: The e ect of the intersection of two semigroups can be explained as follows: let λ ≥ β . If there is an element b = s d ∈ S\T, then the corresponding column of the diagram of S must be deleted and Since u d+ is the number of consecutive gap numbers which are between s d and s d+ , the number u d+ must be added to u d . Thus the previous column has u d+ + more boxes for gaps which are between s d and s d+ . Hence, we obtain u d+ + + u d consecutive gap numbers in S T. Denote α j = |{b ∈ S\T : s i j < b < s i j+ }|, j ≤ l, and let p j denote the number of the consecutive elements of S T which are greater than or equal to s i j . If t j denotes the length of the jth gap block of S T, then we obtain t j = (t j− + p j ) and this number repeats m j = i j+ z=i j + uz + α j times. If λ < β , the proof follows from the similar argument. Then the intersection of two semigroups gives a binary operation which is denoted by in P: α β = γ, where γ = t m t m · · · t m l l , t j = (t j− + p j ) and m j = i j+ z=i j + uz + α j , Associativity is clear as a property of intersection, [ ] is unit which represents N . Proof. Because of the fact that the intersection of two Arf semigroups is an Arf semigroup, the proof follows from Proposition 2.5 and Theorem 2.9.
With the same notation as in the proof of Theorem 2.9, we have that α β = γ = m m · · · l m l , where α, β ∈ A, m j = i j+ −i j z= u i j +z + α j and m j ≠ , for ≤ j ≤ l.

The Young dual of an Arf semigroup
Schubert calculus uses the Young diagrams for polynomials. The rational diagram is used for the calculation of rational Schubert polynomials, see [1]. Note that the block [−vr , ..., −v k+ ] of the rational diagram which corresponds to the values v j < gives an inverted diagram. Therefore, we consider the reverse ordering for calculating the hook lengths of the block [−v r+ , ..., −v k+ ]. The concept of the dual of a numerical semigroup has been viewed in [3]. Now, we de ne a new duality concept for a numerical semigroup. The main motivation comes from Schubert calculus (see duality theorem) and it arises naturally by using partitions.
(2) The set of hook lengths of d λ is called the Young dual of S.
Since S is a numerical semigroup, ∈ G(S) and λ − λ > . Note that sr corresponds to the part λ r+ . Then we may assume λ r+ = , we need this for compatibility of the construction of the dual partition of a numerical semigroup.
Example 3.2. Now we consider λ = [ , ]. The duality relation between Y [ , ] and Y [ , ] can be seen by the following diagrams Y [ , ] where the double line determines the rational diagram. Hence, we have two hook sets, G = { , , , } and T = { , , , }. We obtain G as the hook set of Y [ , ] and T as the hook set of the diagram Y [ , ] . We note that Y [ , ] and Y [ , ] are dual to each other. But T is not the set of gaps of a numerical semigroup. The rational diagram corresponding to λ and hook sets can be seen as follows: (2) c = λ i= b i +b λ −i+ + r i= s i +s r−i .
In general, the set S d λ is a numerical set but it may not be a semigroup.
Proof. Using Proposition 3.4 and the de nition of Arf partition, one can obtain the Corollary 3.6 (1). For (2) Y λ . Here Y λ corresponds to the Arf semigroup of λ and Y [r] is the diagram of the semigroup { , r + , −→} which is also Arf. There are several papers dealing with the polynomial P S (x), see [6] and [16]. We can associate semigroup polynomials with a partition of a natural number n. For G = N \S, we have a partition whose hook set is G . For any hook number j which occurs in the rst column, we form a polynomial involving a sum of powers x j . Adding a column to the left of the diagram of λ means the multiplication of the polynomial of λ by x. We can illustrate this association in the following table. Proof. The complement of the hook set of the rst column of Y S is { , u + , u + u + , u + u + u + , . . . , u + u + · · · + ur + r, −→}.

Arf semigroup polynomial
Hence, we obtain (2) S( )(x) = S(x) and for k ≥ , we have Proof. Let k = v + , t = + a , a ≥ . Proposition 2.8 states the partition of S (k) and we get Y S = · · · v (v + ) · · · (v + a + ). Since S (k) is an Arf semigroup, the type sequence is { , . . . , Therefore, Hence, P S (k) (x) = + (x − ) s∈N \S (k) x s = x k+t +x k +x k+t+ + (x+ )(x + ) and the generating function associated to S (k) is Proof. We see that S (k) is an Arf semigroup and L i (S (k) ) is also Arf, ≤ i < k. Proposition 2.7 states the partition of the semigroup L i (S (k) ), and we get Y L i (S (k) ) = m− m− · · · (k − i) m− . Using Corollary 2.4, we obtain the type