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Decomposition of an integer as a sum of positive integers
combinatorics, a partition of a non-negative integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that
Integer_partition
Number of partitions of an integer
the partition function p(n) represents the number of possible partitions of a non-negative integer n. For instance, p(4) = 5 because the integer 4 has
Partition function (number theory)
Partition_function_(number_theory)
Topics referred to by the same term
computer science Integer partition, a way to write an integer as a sum of other integers Multiplicative partition, a way to write an integer as a product
Partition
Array of nonnegative integers in combinatorics
combinatorics, a plane partition is a two-dimensional array of nonnegative integers π i , j {\displaystyle \pi _{i,j}} (with positive integer indices i and j)
Plane_partition
or an ordered partition of a set, partition of a graph, partition of an integer, partition of an interval, partition of unity, partition of a matrix; see
List_of_partition_topics
solid partitions are natural generalizations of integer partitions and plane partitions defined by Percy Alexander MacMahon. A solid partition of n {\displaystyle
Solid_partition
sequence 1038 = even integer that is an unordered sum of two primes in exactly 40 ways 1039 = prime of the form 8n+7, number of partitions of 30 that do not
1000_(number)
Description of degree sequences of graphs
Erdős–Gallai theorem and the theory of integer partitions. Let m = ∑ d i {\displaystyle m=\sum d_{i}} ; then the sorted integer sequences summing to m {\displaystyle
Erdős–Gallai_theorem
In number theory, the crank of an integer partition is a certain number associated with the partition. It was first introduced without a definition by
Crank_of_a_partition
Strongly NP-complete problem in computer science
The 3-partition problem is a strongly NP-complete problem in computer science. The problem is to decide whether a given multiset of integers can be partitioned
3-partition_problem
Natural number
sum of the first 52 integers, and the sum of four consecutive primes (197 + 199 + 211 + 223). 831 = 3 × 277. There are 831 partitions of 32 into at most
800_(number)
Decomposition of a number into a product
decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater
Integer_factorization
In the number theory of integer partitions, the numbers p k ( n ) {\displaystyle p_{k}(n)} denote both the number of partitions of n {\displaystyle n}
Triangle_of_partition_numbers
Integer partition attribute, in number theory
attribute of an integer partition. A partition of n has a Durfee square of size s if s is the largest number such that the partition contains at least
Durfee_square
Mathematical concept
sum, while they are considered to define the same integer partition of that number. Every integer has finitely many distinct compositions. Negative numbers
Composition_(combinatorics)
On integer partitions from monotonic functions
this construction of partitions from inverse functions is universal, in the sense it can explain any partition of positive integers into two infinite parts
Lambek–Moser_theorem
Number used for counting
2, 3, and so on, possibly excluding 0. The terms positive integers, non-negative integers, whole numbers, and counting numbers are also used. The set
Natural_number
Term in number theory and combinatorics
theory and combinatorics, the rank of an integer partition is a certain number associated with the partition. In fact at least two different definitions
Rank_of_a_partition
NP-complete problem in computer science
science, the partition problem, or number partitioning, is the task of deciding whether a given multiset S of positive integers can be partitioned into two
Partition_problem
Mathematical optimization problem restricted to integers
An integer programming, also known as integer optimization, problem is a mathematical optimization or feasibility program in which some or all of the variables
Integer_programming
Computational method in group theory
Here λ and ρ are both integer partitions of some integer n, the order of the symmetric group under consideration. The partition λ specifies the irreducible
Murnaghan–Nakayama_rule
Branch of discrete mathematics
obtaining asymptotic formulae. Partition theory studies various enumeration and asymptotic problems related to integer partitions, and is closely related to
Combinatorics
Combinatorial object in representation theory
order. Listing the number of boxes in each row gives a partition λ of a non-negative integer n, the total number of boxes of the diagram. The Young diagram
Young_tableau
Probability of shared birthdays
the partition problem, a variant of the knapsack problem from operations research. Some weights are put on a balance scale; each weight is an integer number
Birthday_problem
Ordered list of whole numbers
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified explicitly by giving a formula
Integer_sequence
Theorem in number theory
negative integer). Here the associated sign is (−1)s with s = m − 1 = −k, therefore the sign is again (−1)k. In summary, it has been shown that partitions into
Pentagonal_number_theorem
This is a list of notable integer sequences with links to their entries in the On-Line Encyclopedia of Integer Sequences. OEIS core sequences Index to
List_of_integer_sequences
Natural number
52 + 62. the sum of the first eight prime numbers. the number of integer partitions of the number 12. the largest number that cannot be written as a sum
77_(number)
Natural number
Mertens function, a nontotient and a noncototient. There are 404 integer partitions of 20 with an alternating permutation. The HTTP 404 status code is
400_(number)
Computer data storage partitioning standard
The GUID Partition Table (GPT) is a standard for the layout of partition tables of a physical computer storage device, such as a hard disk drive or solid-state
GUID_Partition_Table
Natural number
is a sphenic number and a square pyramidal number. There are 385 integer partitions of 18. 385 = 102 + 92 + 82 + 72 + 62 + 52 + 42 + 32 + 22 + 12. 386
300_(number)
Natural number
than any before it. 299 is a self number, meaning that it has 298 integer partitions. 299 is the twelfth cake number, the maximum number of pieces to get
299_(number)
Natural number
Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A000219 (Number of planar partitions (or plane partitions) of n)". The
1,000,000,000
Mathematical identities related to integer partitions
identities are two identities related to basic hypergeometric series and integer partitions. The identities were first discovered and proved by Leonard James
Rogers–Ramanujan_identities
On the number of partitions of an integer into parts not divisible by another integer
the study of integer partitions. Proved in 1883 by James Whitbread Lee Glaisher, it states that the number of partitions of an integer n {\displaystyle
Glaisher's_theorem
British mathematician (1829–1903)
this theorem of partitions: "The number of modes of partitioning (n) into (m) parts is equal to the number of modes of partitioning (n) into parts, one
Norman_Macleod_Ferrers
American mathematician (born 1938)
of integer partitions. In 1976 he discovered Ramanujan's Lost Notebook. He is interested in mathematical pedagogy. His book The Theory of Partitions is
George Andrews (mathematician)
George_Andrews_(mathematician)
Natural number
odd composite number with two prime factors. 297 is the number of integer partitions of 17. 297 is a decagonal number which applies the properties of triangular
297_(number)
Way to write a number as a product of other numbers
multiplicative partition or unordered factorization of an integer n {\displaystyle n} is a way of writing n {\displaystyle n} as a product of integers greater
Multiplicative_partition
Group of symmetries of an n-dimensional hypercube
rank k equal to ( n k ) 2 {\displaystyle {\binom {n}{k}}^{2}} . Each integer partition λ = ⟨ λ 1 , … , λ ℓ ⟩ {\displaystyle \lambda =\langle \lambda _{1}
Hyperoctahedral_group
multipartition of a positive integer n is a way of writing n as a sum, each element of which is in turn an integer partition. The concept is also found
Multipartition
Type of group in abstract algebra
is not unique. Conjugacy classes of Sn correspond to integer partitions of n: to the partition μ = (μ1, μ2, ..., μk) with n = ∑ i = 1 k μ i {\textstyle
Symmetric_group
Complex number whose real and imaginary parts are both integers
number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and
Gaussian_integer
Figurate number
Generalized pentagonal numbers are important to Euler's theory of integer partitions, as expressed in his pentagonal number theorem. The number of dots
Pentagonal_number
Online database of integer sequences
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching
On-Line Encyclopedia of Integer Sequences
On-Line_Encyclopedia_of_Integer_Sequences
Topics referred to by the same term
the statistical mechanics concept Partition function (number theory), the number of possible partitions of an integer This disambiguation page lists articles
Partition_function
Natural number
Smith number There are 627 integer partitions of 20. 628 = 22 × 157. It is a nontotient and the totient sum for first 45 integers. 629 = 17 × 37. It is a
600_(number)
Sampling formula which describes the probabilities of alleles in a sample
same. When θ = 1, then the distribution is precisely that of the integer partition induced by a uniformly distributed random permutation. As θ → ∞, the
Ewens's_sampling_formula
inversion formula Divisor function Liouville function Partition function (number theory) Integer partition Bell numbers Landau's function Pentagonal number
List_of_number_theory_topics
Area of mathematics
namely by partitions of n or equivalently Young diagrams of size n. Each such irreducible representation can in fact be realized over the integers (every
Representation theory of the symmetric group
Representation_theory_of_the_symmetric_group
of the number of smallest parts in each integer partition of a positive integer. It is related to the partition function. The first few values of spt(n)
Spt_function
British-American mathematician
sieve of Atkin. Atkin is also known for his work on properties of the integer partition function and the monster module. He was a vocal fan of using computers
A._O._L._Atkin
Game of dice
the equivalent of dice rolls adding up to 11 and 12 pips Pub games Integer partition High Rollers, a game show which used shut the box as its primary mechanic
Shut_the_box
Polynomials in combinatorial mathematics
the number of ways the integer n can be expressed as a summation of k positive integers. This is the same as the integer partition of n into k parts. For
Bell_polynomials
Theorem in graph theory
considered as an integer partition of the same number m = ∑ i = 1 n a i {\displaystyle m=\sum _{i=1}^{n}a_{i}} . It turns out that partition ( a 1 ∗ , …
Gale–Ryser_theorem
Nearest integers from a number
returns the greatest integer less than or equal to x, written ⌊x⌋ or floor(x). Similarly, the ceiling function returns the least integer greater than or equal
Floor_and_ceiling_functions
Natural number
Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A026905 (Partial sums of the partition numbers A000041 of the positive integers)"
2000_(number)
Arithmetic operation
with remainder or Euclidean division of two natural numbers provides an integer quotient, which is the number of times the second number is completely
Division_(mathematics)
Natural number
Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A000041 (a(n) is the number of partitions of n (the partition numbers))"
100,000
First sector of partitioned PC computer disk
more bytes, forming a 64-bit integer, in little-endian notation, which are used to locate the byte offset of this partition. In this case, 00 7E corresponds
Master_boot_record
Natural number
47. It is a nontotient. There are 752 partitions of 11 into parts of 2 kinds 753 = 3 × 251. It is a blum integer. 754 = 2 × 13 × 29. It is a sphenic number
700_(number)
Lattice formed by all integer partitions
In mathematics, Young's lattice is a lattice that is formed by all integer partitions. It is named after Alfred Young, who, in a series of papers On quantitative
Young's_lattice
Mathematical function
symmetric polynomials.) Given an integer partition (that is, a finite non-increasing sequence of positive integers) λ = (λ1, ..., λm), one defines the
Elementary symmetric polynomial
Elementary_symmetric_polynomial
Measure of a scholar's citation impact
of citations among papers as a random integer partition and the h-index as the Durfee square of the partition, Yong arrived at the formula h ≈ 0.54 N
H-index
Tensor quantum field of mixed symmetry
the permutation symmetry of the Young diagram corresponding to the integer partition 3=2+1. That is to say, T α β μ = T [ α β ] μ {\displaystyle T_{\alpha
Curtright_field
British theoretical physicist and mathematician (1923–2020)
theory and combinatorics, the rank of an integer partition is a certain integer associated with the partition. Dyson introduced the concept in a paper
Freeman_Dyson
Discrete math concept
order, natural ordering) is a partial order on the set of partitions of a positive integer n that plays an important role in algebraic combinatorics and
Dominance_order
Stochastic point process in mathematics
Gaussian Unitary Ensemble. The poissonized Plancherel measure on integer partition (and therefore on Young diagrams) plays an important role in the study
Determinantal_point_process
American musician
specializing in number theory and combinatorics, particularly the theory of integer partitions and analytic number theory. After spending the first six years of
Robert_Schneider
American mathematician
unexpected way to identify prime numbers using the properties of integer partitions. In 2025 he was nominated for the Cozzarelli Prize. Beginning in 2016
Ken_Ono
Measure on group representations
with the set of integer partitions of n {\displaystyle n} . For an irreducible representation associated with an integer partition λ {\displaystyle \lambda
Plancherel_measure
{\displaystyle \lambda =(\lambda _{1},\ldots ,\lambda _{\ell })} is an integer partition of n {\displaystyle n} , then the Young subgroup S λ {\displaystyle
Young_subgroup
Type of spring shaped like a washer
unique ways to stack n {\displaystyle {n}} washers is defined by the integer partition function p(n) and increases rapidly with large n {\displaystyle {n}}
Belleville_washer
Equivalence class in mathematics
Possible patterns of bracelets of length n corresponding to the k-th integer partition (set partitions up to rotation and reflection)
Necklace_(combinatorics)
Branch of pure mathematics
rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions
Number_theory
Divide and conquer sorting algorithm
three partitions algorithm partition(A, lo, hi) is // Pivot value pivot := A[(lo + hi) / 2] // Choose the middle element as the pivot (integer division)
Quicksort
Quantum mechanical model
harmonic trap, the degeneracy scales as the number of ways to partition an integer n using integers less than or equal to N. It can be shown that the large-
Quantum_harmonic_oscillator
Australian mathematician
forms, metaplectic forms and their connections to prime numbers and integer partitions." Sloman, Leila (2022-08-15). "A Numerical Mystery From the 19th Century
Alexander Dunn (mathematician)
Alexander_Dunn_(mathematician)
Number-theoretical function
{\displaystyle r_{k}(n),\;k=1,\dots ,8} are listed in the table below: Integer partition Jacobi's four-square theorem Gauss circle problem P. T. Bateman (1951)
Sum_of_squares_function
Study of discrete mathematical structures
intersection properties. Partition theory studies various enumeration and asymptotic problems related to integer partitions, and is closely related to
Discrete_mathematics
parametrized by a positive integer k, and called k-way number partitioning. The input to the problem is a multiset S of numbers (usually integers), whose sum is k*T
Multiway_number_partitioning
\mu }} (depending on two integer partitions λ {\displaystyle \lambda } and μ {\displaystyle \mu } ) is a non-negative integer that is equal to the number
Kostka_number
Function in Chemistry
rotational partition function relates the rotational degrees of freedom to the rotational part of the energy. The total canonical partition function Z
Rotational_partition_function
Mathematical version of an order change
{\displaystyle \sigma } of a set with n elements partition that set; so the lengths of these cycles form an integer partition of n, which is called the cycle type
Permutation
Number of orderings allowing ties
ordered integer partition, a representation of n {\displaystyle n} as an ordered sum of positive integers. For instance, the ordered partition {a,b},{c}
Ordered_Bell_number
Triangle with integer side lengths
integer triangle that is unique up to congruence. So the number of integer triangles (up to congruence) with perimeter p is the number of partitions of
Integer_triangle
Statistical hypothesis test
The function u n {\displaystyle u_{n}} is closely related to the integer partition function. If p n ( t + ) {\displaystyle p_{n}(t^{+})} is the probability
Wilcoxon_signed-rank_test
Abstract strategy game
intermediate positions in an m × n Chomp are integer-partitions (non-increasing sequences of positive integers) λ1 ≥ λ2 ≥···≥ λr, with λ1 ≤ n and r ≤ m.
Chomp
Natural number
number. 231 is palindromic in base 2 (111001112). 231 is the number of integer partitions of 16. The Mertens function of 231 returns 0. A US gallon is defined
231_(number)
English mathematician and astronomer
known for Glaisher's theorem, an important result in the field of integer partitions, and for the Glaisher–Kinkelin constant, a number important in both
James_Whitbread_Lee_Glaisher
British mathematician (1877–1947)
Hardy–Weinberg principle in population genetics, his famous work on integer partitions with his collaborator Ramanujan, known as the Hardy–Ramanujan asymptotic
G._H._Hardy
convergence of the geometric series with first term 1 and ratio 1/2 Integer partition Irrational number irrationality of log23 irrationality of the square
List_of_mathematical_proofs
Mathematical space
i m ( V i ) = i {\displaystyle \mathrm {dim} (V_{i})=i} . For any integer partition λ = ( λ 1 , ⋯ , λ k ) {\displaystyle \lambda =(\lambda _{1},\cdots
Grassmannian
for many well studied sequences of integers, polynomials or power series, such as the number of integer partitions. It is also an important technique
Plethystic_exponential
Natural number
"Sequence A000041 (a(n) is the number of partitions of n (the partition numbers))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane,
10,000
Surname list
2001), American football player Durfee square, an attribute of an integer partition in mathematics This page lists people with the surname Durfee. If
Durfee_(surname)
Mathematical problem
{\displaystyle i} , where a i {\displaystyle a_{i}} is a positive integer. Partition the necklace into k {\displaystyle k} parts (not necessarily contiguous)
Necklace_splitting_problem
Relationship between elements of two sets
in order theory. In 1951 Jacques Riguet adopted the ordering of an integer partition, called a Ferrers diagram, to extend ordering to binary relations
Binary_relation
Natural number
pyramidal number 53174 = number of partitions of 42 53361 = 2312 sum of the cubes of the first 21 positive integers 54205 = Zeisel number 54688 = 2-automorphic
50,000
Natural number
On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A065577 (Number of Goldbach partitions of 10^n)". The On-Line
1,000,000
INTEGER PARTITION
INTEGER PARTITION
Boy/Male
Hindu, Indian, Traditional
Noble Partition
Boy/Male
Muslim
To wait
Boy/Male
Norse
Son's army.
Girl/Female
Scandinavian Teutonic Danish Swedish
Ing's abundance. Feminine of Ing who was Norse mythological god of the earth's fertility.
Boy/Male
Arabic, Muslim
To Wait
Boy/Male
German, Norse, Swedish
Guarded by Ing; Ing's Beauty
Girl/Female
American, Australian, Danish, Finnish, German, Scandinavian, Swedish, Teutonic
Guarded by Ing; Ing is Beautiful; Daughter of Hero; Enclosure
Female
Scandinavian
Scandinavian form of Old Norse Ingigerðr, INGEGERD means "Ing's enclosure."
Girl/Female
Danish, Finnish, German, Swedish
Guarded by Ing; Ing's Beauty; Ing's Place
Female
Swedish
Swedish contracted form of Scandinavian Ingegerd, INGER means "Ing's enclosure."
Boy/Male
Arabic
Partition; Curtain
Boy/Male
Indian, Sikh
A Partition in the World
INTEGER PARTITION
INTEGER PARTITION
Boy/Male
Muslim
Administration
Surname or Lastname
English
English : occupational name denoting the servant (Middle English man) of someone called Hudde (see Hutt).
Girl/Female
Indian
Blessed by Gods Group
Boy/Male
Tamil
Dipanjan | தீபநà¯à®œà®¨
Eye of lamp
Boy/Male
Indian
Scholar
Girl/Female
Hindu
Woman
Girl/Female
Tamil
Beautiful appearance
Girl/Female
English American
Healer.
Girl/Female
Arabic, Muslim, Sindhi
Narrator of Hadith from the Generation After the Companions (an)
Girl/Female
Latin American
Of Mars. Feminine of Marcus. Mars was mythological Roman god of fertility also identified with...
INTEGER PARTITION
INTEGER PARTITION
INTEGER PARTITION
INTEGER PARTITION
INTEGER PARTITION
n.
One who makes an index.
n.
One who gathers the vintage.
n.
That number placed below the line in vulgar fractions which shows into how many parts the integer or unit is divided.
v. t.
To bury; to inter; to entomb; as, obscurely sepulchered.
n.
A complete entity; a whole number, in contradistinction to a fraction or a mixed number.
v. t.
To deposit, as a dead body, in the earth; to bury; to inter.
a.
Essential to completeness; constituent, as a part; pertaining to, or serving to form, an integer; integrant.
n.
One who inters.
v. t.
To deposit or inter in a chapel; to enshrine.
n.
One who makes an entrance or beginning.
imp. & p. p.
of Inter
v. t.
To inter again.
v. t.
To inter with funeral rites; to bury.
v. t.
To deposit and cover in the earth; to bury; to inhume; as, to inter a dead body.
v. t.
To inter.
v. t.
To place in a tomb; to bury; to inter; to entomb.
n.
One who intends.
p. pr. & vb. n.
of Inter
v. t.
To inhume; to bury; to inter.