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...>k</math> positive integers that add to <math>n</math>), and the number of partitions of <math>n</math> into parts of maximum size exactly <math>k</math>. These | contribution = 16.4.1: Unrestricted partitions and partitions into <math>m</math> parts ...

{{short description|Integer partition attribute, in number theory}} In [[number theory]], a '''Durfee square''' is an attribute of an [[integer partition]]. A partition of ''n'' has a Durfee square of size ''s'' if ''s' ...

...ame=Dyson>{{cite journal|last=F. Dyson|title=Some guesses in the theory of partitions|journal=Eureka (Cambridge)|year=1944|volume=8|pages=10–15|url=https://www.a By a ''[[integer partition|partition]]'' of a positive integer ''n'' we mean a finite multiset &lambda; = { &lambda;_''k'', &lam ...

...f>{{cite journal|last=Freeman J. Dyson|title=Some Guesses in The Theory of Partitions|journal=Eureka (Cambridge)|year=1944|volume=8|pages=10–15|isbn=978082180561 ...Srinivasa|first=Ramanujan|title=Some properties of ''p''(''n''), number of partitions of ''n''|journal=Proceedings of the Cambridge Philosophical Society|year=19 ...

...mathematician) |date=2008-11-01 |title=The number of smallest parts in the partitions of n |url=https://www.degruyter.com/document/doi/10.1515/CRELLE.2008.083/ht For example, there are five partitions of 4 (with smallest parts underlined): ...

...[[binomial coefficients]]. For any non-negative integer ''n'' and for any integer ''k'' included between 0 and ''n'', the component in row ''n'' and column ' ...tions]] of ''n''&hairsp;+1 into ''k''&hairsp;+1 '''''or fewer''''' ordered partitions form Bernoulli's triangle]] ...

In mathematics, '''solid partitions''' are natural generalizations of [[integer partition]]s and [[plane partition]]s defined by [[Percy Alexander MacMahon ...imensional partitions and partitions are one-dimensional partitions. Solid partitions and their higher-dimensional generalizations are discussed in the book by [ ...

'''Multiplicative partitions of factorials''' are expressions of values of the [[factorial]] function as ...t 2^1 \cdot 3^1 \cdot 5^1.</math>The number of such "sorted multiplicative partitions" of <math display="inline">n!</math> grows with <math display="inline">n</m ...

...artitions]] <math>\lambda</math> and <math>\mu</math>) is a [[non-negative integer]] that is equal to the number of [[semistandard Young tableaux]] of shape < ...<math>\lambda</math> and <math>\mu</math> are both partitions of the same integer <math>n</math> and <math>\lambda</math> is larger than <math>\mu</math> in ...

...s [[Monotonic function|weakly decreasing]] tuples; for example, two of the partitions of 8 are (5,2,1) and (3,3,1,1). * the boxes of ''T'' are filled with integers, with the integer ''i'' appearing ρ_i times ...

...her denote by <math>C \rightarrow (B)^A_k</math> the property that for all partitions <math>X_1 \cup X_2\cup \dots\cup X_k</math> of <math>\binom{C}{A}</math> th ...class <math>K</math> has the '''A-Ramsey property''' if for ever positive integer <math>k</math> and for every <math>B\in K</math> there is a <math>C \in K</ ...

A sequence of non-negative [[integer]]s <math>d_1\geq\cdots\geq d_n</math> can be represented as the degree sequ ==Relation to integer partitions== ...

...ld visit.<ref name="sarkozy"/> Nicolas is also known for his research on [[integer partition]]s,<ref name="sarkozy">{{citation | title = Jean-Louis Nicolas and the partitions ...

...ancherel measure''' is often thought of as a measure on the set of integer partitions of given order&nbsp;''n'', given by ...lancherel measure''', on the set <math>\mathcal{P}^*</math> of all integer partitions. For any <math>\theta > 0</math>, the '''Poissonized Plancherel measure wit ...

...or power series, such as the number of integer [[List of partition topics|partitions]]. It is also an important technique in the [[enumerative combinatorics]] o In this last example, <math>p(n)</math> is number of partitions of <math>n\in\mathbb{N}</math>. ...

...xt|Madritsch|Wagner|2010}} provide asymptotic formulas for the number of [[integer partition]]s in which all parts are fibbinary.{{r|madwag}} ...rcube graph]] <math>Q_d</math> of dimension <math>d</math> is indexed by [[integer]]s from 0 to <math>2^d-1</math>, so that two [[vertex (graph theory)|vertic ...

...ten functions''' are [[rational function]]s indexed by [[Integer partition|partitions of integers]] that can be used to calculate integrals of products of matrix where the sum is over all partitions λ of ''q'' {{harv|Collins|2003}}. Here &chi;^&lambda; is the char ...

...ension of [[Szemerédi regularity lemma|Szemerédi's regularity lemma]] that partitions any given graph into bounded number parts such that edges between the parts ...keness) and boundedness, as well as to describe the random-like blocks and partitions. ...

...n assignment of a community-index <math>\sigma(v)</math> (here taken as an integer from <math>1</math> to <math>b</math>) to each vertex <math>v\in V(G)</math ...

In Euclidean space, dyadic cubes may be constructed as follows: for each integer ''k'' let Δ_''k'' be the set of cubes in '''R'''^''n'' # For each integer ''k'', Δ_''k'' partitions '''R'''^''n''. ...