Old and new designs via difference multisets and strong difference families

Let X =((x_1,1,x_1,2,…,x_1,k),(x_2,1,x_2,2,…,x_2,k),…,(x_t,1,x_t,2,…,x_t,k)) be a family of t multisets of size k defined on an additive group G. We say that X is a t-(G,k,μ) strong difference family (SDF) if the list of differences (x_h,i-x_h,j∣h=1,…,t;i≠ j) covers all of G exactly μ times. If a SDF consists of a single multiset X, we simply say that X is a (G,k,μ) difference multiset. After giving some constructions for SDF's, we show that they allow us to obtain a very useful method for constructing regular group divisible designs and regular (or 1-rotational) balanced incomplete block designs. In particular cases this construction method has been implicitly used by many authors, but strangely, a systematic treatment seems to be lacking. Among the main consequences of our research, we find new series of regular BIBD's and new series of 1-rotational (in many cases resovable) BIBD's.     

On averaging multisets

We give sufficient conditions to ensure that, given a set , everyxint convM can be represented as a convex combination,x = _{i = 1} ⁿ _i x _i , wherex _i M, _i rational, andn=2s orn=2s–1, respectively.     

Partitions of multisets

A multiset is a set with repeated elements. There are four distinct partition numbers to consider, unlike the classical set partition case which involves only Stirling numbers of the second kind. Using inclusion-exclusion, we obtain generating functions when each element appears exactly r = 1, 2 or 3 times. The case r = 1 is classical and r = 2 was studied by Comtet and Baróti using other methods. Our approach also leads to asymptotic formulae for the total number of partitions of multisets in which the repetition of elements is bounded. Another approach to multiset enumeration, using de Brujin's theorem for group reduced distributions, is described.     

Some combinatorics of multisets

The purpose of this paper is to delineate some insights into the concept of multisets along with a couple of combinatorial results related to multisets. The paper indicates that a general formula needs to be worked out for determining the cardinality of the ‘Set’ (whose elements may be multisets but do not repeat) of all multisubsets of a finite multiset [x, y, z,…] _m , _p , _t , … in which x occurs at most m times, y occurs at most p times, and so on. It outlines some directions provided in the literature and points out that they all turn out to be inefficient. Finally, a relatively more efficient formula to this effect is provided along with a remark that the problem needs further vindication.     

Topological approximations of multisets

Rough set theory is a powerful mathematical tool for dealing with inexact, uncertain or vague information. The core concept of rough set theory are information systems and approximation operators of approximation spaces. In this paper, we define and investigate three types of lower and upper multiset approximations of any multiset. These types based on the multiset base of multiset topology induced by a multiset relation. Moreover, the relationships between generalized rough msets and mset topologies are given. In addition, an illustrative example is given to illustrate the relationships between different types of generalized definitions of rough multiset approximations.     

A generalized Combinatorial Nullstellensatz for multisets

The Combinatorial Nullstellensatz is one of the most powerful algebraic tools in combinatorics. The aim of this paper is to prove an extension of the Combinatorial Nullstellensatz for multisets due to Kós–Rónyai. Our generalization gives an improvement on the size of sets chosen in the statement of Combinatorial Nullstellensatz for some polynomials.     

Projection-forcing multisets of weight changes

Let F be a finite field. A multiset S of integers is projection-forcing if for every linear function ?:Fn→Fm whose multiset of weight changes is S, ? is a coordinate projection up to permutation and scaling of entries. The MacWilliams Extension Theorem from coding theory says that S={0,0,…,0} is projection-forcing. We give a (super-polynomial) algorithm to determine whether or not a given S is projection-forcing. We also give a condition that can be checked in polynomial time that implies that S is projection-forcing. This result is a generalization of the MacWilliams Extension Theorem and work by the first author.     

On universal cycles for multisets

A Universal Cycle for t-multisets of [n]={1,…,n} is a cyclic sequence of integers from [n] with the property that each t-multiset of [n] appears exactly once consecutively in the sequence. For such a sequence to exist it is necessary that n divides , and it is reasonable to conjecture that this condition is sufficient for large enough n in terms of t. We prove the conjecture completely for t{2,3} and partially for t{4,6}. These results also support a positive answer to a question of Knuth.     

A Snevily-type inequality for multisets

Acta Mathematica Hungarica - Alon [1] proved that if $$p$$ is an odd prime, $$1\le n < p$$ and $$a_1,\ldots,a_n$$ are distinct elements in $$Z_p$$ and $$b_1,\ldots,b_n$$ are arbitrary...     

Mutually describing multisets and integer partitions

To each finite multiset $A$, with underlying set $S\left(A\right)$, we associate a new multiset $d\left(A\right)$, obtained by adjoining to $S\left(A\right)$ the multiplicities of its elements in $A$. We study the orbits of the map $d$ under iteration, and show that if $A$ consists of nonnegative integers, then its orbit under $d$ converges to a cycle. Moreover, we prove that all cycles of $d$ over $\mathbb{Z}$ are of length at most $3$, and we completely determine them. This amounts to finding all systems of mutually describing multisets. In the process, we are led to introduce and study a related discrete dynamical system on the set of integer partitions of $n$ for each $n\ge 1$.     

On k-partitions of multisets with equal sums

The Ramanujan Journal - We study the number of ordered k-partitions of a multiset with equal sums, having elements $$\alpha _1,\ldots ,\alpha _n$$ and multiplicities $$m_1,\ldots ,m_n$$ . Denoting...     

Topological approaches to generalized rough multisets

This paper proposes new definitions of lower and upper mset approximations, which are basic concepts of the rough mset theory. These definitions come naturally from the concepts of multiset topologies and of ambiguity introduced in this paper. The new definitions are compared to classical definitions and are shown to be more general. In the sense, they are the only ones which can be used for any type of indiscernibility or similarity mset relation.     

Optimal pairs of incomparable clouds in multisets

We consider the partially ordered set ([k] ⁿ, ), which is defined asn-th product of the chain [k] = {0, 1, 2,...,k – 1}, and study pairs (A, B) of incomparable setsA, B [k] ⁿ, that is,a b, a b for alla A, b B or (in short notation) A BWe are concerned with the growth of the functionsf _n: {0, 1,...,k ⁿ} {0, 1,...,k ⁿ},n , defined byf _n() = max {|B|: A, B [k] ⁿ with|A| = and A B} and a characterisation of pairs (A, B), which assume this bound.In the previously studied casek = 2 our results are considerably sharper than earlier results by Seymour, Hilton, Ahlswede and Zhang.     

Semiparametric analysis in double-sampling designs via empirical likelihood

Double-sampling designs are commonly used in real applications when it is infeasible to collect exact measurements on all variables of interest. Two samples, a primary sample on proxy measures and a validation subsample on exact measures, are available in these designs. We assume that the validation sample is drawn from the primary sample by the Bernoulli sampling with equal selection probability. An empirical likelihood based approach is proposed to estimate the parameters of interest. By allowing the number of constraints to grow as the sample size goes to infinity, the resulting maximum empirical likelihood estimator is asymptotically normal and its limiting variance-covariance matrix reaches the semiparametric efficiency bound. Moreover, the Wilks-type result of convergence to chi-squared distribution for the empirical likelihood ratio based test is established. Some simulation studies are carried out to assess the finite sample performances of the new approach.     

Intersection sets,three-character multisets and associated codes

In this article we construct new minimal intersection sets in \(\mathrm {AG}(r,q^2)\) sporting three intersection numbers with hyperplanes; we then use these sets to obtain linear error correcting codes with few weights, whose weight enumerator we also determine. Furthermore, we provide a new family of three-character multisets in \(\mathrm {PG}(r,q^2)\) with r even and we also compute their weight distribution.     

A variant of the Stanley depth for multisets

We define and study a variant of the Stanley depth which we call total depth for partially ordered sets (posets). This total depth is the most natural variant of Stanley depth from $?S_k?$ – the poset of nonempty subsets of $\{1,2,\dots ,k\}$ ordered by inclusion – to any finite poset. In particular, the total depth can be defined for the poset of nonempty submultisets of a multiset ordered by inclusion, which corresponds to a product of chains with the bottom element deleted. We show that the total depth agrees with Stanley depth for $?S_k?$ but not for such posets in general. We also prove that the total depth of the product of chains ${\mathit{n}}^k$ with the bottom element deleted is $(n?1)?k∕2?$, which generalizes a result of Biró, Howard, Keller, Trotter, and Young (2010). Further, we provide upper and lower bounds for a general multiset and find the total depth for any multiset with at most five distinct elements. In addition, we can determine the total depth for any multiset with $k$ distinct elements if we know all the interval partitions of $?S_k?$.     

Multiple blocking sets and multisets in Desarguesian planes

In AG(2, q ²), the minimum size of a minimal (q ? 1)-fold blocking set is known to be q ³ ? 1. Here, we construct minimal (q ? 1)-fold blocking sets of size q ³ in AG(2, q ²). As a byproduct, we also obtain new two-character multisets in PG(2, q ²). The essential idea in this paper is to investigate q ³-sets satisfying the opposite of Ebert’s discriminant condition.