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Mathematical set with repetitions allowed
In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its
Multiset
Abstract data type for storing distinct values
sets, allow also the insertion and deletion of elements from the set. A multiset is a special kind of set in which an element can appear multiple times
Set_(abstract_data_type)
Approximate distinct counting algorithm
number of distinct elements in a multiset. Calculating the exact cardinality of the distinct elements of a multiset requires an amount of memory proportional
HyperLogLog
Graphical aid for deriving some concepts in combinatorics
is equal to the number of multisets of size k − 1 taken from a set of size n + 1, or equivalently, the number of multisets of size n taken from a set
Stars and bars (combinatorics)
Stars_and_bars_(combinatorics)
Total order in computer science
The multiset path ordering (>) can be defined as follows: where (≥) denotes the reflexive closure of the mpo (>), { s1,...,sm } denotes the multiset of
Path ordering (term rewriting)
Path_ordering_(term_rewriting)
Group of class templates in the C++ Standard Library
standard: std::unordered_set<T> std::unordered_map<K, V> std::unordered_multiset<T> std::unordered_multimap<K, V>. Each of these containers differ only
Unordered associative containers (C++)
Unordered_associative_containers_(C++)
Selection of items from a set
which repetition is allowed, the terms k-combination with repetition, k-multiset, or k-selection, are often used. If, in the above example, it were possible
Combination
Mathematical problem
any multiset of 2n − 1 integers has a subset of size n the sum of whose elements is a multiple of n, but that the same is not true of multisets of size
Zero-sum_problem
Multiset of prime exponents in a prime factorization
In mathematics, the prime signature of a number is the multiset of (nonzero) exponents of its prime factorization. The prime signature of a number having
Prime_signature
Number of times an object must be counted for making true a general formula
mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial
Multiplicity_(mathematics)
Class templates in the C++ programming language
current revision of the C++ standard: std::set<T> std::map<K, V> std::multiset<T> std::multimap<K, V> Each of these containers differ only on constraints
Associative_containers_(C++)
mathematics, the Dershowitz–Manna ordering is a well-founded ordering on multisets named after Nachum Dershowitz and Zohar Manna. It is often used in context
Dershowitz–Manna_ordering
Number of subsets of a given size
choose k elements from a set of n elements if repetitions are allowed. See Multiset. There are ( n + k k ) {\displaystyle {\tbinom {n+k}{k}}} strings containing
Binomial_coefficient
Data organization and storage formats
type the data is Container List Tuple Associative array, Map Multimap Set Multiset (bag) Stack Queue (example Priority queue) Double-ended queue Graph (example
List_of_data_structures
Technique in topological data analysis
growing family of spaces. Formally, a persistence barcode consists of a multiset of intervals in the extended real line, where the length of each interval
Persistence_barcode
Set of a matrix's eigenvalues
of a matrix is the set of its eigenvalues. (More precisely, it is its multiset of eigenvalues, where each eigenvalue comes with an associated multiplicity
Spectrum_of_a_matrix
Mathematical version of an order change
k-tuples over S is n k . {\displaystyle n^{k}.} If M is a finite multiset, then a multiset permutation is an ordered arrangement of elements of M in which
Permutation
Mathematical space with a notion of distance
to a multiset of elements. A multiset is a generalization of the notion of a set in which an element can occur more than once. Define the multiset union
Metric_space
Punctuation and accent mark (~, ◌̃)
logical function NOT. and in APL it additionally represents the dyadic multiset function without (set difference). In C the tilde character is used as
Tilde
Type of permutation in combinatorial mathematics
mathematics, a Stirling permutation of order k is a permutation of the multiset 1, 1, 2, 2, ..., k, k (with two copies of each value from 1 to k) with
Stirling_permutation
Estimate of the importance of a word in a document
frequently in general. Like the bag-of-words model, it models a document as a multiset of words, without word order. It is a refinement over the simple bag-of-words
Tf–idf
German computer scientist
inaugural dissertation (Habilitationsschrift) on Quicksort variants for multisets. Examiners were Thomas Ottmann, Wolfgang Janko and Jan van Leeuwen (Utrecht)
Lutz_Michael_Wegner
Character encoding standard
two-character prefix U+ was chosen as an ASCII approximation of U+228E ⊎ MULTISET UNION. a code point is an abstract representation of an UCS character by
Unicode
Multiset analogue of matroids
integer polymatroid. In this sense, a polymatroid can be considered a multiset analogue of a matroid. Let E {\displaystyle E} be a finite set. If u ,
Polymatroid
Branch of discrete mathematics
Inclusion–exclusion principle Combinatorics basics Combination · Permutation · Multiset · Combinatorial number system · List of permutation topics · Antichain
Combinatorics
Software library for the C++ programming language
The standard associative containers are set, multiset, map, multimap, hash_set, hash_map, hash_multiset and hash_multimap. There are also container adaptors
Standard_Template_Library
Number of occurrences in an experiment or study
Empirical distribution function Law of large numbers Multiset multiplicity, analogous to frequency in multiset theory Probability density function Probability
Frequency_(statistics)
22 × 3 × 97. There are 1164 chains of multisets that partition a normal multiset of weight 8, where a multiset is normal if it spans an initial interval
1000_(number)
Type of positive integer
An amenable number is a positive integer for which there exists a multiset of as many integers as the original number that both add up to the original
Amenable_number
Natural number
i and j have the same number of digits and the multiset of the digits of n coincides with the multiset of the digits of i and j.)". The On-Line Encyclopedia
6000_(number)
Israeli computer scientist
computer scientist, known e.g. for the Dershowitz–Manna ordering and the multiset path ordering used to prove termination of term rewrite systems. He obtained
Nachum_Dershowitz
Mathematical technique
applied to the nth slot. This creates multisets in the unlabelled case and sets in the labelled case (there are no multisets in the labelled case because the
Symbolic method (combinatorics)
Symbolic_method_(combinatorics)
Abstract data type
Queue Double-ended queue Priority queue Double-ended priority queue Set Multiset Disjoint-set Arrays Bit array Circular buffer Dynamic array Hash table
Stack_(abstract_data_type)
Type of programming language
singleton multiset, and ⊎ {\displaystyle \uplus } is the multiset union operation. The semantics is then defined as a reduction relation on a multiset M → M
Linda_(coordination_language)
Decision problem in computer science
problem in computer science. In its most general formulation, there is a multiset S {\displaystyle S} of integers and a target-sum T {\displaystyle T} ,
Subset_sum_problem
Musical scale in intonation
whose pitches can be expressed as products of some of the members of some multiset of generating prime factors. Powers of two are usually ignored, because
Euler–Fokker_genus
Finite ordered list of elements
3\}=\{3,2,1\}} . A tuple has a finite number of elements, while a set or a multiset may have an infinite number of elements. There are several definitions
Tuple
Design pattern in functional programming to build generic types
properties Commutative? Idempotent? Details Ordered? Unique items? List No No Free monoid Yes No Finite multiset Yes No No No Finite set Yes Yes No Yes
Monad (functional programming)
Monad_(functional_programming)
Cycles in a graph that cover each edge twice
are actually looking for a multiset of cycles so that every edge of the graph belongs to precisely two elements of the multiset. On July 10, 2026, OpenAI
Cycle_double_cover
Model to describe distributed systems
the elementary net so that N = (P, T, F) is a net. M: P → Z is a place multiset, where Z is a countable set. M extends the concept of configuration and
Petri_net
Algebraic formula
a given ideal may be present more than once, so the list represents a multiset of primary ideals); the elements q i {\displaystyle q_{i}} are unique only
Elementary_divisors
Set of data elements in databases
"relation"; although there is the difference in that a table is usually a multiset (bag) of rows where a relation is a set and does not allow duplicates.
Table_(database)
Concept in mathematics
all finite multisets with elements drawn from A, with the monoid operation being multiset sum and the monoid unit being the empty multiset. For example
Free_monoid
Algebra of formal sums
thought of as signed multisets containing finitely many elements of B {\displaystyle B} , with the multiplicity of an element in the multiset equal to its coefficient
Free_abelian_group
Heuristic test for graph isomorphism
refinement, repeatedly updates the color of each vertex according to the multiset of colors of its neighbors until a stable color configuration is reached
Weisfeiler Leman graph isomorphism test
Weisfeiler_Leman_graph_isomorphism_test
initialization expression i. The initialization expression must evaluate to multiset of tokens with a color corresponding to the color of the place C(p). A
Coloured_Petri_net
NP-complete problem in computer science
problem, or number partitioning, is the task of deciding whether a given multiset S of positive integers can be partitioned into two subsets S1 and S2 such
Partition_problem
Area within computer science
computing deals with distributed and parallel computing models, processing multisets of symbol objects in a localized manner. Thus, evolution rules allow for
Membrane_computing
Description of the behaviour of bosons
{\displaystyle S(4,3)} can be thought of as a multiset of cardinality n = 4 {\displaystyle n=4} ; the elements of such multiset are taken from the set { 1 , 2 , 3
Bose–Einstein_statistics
Mathematical optimization problem
generalization of the subset sum problem. The input to the problem is a multiset S {\displaystyle S} of n integers and a positive integer m representing
Multiple_subset_sum
Linear programming for Combinatorial optimization
configuration-LP, there is a variable for each possible configuration - each possible multiset of items that can fit in a single bin (these configurations are also known
Configuration_linear_program
Natural number
tilings of a 12-gon. 909 = 32 × 101. There are 909 non-isomorphic aperiodic multiset partitions of weight 7. 910 = 2 × 5 × 7 × 13. It is a Harshad number, a
900_(number)
Branch of mathematics that studies sets
languages. Likewise, sets and other collection-like objects, such as multisets and lists, are common datatypes in computer science and programming. In
Set_theory
Analysis of datasets using techniques from topology
persistence barcode is a multiset of intervals in R {\displaystyle \mathbb {R} } , and a persistence diagram is a multiset of points in Δ {\displaystyle
Topological_data_analysis
Unsolved problem about sums of powers
two disjoint multisets A and B of n integers each, whose first k power sum symmetric polynomials are all equal. That is, the two multisets should satisfy
Prouhet–Tarry–Escott_problem
Continuous range of values, such as wavelengths in physics
neurodevelopmental disorders. In mathematics, the spectrum of a matrix is the multiset of the eigenvalues of the matrix. In functional analysis, the concept of
Spectrum
Group of standard library class templates
std::flat_multiset, and std::flat_multimap) use std::vector as the underlying container, unlike their "non-flat" equivalents (std::set, std::map, std::multiset
Sequence_container_(C++)
German geometer (1861–1927)
German geometer, known for Eberhard's theorem partially characterizing the multisets of faces that can form convex polyhedra. Eberhard was born on 17 January
Victor_Eberhard
Any collection of sets, or subsets of a set
of mathematics, family or collection is used to mean set, indexed set, multiset, tuple, or class. It is usually used in phrases like "family of sets" because
Family_of_sets
Algorithm for estimating a count of distinct elements
Now the Flajolet–Martin algorithm for estimating the cardinality of a multiset M {\displaystyle M} is as follows: Initialize a bit-vector BITMAP to be
Flajolet–Martin_algorithm
Logical proof involving antecedents and consequents
the antecedent. In many cases, sequents are also assumed to consist of multisets or sets instead of sequences. Thus one disregards the order or even the
Sequent
Middle quantile of a data set or probability distribution
may be measurement or transcription errors. For example, consider the multiset 1, 2, 2, 2, 3, 14. The median is 2 in this case, as is the mode, and it
Median
2011 edition of the C++ programming language standard
table Associated values Equivalent keys std::unordered_set No No std::unordered_multiset No Yes std::unordered_map Yes No std::unordered_multimap Yes Yes
C++11
Infinite Cardinal number
of integers or of rationals or of algebraic numbers the set of finite multisets of integers the set of finite sequences of integers. Sets with cardinality
Beth_number
Kazakh mathematician and physicist (born 1956)
849–869. Dzhumadildaev A.S., Yeliussizov D., Stirling permutations on multisets // European Journal of Combinatorics. – 2014. – V. 36. – P. 377–392. Dzhumadildaev
Askar_Dzhumadildayev
Multiscale analog of homology group
not give meaningful multiplicities for a multiset of intervals. Instead, such multiplicities and a multiset of intervals in the extended real line are
Persistent_homology_group
Counting technique in combinatorics
must be generalized to multisets. For multisets instead of sets, (2) becomes where A − S {\displaystyle A-S} is the multiset for which ( A − S ) ⊎ S
Inclusion–exclusion_principle
Strongly NP-complete problem in computer science
whether a given multiset of integers can be partitioned into triplets that all have the same sum. More precisely: Input: a multiset S containing n positive
3-partition_problem
respectively. The imbalance sequence M G {\displaystyle M_{G}} is the multiset of all edge imbalances in G {\displaystyle G} . A sequence of non-negative
Imbalance_conjecture
Branch of non-classical logic
they are composed into more fine-grained structures, such as trees or multisets (sets that distinguish multiple occurrences of elements) or sequences
Substructural_logic
Theory of relational databases
Note: when implemented in SQL standard the "default projection" returns a multiset instead of a set, and the Π projection to eliminate duplicate data is obtained
Relational_algebra
Graph with oriented edges
graphs to have such multiple arcs (namely, they allow the arc set to be a multiset). Sometimes these entities are called directed multigraphs (or multidigraphs)
Directed_graph
Matrix with exactly one 1 per row and column
_{i}^{\,{\text{th}}}} roots of unity. The multiset union of the L i {\displaystyle L_{i}} is then the multiset of eigenvalues of P. Since writing ρ P {\displaystyle
Permutation_matrix
Natural number
series-reduced rooted trees whose leaves are integer partitions whose multiset union is an integer partition of 7. 886 = 2 × 443. It is a zero of Mertens
800_(number)
Systematic classification of 12 related enumerative problems concerning two finite sets
include the classical problems of counting permutations, combinations, multisets, and partitions either of a set or of a number. The idea of the classification
Twelvefold_way
Geometry of the location of polynomial roots
has n complex roots (if counted with their multiplicities). They form a multiset of n points in the complex plane, whose geometry can be deduced from the
Geometrical properties of polynomial roots
Geometrical_properties_of_polynomial_roots
Associative algebra used in combinatorics
Finite sub-multisets of some multiset E, ordered by inclusion The above three examples can be unified and generalized by considering a multiset E, and finite
Incidence_algebra
rewriting can be based on relational algebra or an extension thereof (e.g. multiset relational algebra with sorting, aggregation and three-valued predicates
Query_rewriting
conjecture: If the imbalance for each edge of a graph is at least 1, is the multiset of all edge imbalances always graphic? The implicit graph conjecture on
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Model from mathematical physics
field theory. This Fock space has an orthonormal basis given by finite multisets of primes. In other words, to specify one of these basis elements we can
Primon_gas
Mathematical concept in algebra
such a way that the multiset of eigenvalues of any polynomial P ( A , B ) {\displaystyle P(A,B)} in the two matrices is the multiset of the values P ( α
Commuting_matrices
Software testing method
of choices of ranges X = { n i } {\displaystyle X=\{n_{i}\}} can be a multiset[clarification needed], because there can be multiple parameters having
All-pairs_testing
Data structure holding key/value pairs
Template library ... some of its containers — the set<T>, map<T1, T2>, multiset<T>, and multimap<T1, T2> templates — are generally built using a special
Associative_array
Type of function in database management
"null") Stddev Formally, an aggregate function takes as input a set, a multiset (bag), or a list from some input domain I and outputs an element of an
Aggregate_function
Programming language
includes mathematical integers and reals, bit-vectors, sequences, sets, multisets, infinite sequences and sets, induction, co-induction, and calculational
Dafny
Topics referred to by the same term
oriented manifold Signature of a knot, in knot theory Prime signature, the multiset of exponents in the prime factorisation of a number Signature (matrix)
Signature_(disambiguation)
Biological System
using the rules means that in each step a maximal multiset of rules is applied, namely a multiset of rules such that no further rule can be added to
Mobile_membranes
the sum of values for each agent is normalized to 1); 2 Multiset - algorithm knows the multiset of values for each agent (but not their order). 3 Complete
Temporal_fair_division
Matrix in graph theory (mathematics)
adjacency matrix of G from the adjacency matrix of the complement of G. The multiset of eigenvalues of this matrix is called the Seidel spectrum. The Seidel
Seidel_adjacency_matrix
Term in mathematics
intersection has a multidegree, written as the tuple (properly though a multiset) of the degrees of defining hypersurfaces. For example, taking quadrics
Complete_intersection
Measure of similarity and diversity between sets
with the Jaccard index. Jaccard similarity also applies to bags, i.e., multisets. This has a similar formula, but the symbols used represent bag intersection
Jaccard_index
Sets whose elements have degrees of membership
sets (Atanassov, 1983), fuzzy multisets (Yager, 1986), intuitionistic L-fuzzy sets (Atanassov, 1986), rough multisets (Grzymala-Busse, 1987), fuzzy rough
Fuzzy_set
Class of binary number
is the unique partition of these numbers into two sets that have equal multisets of pairwise sums. As 19th-century mathematician Eugène Prouhet showed
Evil_number
Topics referred to by the same term
method Set (mathematics) Class (set theory) Family of sets Indexed family Multiset Parametric family Collection (Soccer Mommy album), 2017 Collection (2NE1
Collection
Relational database programming language
constructed types user-defined types. Constructed types are one of ARRAY, MULTISET, REF (reference), or ROW. User-defined types are comparable to classes
SQL
Repeated application of an operation to a sequence
then F can operate on any non-empty finite multiset by applying it to an arbitrary enumeration of the multiset. If f moreover has an identity element e
Iterated_binary_operation
Conjecture in graph theory
{\displaystyle G} , the deck of G, denoted D ( G ) {\displaystyle D(G)} , is the multiset of isomorphism classes of all vertex-deleted subgraphs of G {\displaystyle
Reconstruction_conjecture
Statistical measure of inter-rater agreement
all observers for an example is called a unit (it forms a multiset). We denote a multiset with these units as the items, U {\displaystyle U} . Alpha
Krippendorff's_alpha
Natural number
Eisenstein prime with no imaginary part. There are 677 non-isomorphic self-dual multiset partitions of weight 10. 678 = 2 × 3 × 113. It is a sphenic number, a nontotient
600_(number)
set Analytic set, Coanalytic set Suslin set Projective set Inhabited set Multiset List of set identities and relations – Equalities for combinations of sets
List_of_types_of_sets
MULTISET
MULTISET
MULTISET
MULTISET
MULTISET
MULTISET
MULTISET
MULTISET
MULTISET