# Multiset

Last updated

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 elements. 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 a and b, but vary in the multiplicities of their elements:

## Contents

• The set {a, b} contains only elements a and b, each having multiplicity 1 when {a, b} is seen as a multiset.
• In the multiset {a, a, b}, the element a has multiplicity 2, and b has multiplicity 1.
• In the multiset {a, a, a, b, b, b}, a and b 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 tuples, order does not matter in discriminating multisets, so {a, a, b} and {a, b, a} denote the same multiset. To distinguish between sets and multisets, a notation that incorporates square brackets is sometimes used: the multiset {a, a, b} can be denoted as [a, a, b]. 

The cardinality of a multiset is constructed by summing up the multiplicities of all its elements. For example, in the multiset {a, a, b, b, b, c} the multiplicities of the members a, b, and c are respectively 2, 3, and 1, and therefore the cardinality of this multiset is 6.

Nicolaas Govert de Bruijn coined the word multiset in the 1970s, according to Donald Knuth.  :694 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.  :694

## 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.  :323 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).  :320 

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.  :694 The work of Marius Nizolius (1498–1576) contains another early reference to the concept of multisets.  Athanasius Kircher 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 explained this rule in more detail in 1685. 

Multisets appeared explicitly in the work of Richard Dedekind.  :114 

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 functions may take any integer value - positive, negative or zero).  :326  :405 Monro (1987) investigated the category 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 numbers, 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.  :327–328 

## Examples

One of the simplest and most natural examples is the multiset of prime factors of a natural number n. Here the underlying set of elements is the set of prime factors of n. For example, the number 120 has the prime factorization

$120=2^{3}3^{1}5^{1}$ which gives the multiset {2, 2, 2, 3, 5}.

A related example is the multiset of solutions of an algebraic equation. A quadratic equation, 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 {3, 5}, or it could be {4, 4}. In the latter case it has a solution of multiplicity 2. More generally, the fundamental theorem of algebra asserts that the complex solutions of a polynomial equation of degree d always form a multiset of cardinality d.

A special case of the above are the eigenvalues of a matrix, whose multiplicity is usually defined as their multiplicity as roots of the characteristic polynomial. However two other multiplicities are naturally defined for eigenvalues, their multiplicities as roots of the minimal polynomial, and the geometric multiplicity, which is defined as the dimension of the kernel of AλI (where λ is an eigenvalue of the matrix A). These three multiplicities define three multisets of eigenvalues, which may be all different: Let A be a n×n matrix in Jordan normal form that has a single eigenvalue. Its multiplicity is n, 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 (A, m) where A is the underlying set of the multiset, formed from its distinct elements, and $m\colon A\to \mathbb {Z} ^{+}$ is a function from A to the set of the positive integers, giving the multiplicity, that is, the number of occurrences, of the element a in the multiset as the number m(a).

Representing the function m by its graph (the set of ordered pairs $\left\{\left(a,m\left(a\right)\right):a\in A\right\}$ ) allows for writing the multiset {a, a, b} as ({a, b}, {(a, 2), (b, 1)}), and the multiset {a, b} as ({a, b}, {(a, 1), (b, 1)}). This notation is however not commonly used and more compact notations are employed.

If $A=\{a_{1},\ldots ,a_{n}\}$ is a finite set, the multiset (A, m) is often represented as

$\left\{a_{1}^{m(a_{1})},\ldots ,a_{n}^{m(a_{n})}\right\},\quad$ sometimes simplified to $\quad a_{1}^{m(a_{1})}\cdots a_{n}^{m(a_{n})},$ where upper indices equal to 1 are omitted. For example, the multiset {a, a, b} may be written $\{a^{2},b\}$ or $a^{2}b.$ If the elements of the multiset are numbers, a confusion is possible with ordinary arithmetic operations, those normally can be excluded from the context. On the other hand, the latter notation is coherent with the fact that the prime factorization of a positive integer is a uniquely defined multiset, as asserted by the fundamental theorem of arithmetic. Also, a monomial is a multiset of indeterminates; for example, the monomial x3y2 corresponds to the multiset {x, x, x, y, y}.

A multiset corresponds to an ordinary set if the multiplicity of every element is one (as opposed to some larger positive integer). An indexed family, (ai)iI, where i varies over some index-set I, may define a multiset, sometimes written {ai}. In this view the underlying set of the multiset is given by the image of the family, and the multiplicity of any element x is the number of index values i 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 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 U, sometimes called a universe, which is often the set of natural numbers. An element of U 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 U to the set $\mathbb {N}$ of nonnegative integers. This defines a one-to-one correspondence between these functions and the multisets that have their elements in U.

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 of a subset, and shares some properties with it.

The support of a multiset $A$ in a universe U is the underlying set of the multiset. Using the multiplicity function $m$ , it is characterized as

$\operatorname {Supp} (A):=\left\{x\in U\mid m_{A}(x)>0\right\}$ .

A multiset is finite if its support is finite, or, equivalently, if its cardinality

$|A|=\sum _{x\in \operatorname {Supp} (A)}m_{A}(x)=\sum _{x\in U}m_{A}(x)$ is finite. The empty multiset is the unique multiset with an empty 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, A and B are multisets in a given universe U, with multiplicity functions $m_{A}$ and $m_{B}.$ • Inclusion:A is included in B, denoted AB, if
$\forall x\in U,m_{A}(x)\leq m_{B}(x).$ • Intersection: the intersection (called, in some contexts, the infimum or greatest common divisor) of A and B is the multiset C with multiplicity function
$m_{C}(x)=\min(m_{A}(x),m_{B}(x))\quad \forall x\in U.$ • Union: the union (called, in some contexts, the maximum or lowest common multiple) of A and B is the multiset C with multiplicity function
$m_{C}(x)=\max(m_{A}(x),m_{B}(x))\quad \forall x\in U.$ • Sum: the sum of multisets may be viewed as a generalization of the disjoint union of sets, and is defined as the multiset C with multiplicity function
$m_{C}(x)=m_{A}(x)+m_{B}(x)\quad \forall x\in U.$ The sum defines a commutative monoid structure on the finite multisets in a given universe. This monoid is a free commutative monoid, with the universe as a basis.

Two multisets are disjoint if their supports are disjoint sets. 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.[ citation needed ]

## Counting multisets Bijection between 3-subsets of a 7-set (left)and 3-multisets with elements from a 5-set (right)So this illustrates that (73)=((53)){\displaystyle \textstyle {7 \choose 3}=\left(\!\!{5 \choose 3}\!\!\right)}.

The number of multisets of cardinality k, with elements taken from a finite set of cardinality n, is called the multiset coefficient or multiset number. This number is written by some authors as $\textstyle \left(\!\!{n \choose k}\!\!\right)$ , a notation that is meant to resemble that of binomial coefficients; it is used for instance in (Stanley, 1997), and could be pronounced "n multichoose k" to resemble "n choose k" for ${\tbinom {n}{k}}$ . 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 coefficients that occur in the multinomial theorem.

The value of multiset coefficients can be given explicitly as

$\left(\!\!{n \choose k}\!\!\right)={n+k-1 \choose k}={\frac {(n+k-1)!}{k!\,(n-1)!}}={n(n+1)(n+2)\cdots (n+k-1) \over k!},$ 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 k in a set of cardinality n + k 1. The analogy with binomial coefficients can be stressed by writing the numerator in the above expression as a rising factorial power

$\left(\!\!{n \choose k}\!\!\right)={n^{\overline {k}} \over k!},$ to match the expression of binomial coefficients using a falling factorial power:

${n \choose k}={n^{\underline {k}} \over k!}.$ There are for example 4 multisets of cardinality 3 with elements taken from the set {1, 2} of cardinality 2 (n = 2, k = 3), namely {1, 1, 1}, {1, 1, 2}, {1, 2, 2}, {2, 2, 2}. There are also 4 subsets of cardinality 3 in the set {1, 2, 3, 4} of cardinality 4 (n + k 1), namely {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}.

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 {a, a, a, a, a, a, b, b, c, c, c, d, d, d, d, d, d, d} (6 as, 2 bs, 3 cs, 7 ds) in this form:

|    |     |

This is a multiset of cardinality k = 18 made of elements of a set of cardinality n = 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

${4+18-1 \choose 4-1}={4+18-1 \choose 18}=1330,$ thus is the value of the multiset coefficient and its equivalencies:

$\left(\!\!{4 \choose 18}\!\!\right)={21 \choose 18}={\frac {21!}{18!\,3!}}={21 \choose 3}=\left(\!\!{19 \choose 3}\!\!\right),$ $={\frac {{\color {red}{\mathfrak {4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9\cdot 10\cdot 11\cdot 12\cdot 13\cdot 14\cdot 15\cdot 16\cdot 17\cdot 18}}}\cdot \mathbf {19\cdot 20\cdot 21} }{\mathbf {1\cdot 2\cdot 3} \cdot {\color {red}{\mathfrak {4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9\cdot 10\cdot 11\cdot 12\cdot 13\cdot 14\cdot 15\cdot 16\cdot 17\cdot 18}}}}},$ $={\frac {1\cdot 2\cdot 3\cdot 4\cdot 5\cdots 16\cdot 17\cdot 18\;\mathbf {\cdot \;19\cdot 20\cdot 21} }{\,1\cdot 2\cdot 3\cdot 4\cdot 5\cdots 16\cdot 17\cdot 18\,\times \,\mathbf {1\cdot 2\cdot 3\quad } }},$ $={\frac {19\cdot 20\cdot 21}{1\cdot 2\cdot 3}}.$ One may define a generalized binomial coefficient

${n \choose k}={n(n-1)(n-2)\cdots (n-k+1) \over k!}$ in which n is not required to be a nonnegative integer, but may be negative or a non-integer, or a non-real complex number. (If k = 0, then the value of this coefficient is 1 because it is the empty product.) Then the number of multisets of cardinality k in a set of cardinality n is

$\left(\!\!{n \choose k}\!\!\right)=(-1)^{k}{-n \choose k}.$ ### Recurrence relation

A recurrence relation for multiset coefficients may be given as

$\left(\!\!{n \choose k}\!\!\right)=\left(\!\!{n \choose k-1}\!\!\right)+\left(\!\!{n-1 \choose k}\!\!\right)\quad {\mbox{for }}n,k>0$ with

$\left(\!\!{n \choose 0}\!\!\right)=1,\quad n\in \mathbb {N} ,\quad {\mbox{and}}\quad \left(\!\!{0 \choose k}\!\!\right)=0,\quad k>0.$ The above recurrence may be interpreted as follows. Let [n] := $\{1,\dots ,n\}$ be the source set. There is always exactly one (empty) multiset of size 0, and if n = 0 there are no larger multisets, which gives the initial conditions.

Now, consider the case in which n, k > 0. A multiset of cardinality k with elements from [n] might or might not contain any instance of the final element n. If it does appear, then by removing n once, one is left with a multiset of cardinality k  1 of elements from [n], and every such multiset can arise, which gives a total of

$\left(\!\!{n \choose k-1}\!\!\right)$ possibilities.

If n does not appear, then our original multiset is equal to a multiset of cardinality k with elements from [n  1], of which there are

$\left(\!\!{n-1 \choose k}\!\!\right).$ Thus,

$\left(\!\!{n \choose k}\!\!\right)=\left(\!\!{n \choose k-1}\!\!\right)+\left(\!\!{n-1 \choose k}\!\!\right).$ ### Generating series

The generating function of the multiset coefficients is very simple, being

$\sum _{d=0}^{\infty }\left(\!\!{n \choose d}\!\!\right)t^{d}={\frac {1}{(1-t)^{n}}}.$ As multisets are in one-to-one correspondence with monomials, $\left(\!\!{n \choose d}\!\!\right)$ is also the number of monomials of degree d in n indeterminates. Thus, the above series is also the Hilbert series of the polynomial ring $k[x_{1},\ldots ,x_{n}].$ As $\left(\!\!{n \choose d}\!\!\right)$ is a polynomial in n, it is defined for any complex value of n.

### 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):

$\left(\!\!{\alpha \choose k}\!\!\right)={\frac {\alpha ^{\overline {k}}}{k!}}={\frac {\alpha (\alpha +1)(\alpha +2)\cdots (\alpha +k-1)}{k(k-1)(k-2)\cdots 1}}\quad {\text{for }}k\in \mathbb {N} {\text{ and arbitrary }}\alpha .$ With this definition one has a generalization of the negative binomial formula (with one of the variables set to 1), which justifies calling the $\left(\!\!{\alpha \choose k}\!\!\right)$ negative binomial coefficients:

$(1-X)^{-\alpha }=\sum _{k=0}^{\infty }\left(\!\!{\alpha \choose k}\!\!\right)X^{k}.$ This Taylor series formula is valid for all complex numbers α and X with |X| < 1. It can also be interpreted as an identity of formal power series 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, notably

$(1-X)^{-\alpha }(1-X)^{-\beta }=(1-X)^{-(\alpha +\beta )}\quad {\text{and}}\quad ((1-X)^{-\alpha })^{-\beta }=(1-X)^{-(-\alpha \beta )}$ ,

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 numbers, the series is infinite.

## Applications

Multisets have various applications.  They are becoming fundamental in combinatorics.     Multisets have become an important tool in the theory of relational databases, 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 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 multigraphs. 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 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."  :328–329

## 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 ([0, 1] or ℕ = {0, 1, 2, 3, ...} respectively) by ℝ0+ = [0, ∞). However, this approach cannot be easily extended for generalized fuzzy sets which use a poset or lattice instead of a simple degree of membership. Several other approaches for fuzzy multisets have been developed that don't have this restriction.
• 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)    

## Related Research Articles In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers nk ≥ 0 and is written It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n, and is given by the formula

In elementary algebra, the binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial (x + y)n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b. For example,

In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.

In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in India, Persia, China, Germany, and Italy.

In mathematics, a formal power series is a generalization of a polynomial, where the number of terms is allowed to be infinite, with no requirements of convergence. Thus, the series may no longer represent a function of its variable, merely a formal sequence of coefficients, in contrast to a power series, which defines a function by taking numerical values for the variable within a radius of convergence. In a formal power series, the powers of the variable are used only as position-holders for the coefficients, so that the coefficient of is the fifth term in the sequence. In combinatorics, the method of generating functions uses formal power series to represent numerical sequences and multisets, for instance allowing concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved. More generally, formal power series can include series with any finite number of variables, and with coefficients in an arbitrary ring. In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors and , denoted by , is called a bivector and lives in a space called the exterior square, a vector space that is distinct from the original space of vectors. The magnitude of can be interpreted as the area of the parallelogram with sides and , which in three dimensions can also be computed using the cross product of the two vectors. More generally, all parallel plane surfaces with the same orientation and area have the same bivector as a measure of their oriented area. Like the cross product, the exterior product is anticommutative, meaning that for all vectors and , but, unlike the cross product, the exterior product is associative.

In mathematics, a free abelian group or free Z-module is an abelian group with a basis, or, equivalently, a free module over the integers. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as a linear combination of basis elements with integer coefficients. For instance, the integers with addition form a free abelian group with basis {1}. Free abelian groups have properties which make them similar to vector spaces. They have applications in algebraic topology, where they are used to define chain groups, and in algebraic geometry, where they are used to define divisors. Integer lattices also form examples of free abelian groups, and lattice theory studies free abelian subgroups of real vector spaces. In combinatorics, a branch of mathematics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates with coefficients in another ring, often a field.

In mathematics, the multinomial theorem describes how to expand a 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.

In mathematics, the binomial series is the Taylor series for the function given by where is an arbitrary complex number and |x| < 1. Explicitly,

In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers that describes the vector in terms of a particular ordered basis. Coordinates are always specified relative to an ordered basis. Bases and their associated coordinate representations let one realize vector spaces and linear transformations concretely as column vectors, row vectors, and matrices; hence, they are useful in calculations.

In commutative algebra and field theory, the Frobenius endomorphism is a special endomorphism of commutative rings with prime characteristic p, an important class which includes finite fields. The endomorphism maps every element to its p-th power. In certain contexts it is an automorphism, but this is not true in general.

In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an explicit well-ordered set that represents the result of the operation or by using transfinite recursion. Cantor normal form provides a standardized way of writing ordinals. In addition to these usual ordinal operations, there are also the "natural" arithmetic of ordinals and the nimber operations.

In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles.

In combinatorics, the twelvefold way is a systematic classification of 12 related enumerative problems concerning two finite sets, which include the classical problems of counting permutations, combinations, multisets, and partitions either of a set or of a number. The idea of the classification is credited to Gian-Carlo Rota, and the name was suggested by Joel Spencer.

In mathematics, the Nørlund–Rice integral, sometimes called Rice's method, relates the nth forward difference of a function to a line integral on the complex plane. It commonly appears in the theory of finite differences and has also been applied in computer science and graph theory to estimate binary tree lengths. It is named in honour of Niels Erik Nørlund and Stephen O. Rice. Nørlund's contribution was to define the integral; Rice's contribution was to demonstrate its utility by applying saddle-point techniques to its evaluation.

In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by Hermann Weyl. There is a closely related formula for the character of an irreducible representation of a semisimple Lie algebra. In Weyl's approach to the representation theory of connected compact Lie groups, the proof of the character formula is a key step in proving that every dominant integral element actually arises as the highest weight of some irreducible representation. Important consequences of the character formula are the Weyl dimension formula and the Kostant multiplicity formula.

In 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 has a root at a given point is the multiplicity of that root.

In mathematics, a transformation of a sequence's generating function provides a method of converting the generating function for one sequence into a generating function enumerating another. These transformations typically involve integral formulas applied to a sequence generating function or weighted sums over the higher-order derivatives of these functions.

1. Hein, James L. (2003). . Jones & Bartlett Publishers. pp.  29–30. ISBN   0-7637-2210-3.
2. Knuth, Donald E. (1998). Seminumerical Algorithms. The Art of Computer Programming. 2 (3rd ed.). Addison Wesley. ISBN   0-201-89684-2.
3. Blizard, Wayne D (1989). "Multiset theory". Notre Dame Journal of Formal Logic. 30 (1): 36–66. doi:.
4. Blizard, Wayne D. (1991). "The Development of Multiset Theory". Modern Logic. 1 (4): 319–352.
5. Rulifson, J. F.; Derkson, J. A.; Waldinger, R. J. (November 1972). QA4: A Procedural Calculus for Intuitive Reasoning (Technical report). SRI International. 73.
6. Singh, D.; Ibrahim, A. M.; Yohanna, T.; Singh, J. N. (2007). "An overview of the applications of multisets". Novi Sad Journal of Mathematics. 37 (2): 73–92.
7. Angelelli, I. (1965). "Leibniz's misunderstanding of Nizolius' notion of 'multitudo'". Notre Dame Journal of Formal Logic (6): 319–322.
8. Kircher, Athanasius (1650). Musurgia Universalis. Rome: Corbelletti.
9. Prestet, Jean (1675). Elemens des Mathematiques. Paris: André Pralard.
10. Wallis, John (1685). A treatise of algebra. London: John Playford.
11. Dedekind, Richard (1888). Was sind und was sollen die Zahlen?. Braunschweig: Vieweg.
12. Syropoulos, Apostolos (2001). "Mathematics of Multisets". In Calude, C. S.; et al. (eds.). Multiset processing: Mathematical, computer science, and molecular computing points of view. Springer-Verlag. pp. 347–358.
13. Whitney, H. (1933). "Characteristic Functions and the Algebra of Logic". Annals of Mathematics. 34: 405–414. doi:10.2307/1968168.
14. Monro, G. P. (1987). "The Concept of Multiset". Zeitschrift für Mathematische Logik und Grundlagen der Mathematik. 33: 171–178. doi:10.1002/malq.19870330212.
15. Syropoulos, Apostolos (2000). "Mathematics of Multisets". Lecture Notes in Computer Science. 2235: 347--358. doi:10.1007/3-540-45523-X_17 . Retrieved 16 February 2021.
16. Aigner, M. (1979). Combinatorial Theory. New York/Berlin: Springer Verlag.
17. Anderson, I. (1987). . Oxford: Clarendon Press.
18. Stanley, Richard P. (1997). Enumerative Combinatorics. 1. Cambridge University Press. ISBN   0-521-55309-1.
19. Stanley, Richard P. (1999). Enumerative Combinatorics. 2. Cambridge University Press. ISBN   0-521-56069-1.
20. Grumbach, S.; Milo, T (1996). "Towards tractable algebras for bags". Journal of Computer and System Sciences. 52 (3): 570–588. doi:10.1006/jcss.1996.0042.
21. Libkin, L.; Wong, L. (1994). "Some properties of query languages for bags". Proceedings of the Workshop on Database Programming Languages. Springer Verlag. pp. 97–114.
22. Libkin, L.; Wong, L. (1995). "On representation and querying incomplete information in databases with bags". Information Processing Letters. 56 (4): 209–214. doi:10.1016/0020-0190(95)00154-5.
23. Blizard, Wayne D. (1989). "Real-valued Multisets and Fuzzy Sets". Fuzzy Sets and Systems. 33: 77–97. doi:10.1016/0165-0114(89)90218-2.
24. Blizard, Wayne D. (1990). "Negative Membership". Notre Dame Journal of Formal Logic. 31 (1): 346–368.
25. Yager, R. R. (1986). "On the Theory of Bags". International Journal of General Systems. 13: 23–37. doi:10.1080/03081078608934952.
26. Grzymala-Busse, J. (1987). "Learning from examples based on rough multisets". Proceedings of the 2nd International Symposium on Methodologies for Intelligent Systems. Charlotte, North Carolina. pp. 325–332.
27. Loeb, D. (1992). "Sets with a negative numbers of elements". Advances in Mathematics . 91: 64–74. doi:.
28. Miyamoto, S. (2001). "Fuzzy Multisets and their Generalizations". Multiset Processing. 2235: 225–235.
29. Alkhazaleh, S.; Salleh, A. R.; Hassan, N. (2011). "Soft Multisets Theory". Applied Mathematical Sciences. 5 (72): 3561–3573.
30. Alkhazaleh, S.; Salleh, A. R. (2012). "Fuzzy Soft Multiset Theory". Abstract and Applied Analysis.
31. Burgin, Mark (1990). "Theory of Named Sets as a Foundational Basis for Mathematics". Structures in Mathematical Theories. San Sebastian. pp. 417–420.
32. Burgin, Mark (1992). "On the concept of a multiset in cybernetics". Cybernetics and System Analysis. 3: 165–167.
33. Burgin, Mark (2004). "Unified Foundations of Mathematics". arXiv:.
34. Burgin, Mark (2011). Theory of Named Sets. Mathematics Research Developments. Nova Science Pub Inc. ISBN   978-1-61122-788-8.