In

^{3}''y''^{2} corresponds to the multiset .
A multiset corresponds to an ordinary set if the multiplicity of every element is one (as opposed to some larger positive integer). An

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* Fuzzy multisets
* Rough multisets
* Hybrid sets
* Multisets whose multiplicity is any real-valued step function
* Soft multisets
* Soft fuzzy multisets
* Named sets (unification of all generalizations of sets)

Basic concepts in set theory
Factorial and binomial topics

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...

, 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 element
Element may refer to:
Science
* Chemical element
Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements
In chemistry, an element is a pure substance consisting only of atoms that all ...

s. The number of instances given for each element is called the multiplicity of that element in the multiset. As a consequence, an infinite number of multisets exist which contain only elements and , but vary in the multiplicities of their elements:
* The set contains only elements and , each having multiplicity 1 when is seen as a multiset.
* In the multiset , the element has multiplicity 2, and has multiplicity 1.
* In the multiset , and both have multiplicity 3.
These objects are all different, when viewed as multisets, although they are the same set, since they all consist of the same elements. As with sets, and in contrast to tuple
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s, order
Order or ORDER or Orders may refer to:
* Orderliness
Orderliness is associated with other qualities such as cleanliness
Cleanliness is both the abstract state of being clean and free from germs, dirt, trash, or waste, and the habit of achieving a ...

does not matter in discriminating multisets, so and denote the same multiset. To distinguish between sets and multisets, a notation that incorporates square brackets is sometimes used: the multiset can be denoted as .
The cardinality
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of a multiset is constructed by summing up the multiplicities of all its elements. For example, in the multiset the multiplicities of the members , , and are respectively 2, 3, and 1, and therefore the cardinality of this multiset is 6.
Nicolaas Govert de Bruijn
Nicolaas Govert (Dick) de Bruijn (; 9 July 1918 – 17 February 2012) was a Dutch mathematician, noted for his many contributions in the fields of analysis, number theory, combinatorics and logic
Logic (from Ancient Greek, Greek: grc, wi ...

coined the word ''multiset'' in the 1970s, according to Donald Knuth
Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist
A computer scientist is a person who has acquired the knowledge of computer science
Computer science deals with the theoretical foundations of information, ...

. However, the use of the concept of multisets predates the coinage of the word ''multiset'' by many centuries. Knuth himself attributes the first study of multisets to the Indian mathematician Bhāskarāchārya, who described permutations of multisets around 1150. Other names have been proposed or used for this concept, including ''list'', ''bunch'', ''bag'', ''heap'', ''sample'', ''weighted set'', ''collection'', and ''suite''.
History

Wayne Blizard traced multisets back to the very origin of numbers, arguing that "in ancient times, the number ''n'' was often represented by a collection of ''n'' strokes, tally marks, or units." These and similar collections of objects are multisets, because strokes, tally marks, or units are considered indistinguishable. This shows that people implicitly used multisets even before mathematics emerged. Practical needs for this structure have caused multisets to be rediscovered several times, appearing in literature under different names. For instance, they were important in early AI languages, such as QA4, where they were referred to as ''bags,'' a term attributed to Peter Deutsch. A multiset has been also called an aggregate, heap, bunch, sample, weighted set, occurrence set, and fireset (finitely repeated element set). Although multisets were used implicitly from ancient times, their explicit exploration happened much later. The first known study of multisets is attributed to the Indian mathematician Bhāskarāchārya circa 1150, who described permutations of multisets. The work ofMarius Nizolius
Marius Nizolius ( it, Mario Nizolio; 1498–1576) was an Italian humanist scholar, known as a proponent of Cicero. He considered rhetoric to be the central intellectual discipline, slighting other aspects of the philosophical tradition. He is descri ...

(1498–1576) contains another early reference to the concept of multisets. Athanasius Kircher
Athanasius Kircher (2 May 1602 – 28 November 1680) was a German Jesuit
, image = Ihs-logo.svg
, caption = Christogram
A Christogram (Latin ') is a monogram or combination of letters that forms an abbre ...

found the number of multiset permutations when one element can be repeated. Jean Prestet published a general rule for multiset permutations in 1675. John Wallis
John Wallis (; la, Wallisius; ) was an English clergyman and mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), f ...

explained this rule in more detail in 1685.
Multisets appeared explicitly in the work of Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory
In algebra, ring theory is the study of ring (mathematics), rings ...

.
Other mathematicians formalized multisets and began to study them as precise mathematical structures in the 20th century. For example, Whitney (1933) described ''generalized sets'' ("sets" whose characteristic functionIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

s may take any integer
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

value - positive, negative or zero). Monro (1987) investigated the category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...

Mul of multisets and their morphisms, defining a ''multiset'' as a set with an equivalence relation between elements "of the same ''sort''", and a ''morphism'' between multisets as a function which respects ''sorts''. He also introduced a ''multinumber'': a function ''f''(''x'') from a multiset to the natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

s, giving the ''multiplicity'' of element ''x'' in the multiset. Monro argued that the concepts of multiset and multinumber are often mixed indiscriminately, though both are useful.
Examples

One of the simplest and most natural examples is the multiset ofprime
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

factors of a natural number . Here the underlying set of elements is the set of prime factors of . For example, the number 120 has the prime factorization
In number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number ...

:$120\; =\; 2^3\; 3^1\; 5^1$
which gives the multiset .
A related example is the multiset of solutions of an algebraic equation. A quadratic equation
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...

, for example, has two solutions. However, in some cases they are both the same number. Thus the multiset of solutions of the equation could be , or it could be . In the latter case it has a solution of multiplicity 2. More generally, the fundamental theorem of algebra
The fundamental theorem of algebra states that every non- constant single-variable polynomial
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, s ...

asserts that the complex
The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London
, mottoeng = Let all come who by merit deserve the most reward
, established =
, type = Public university, Public rese ...

solutions of a polynomial equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

of degree always form a multiset of cardinality .
A special case of the above are the eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a Linear map, linear transformation is a nonzero Vector space, vector that changes at most by a Scalar (mathematics), scalar factor when that linear transformation is applied to i ...

s of a matrix
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics), a rectangular array of numbers, symbols, or expressions
* Matrix (logic), part of a formula in prenex normal form
* Matrix (biology), the material in between a eukaryoti ...

, whose multiplicity is usually defined as their multiplicity as roots of the characteristic polynomial
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces a ...

. However two other multiplicities are naturally defined for eigenvalues, their multiplicities as roots of the minimal polynomial
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (mathematics), ring (which is also a commutative algebra (structure), commutative algebra) formed from the Set (mathematics), set of polynomial ...

, and the geometric multiplicity
In linear algebra, an eigenvector () or characteristic vector of a linear transformation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, str ...

, which is defined as the dimension
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

of the kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learnin ...

of (where is an eigenvalue of the matrix ). These three multiplicities define three multisets of eigenvalues, which may be all different: Let be a matrix in Jordan normal form
In linear algebra, a Jordan normal form, also known as a Jordan canonical form
or JCF,
is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to s ...

that has a single eigenvalue. Its multiplicity is , its multiplicity as a root of the minimal polynomial is the size of the largest Jordan block, and its geometric multiplicity is the number of Jordan blocks.
Definition

A multiset may be formally defined as a 2-tuple
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

where is the ''underlying set'' of the multiset, formed from its distinct elements, and $m\backslash colon\; A\; \backslash to\; \backslash mathbb^$ is a function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

from to the set of the positive
Positive is a property of Positivity (disambiguation), positivity and may refer to:
Mathematics and science
* Converging lens or positive lens, in optics
* Plus sign, the sign "+" used to indicate a positive number
* Positive (electricity), a po ...

integer
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ...

s, giving the ''multiplicity'', that is, the number of occurrences, of the element in the multiset as the number .
Representing the function by its graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discret ...

(the set of ordered pair
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s $\backslash left\backslash $) allows for writing the multiset as , and the multiset as . This notation is however not commonly used and more compact notations are employed.
If $A=\backslash $ is a finite set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

, the multiset is often represented as
:$\backslash left\backslash ,\backslash quad$ sometimes simplified to $\backslash quad\; a\_1^\; \backslash cdots\; a\_n^,$
where upper indices equal to 1 are omitted. For example, the multiset may be written $\backslash $ or $a^2b.$ If the elements of the multiset are numbers, a confusion is possible with ordinary arithmetic operations
Arithmetic (from the Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is app ...

, those normally can be excluded from the context. On the other hand, the latter notation is coherent with the fact that the prime factorization
In number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number ...

of a positive integer is a uniquely defined multiset, as asserted by the fundamental theorem of arithmetic
In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "wh ...

. Also, a monomial
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

is a multiset of indeterminates; for example, the monomial ''x''indexed family
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

, , where varies over some index-set ''I'', may define a multiset, sometimes written . In this view the underlying set of the multiset is given by the image
An image (from la, imago) is an artifact that depicts visual perception
Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...

of the family, and the multiplicity of any element is the number of index values such that $a\_i\; =\; x$. In this article the multiplicities are considered to be finite, i.e. no element occurs infinitely many times in the family: even in an infinite multiset, the multiplicities are finite numbers.
It is possible to extend the definition of a multiset by allowing multiplicities of individual elements to be infinite cardinals
Cardinal or The Cardinal may refer to:
Christianity
* Cardinal (Catholic Church), a senior official of the Catholic Church
* Cardinal (Church of England), two members of the College of Minor Canons of St. Paul's Cathedral
Navigation
* Cardina ...

instead of positive integers, but not all properties carry over to this generalization.
Basic properties and operations

Elements of a multiset are generally taken in a fixed set , sometimes called a ''universe'', which is often the set ofnatural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, w ...

s. An element of that does not belong to a given multiset is said to have a multiplicity 0 in this multiset. This extends the multiplicity function of the multiset to a function from to the set $\backslash N$ of nonnegative integer
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s. This defines a one-to-one correspondence between these functions and the multisets that have their elements in .
This extended multiplicity function is commonly called simply the multiplicity function, and suffices for defining multisets when the universe containing the elements has been fixed. This multiplicity function is a generalization of the indicator function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of a subset, and shares some properties with it.
The support of a multiset $A$ in a universe is the underlying set of the multiset. Using the multiplicity function $m$, it is characterized as
:$\backslash operatorname(A)\; :=\; \backslash left\backslash $.
A multiset is ''finite'' if its support is finite, or, equivalently, if its cardinality
:$,\; A,\; =\; \backslash sum\_m\_A(x)=\; \backslash sum\_m\_A(x)$
is finite. The ''empty multiset'' is the unique multiset with an support (underlying set), and thus a cardinality 0.
The usual operations of sets may be extended to multisets by using the multiplicity function, in a similar way as using the indicator function for subsets. In the following, and are multisets in a given universe , with multiplicity functions $m\_A$ and $m\_B.$
* Inclusion: is ''included'' in , denoted , if
:: $m\_A(x)\; \backslash le\; m\_B(x)\backslash quad\backslash forall\; x\backslash in\; U.$
* Union: the ''union'' (called, in some contexts, the ''maximum'' or ''lowest common multiple'') of and is the multiset with multiplicity function
:: $m\_C(x)\; =\; \backslash max(m\_A(x),m\_B(x))\backslash quad\backslash forall\; x\backslash in\; U.$
* Intersection: the ''intersection'' (called, in some contexts, the ''infimum'' or ''greatest common divisor'') of and is the multiset with multiplicity function
:: $m\_C(x)\; =\; \backslash min(m\_A(x),m\_B(x))\backslash quad\backslash forall\; x\backslash in\; U.$
* Sum: the ''sum'' of and is the multiset with multiplicity function
:: $m\_C(x)\; =\; m\_A(x)\; +\; m\_B(x)\backslash quad\backslash forall\; x\backslash in\; U.$
: It may be viewed as a generalization of the disjoint union
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

of sets. It defines a commutative monoid
In abstract algebra, a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (ma ...

structure on the finite multisets in a given universe. This monoid is a free commutative monoidIn abstract algebra, the free monoid on a Set (mathematics), set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequ ...

, with the universe as a basis.
* Difference: the ''difference'' of and is the multiset with multiplicity function
:: $m\_C(x)\; =\; \backslash max(m\_A(x)\; -\; m\_B(x),\; 0)\backslash quad\backslash forall\; x\backslash in\; U.$
Two multisets are ''disjoint'' if their supports are disjoint sets
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

. This is equivalent to saying that their intersection is the empty multiset or that their sum equals their union.
There is an inclusion–exclusion principle for finite multisets (similar to the one for sets), stating that a finite union of finite multisets is the difference of two sums of multisets: in the first sum we consider all possible intersections of an odd number of the given multisets, while in the second sum we consider all possible intersections of an even number of the given multisets.
Counting multisets

The number of multisets of cardinality , with elements taken from a finite set of cardinality , is called the multiset coefficient or multiset number. This number is written by some authors as $\backslash textstyle\backslash left(\backslash !\backslash !\backslash !\backslash !\backslash right)$, a notation that is meant to resemble that ofbinomial coefficient
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s; it is used for instance in (Stanley, 1997), and could be pronounced " multichoose " to resemble " choose " for $\backslash tbinom\; nk$. Unlike for binomial coefficients, there is no "multiset theorem" in which multiset coefficients would occur, and they should not be confused with the unrelated multinomial coefficient In mathematics, the multinomial theorem describes how to expand a power (mathematics), power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials.
Theorem
For any posi ...

s that occur in the multinomial theorem In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

.
The value of multiset coefficients can be given explicitly as
:$\backslash left(\backslash !\backslash !\backslash !\backslash !\backslash right)\; =\; =\; \backslash frac\; =\; ,$
where the second expression is as a binomial coefficient; many authors in fact avoid separate notation and just write binomial coefficients. So, the number of such multisets is the same as the number of subsets of cardinality in a set of cardinality . The analogy with binomial coefficients can be stressed by writing the numerator in the above expression as a rising factorial power
:$\backslash left(\backslash !\backslash !\backslash !\backslash !\backslash right)\; =\; ,$
to match the expression of binomial coefficients using a falling factorial power:
:$=\; .$
There are for example 4 multisets of cardinality 3 with elements taken from the set of cardinality 2 (, ), namely , , , . There are also 4 ''subsets'' of cardinality 3 in the set of cardinality 4 (), namely , , , .
One simple way to prove the equality of multiset coefficients and binomial coefficients given above, involves representing multisets in the following way. First, consider the notation for multisets that would represent (6 s, 2 s, 3 s, 7 s) in this form:
:
This is a multiset of cardinality = 18 made of elements of a set of cardinality = 4. The number of characters including both dots and vertical lines used in this notation is 18 + 4 − 1. The number of vertical lines is 4 − 1. The number of multisets of cardinality 18 is then the number of ways to arrange the 4 − 1 vertical lines among the 18 + 4 − 1 characters, and is thus the number of subsets of cardinality 4 − 1 in a set of cardinality 18 + 4 − 1. Equivalently, it is the number of ways to arrange the 18 dots among the 18 + 4 − 1 characters, which is the number of subsets of cardinality 18 of a set of cardinality 18 + 4 − 1. This is
:$=\; =\; 1330,$
thus is the value of the multiset coefficient and its equivalencies:
:$\backslash left(\backslash !\backslash !\backslash !\backslash !\backslash right)\backslash frac\backslash left(\backslash !\backslash !\backslash !\backslash !\backslash right),$
:
::$=\backslash frac,$
:
::$=\backslash frac\; ,$
:
::$=\backslash frac.$
One may define a generalized binomial coefficient
:$=$
in which is not required to be a nonnegative integer, but may be negative or a non-integer, or a non-real complex number
In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

. (If = 0, then the value of this coefficient is 1 because it is the empty product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

.) Then the number of multisets of cardinality in a set of cardinality is
:$\backslash left(\backslash !\backslash !\backslash !\backslash !\backslash right)=(-1)^k.$
Recurrence relation

Arecurrence relation
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

for multiset coefficients may be given as
:$\backslash left(\backslash !\backslash !\backslash !\backslash !\backslash right)\; =\; \backslash left(\backslash !\backslash !\backslash !\backslash !\backslash right)\; +\; \backslash left(\backslash !\backslash !\backslash !\backslash !\backslash right)\; \backslash quad\; \backslash mbox\; n,k>0$
with
:$\backslash left(\backslash !\backslash !\backslash !\backslash !\backslash right)\; =\; 1,\backslash quad\; n\backslash in\backslash N,\; \backslash quad\backslash mbox\backslash quad\; \backslash left(\backslash !\backslash !\backslash !\backslash !\backslash right)\; =\; 0,\backslash quad\; k>0.$
The above recurrence may be interpreted as follows.
Let := $\backslash $ be the source set. There is always exactly one (empty) multiset of size 0, and if there are no larger multisets, which gives the initial conditions.
Now, consider the case in which . A multiset of cardinality with elements from might or might not contain any instance of the final element . If it does appear, then by removing once, one is left with a multiset of cardinality of elements from , and every such multiset can arise, which gives a total of
:$\backslash left(\backslash !\backslash !\backslash !\backslash !\backslash right)$ possibilities.
If does not appear, then our original multiset is equal to a multiset of cardinality with elements from , of which there are
:$\backslash left(\backslash !\backslash !\backslash !\backslash !\backslash right).$
Thus,
:$\backslash left(\backslash !\backslash !\backslash !\backslash !\backslash right)\; =\; \backslash left(\backslash !\backslash !\backslash !\backslash !\backslash right)\; +\; \backslash left(\backslash !\backslash !\backslash !\backslash !\backslash right).$
Generating series

Thegenerating function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers (''a'n'') by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinar ...

of the multiset coefficients is very simple, being
:$\backslash sum\_^\backslash infty\; \backslash left(\backslash !\backslash !\backslash !\backslash !\backslash right)t^d\; =\; \backslash frac.$
As multisets are in one-to-one correspondence with monomials, $\backslash left(\backslash !\backslash !\backslash !\backslash !\backslash right)$ is also the number of monomial
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s of degree in indeterminates. Thus, the above series is also the Hilbert seriesIn commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded algebra, graded commutative algebra finitely generated over a field (mathematics), field are three strongly related notions which measure the gro ...

of the polynomial ring
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

$k;\; href="/html/ALL/s/\_1,\_\backslash ldots,\_x\_n.html"\; ;"title="\_1,\; \backslash ldots,\; x\_n">\_1,\; \backslash ldots,\; x\_n$
As $\backslash left(\backslash !\backslash !\backslash !\backslash !\backslash right)$ is a polynomial in , it is defined for any complex
The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London
, mottoeng = Let all come who by merit deserve the most reward
, established =
, type = Public university, Public rese ...

value of .
Generalization and connection to the negative binomial series

The multiplicative formula allows the definition of multiset coefficients to be extended by replacing ''n'' by an arbitrary number ''α'' (negative, real, complex): :$\backslash left(\backslash !\backslash !\backslash !\backslash !\backslash right)\; =\; \backslash frac\; =\; \backslash frac\; \backslash quad\backslash text\; k\backslash in\backslash N\; \backslash text\; \backslash alpha.$ With this definition one has a generalization of the negative binomial formula (with one of the variables set to 1), which justifies calling the $\backslash left(\backslash !\backslash !\backslash !\backslash !\backslash right)$ negative binomial coefficients: :$(1-X)^\; =\; \backslash sum\_^\backslash infty\; \backslash left(\backslash !\backslash !\backslash !\backslash !\backslash right)\; X^k.$ ThisTaylor series
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

formula is valid for all complex numbers ''α'' and ''X'' with < 1. It can also be interpreted as an identity of formal power series
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

in ''X'', where it actually can serve as definition of arbitrary powers of series with constant coefficient equal to 1; the point is that with this definition all identities hold that one expects for exponentiation
Exponentiation is a mathematical
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...

, notably
:$(1-X)^(1-X)^=(1-X)^\; \backslash quad\backslash text\backslash quad\; ((1-X)^)^=(1-X)^$,
and formulas such as these can be used to prove identities for the multiset coefficients.
If ''α'' is a nonpositive integer ''n'', then all terms with ''k'' > −''n'' are zero, and the infinite series becomes a finite sum. However, for other values of ''α'', including positive integers and rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...

s, the series is infinite.
Applications

Multisets have various applications. They are becoming fundamental incombinatorics
Combinatorics is an area of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geom ...

. Multisets have become an important tool in the theory of relational database
A relational database is a digital database
In , a database is an organized collection of stored and accessed electronically from a . Where databases are more complex they are often developed using formal techniques.
The (DBMS) is the tha ...

s, which often uses the synonym ''bag''. For instance, multisets are often used to implement relations in database systems. In particular, a table (without a primary key) works as a multiset, because it can have multiple identical records. Similarly, SQL
SQL ( ''S-Q-L'', "sequel"; Structured Query Language) is a domain-specific languageA domain-specific language (DSL) is a computer languageA computer language is a method of communication with a computer
A computer is a machine that can b ...

operates on multisets and return identical records. For instance, consider "SELECT name from Student". In the case that there are multiple records with name "sara" in the student table, all of them are shown. That means the result set of SQL is a multiset. If it was a set, the repetitive records in the result set were eliminated. Another application of multiset is in modeling multigraph
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s. In multigraphs there can be multiple edges between any two given vertices. As such, the entity that shows edges is a multiset, and not a set.
There are also other applications. For instance, Richard Rado
Richard Rado Fellow of the Royal Society, FRS (28 April 1906 – 23 December 1989) was a Germany, German-born United Kingdom, British mathematician whose research concerned combinatorics and graph theory. He was Jewish and left Germany to escape Na ...

used multisets as a device to investigate the properties of families of sets. He wrote, "The notion of a set takes no account of multiple occurrence of any one of its members, and yet it is just this kind of information which is frequently of importance. We need only think of the set of roots of a polynomial ''f''(''x'') or the spectrum of a linear operator."
Generalizations

Different generalizations of multisets have been introduced, studied and applied to solving problems. * Real-valued multisets (in which multiplicity of an element can be any real number) :This seems straightforward, as many definitions for fuzzy sets and multisets are very similar and can be taken over for real-valued multisets by just replacing the value range of the characteristic function (, 1
The comma is a punctuation
Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...

or $\backslash mathbb$ = respectively) by $\backslash mathbb$See also

* Frequency (statistics) as multiplicity analog * Quasi-set theory, Quasi-sets * Set theory * {{Wikiversity-inline, Partitions of multisetsReferences