Related papers: Counting sub-multisets of fixed cardinality
Every conditionally convergent series of real numbers has a divergent subseries. How many subsets of the natural numbers are needed so that every conditionally convergent series diverges on the subseries corresponding to one of these sets?…
The article contains some important classes of multisets. Combinatorial proofs of problems on the number of m-submultisets and m-permutations of multiset elements are considered and effective algorithms for their calculation are given. In…
We find all subsets of $\mathbb{N}$ which occur as the set of possible cardinalities of preimages of a continuous function. We also study and answer this question for various subclasses of continuous functions.
Every conditionally convergent series of real numbers has a subseries that diverges. The subseries numbers, previously studied in arXiv:1801.06206 , answer the question how many subsets of the natural numbers are necessary, such that every…
Four constructions result from a desire to create enhancements to Cantor's infinite real set cardinality. Each continues to keep Cantor's cardinality formulation in place while providing new comparisons of arbitrary infinite sets. To…
This paper describes problems concerning the range of cardinalities of sumsets and restricted sumsets of finite subsets of the integers and finite subsets of ordered abelian groups.
We consider two problems regarding some divisibility properties of the subset sums of a set $A\subseteq \{1, 2, \ldots ,n\}$. At the beginning, we study the cardinality of $A$ which has the following property: For every $d\le n$ there is a…
We consider a notion of "numerosity" for sets of tuples of natural numbers, that satisfies the five common notions of Euclid's Elements, so it can agree with cardinality only for finite sets. By suitably axiomatizing such a notion, we show…
An algorith to count, or alternatively generate, all k-element transversals of a set system is presented and compared with three known methods. For special cases it works in output-linear time.
A clonoid is a set of finitary functions from a set $A$ to a set $B$ that is closed under taking minors. Hence clonoids are generalizations of clones. By a classical result of Post, there are only countably many clones on a 2-element set.…
For any finite totally ordered set, the multisets of intervals form an abelian category. Various classes of subcategories admit natural combinatorial descriptions, and counting them yields familiar integer sequences. Surprisingly, in some…
Let $\mathbf{x}$ be a (non-empty) sequence of positive real numbers. Its achievement set $\mathcal{\mathbf{x}}$ is the set of all the possible sums of the elements of $\mathbf{x}$. The cardinal function of $\mathbf{x}$ is the function…
In this paper we introduce the notion of elementary numerosity as a special function defined on all subsets of a given set X which takes values in a suitable non-Archimedean field, and satisfies the same formal properties of finite…
We calculate the number of unary clones (submonoids of the full transformation monoid) containing the permutations, on an infinite base set. It turns out that this number is quite large, on some cardinals as large as the whole clone…
Cardinality matching is a computational method for finding the largest possible number of matched pairs of exposed and unexposed individuals from an observational dataset, with specified patterns of baseline characteristics that represent a…
It was shown that in a group of bijections of an infinite set some families of subsets, related to the cardinality of some eigenspaces, are generating. Besides, we derived a criterion for generating by sets of this kind.
For finite sets of integers $A_1, A_2 ... A_n$ we study the cardinality of the $n$-fold sumset $A_1+... +A_n$ compared to those of $n-1$-fold sumsets $A_1+... +A_{i-1}+A_{i+1}+... A_n$. We prove a superadditivity and a submultiplicativity…
In this note, we give the explicit formula for the number of multisubsets of a finite abelian group $G$ with any given size such that the sum is equal to a given element $g\in G$. This also gives the number of partitions of $g$ into a given…
Let X be an infinite set of regular cardinality. We determine all clones on X which contain all almost unary functions. It turns out that independently of the size of X, these clones form a countably infinite descending chain. Moreover, all…
Multisets are sets that allow repetition of elements. As such, multisets pave the way to a number of interesting possibilities of theoretical and applied nature. In the present work, after revising the main aspects of traditional sets, we…