Partition function (number theory) (original) (raw)
数論における分割数（ぶんかつすう、英: partition function） p(n) は自然数 n の分割（n をその順番の違いを除いて自然数の和として表す方法）の総数を表す数論的函数である。ただし、規約として p(0) = 1 および負の整数に対して p(n) = 0 と定める。
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| dbo:abstract | Die Partitionsfunktionen geben die Anzahl der Möglichkeiten an, positive, ganze Zahlen in positive, ganze Summanden zu zerlegen. Üblicherweise betrachtet man die Zerlegungen ohne Berücksichtigung der Reihenfolge. Jede solche Zerlegung wird in der Kombinatorik als (ungeordnete) Zahlpartition oder manchmal kurz Partition bezeichnet. Die Bestimmung aller Zahlpartitionen für eine bestimmte (große) natürliche Zahl ist ein wichtiges Problem sowohl in der theoretischen als auch der praktischen Informatik. Siehe dazu den Artikel Partitionierungsproblem. Die Partitionsfunktion ohne Nebenbedingungen (Anzahl der ungeordneten Zahlpartitionen von ) wird als , manchmal auch als notiert und ist Folge in OEIS. Es gibt eine Reihe von Funktionen, bei denen an die Summanden zusätzliche Bedingungen gestellt werden, zum Beispiel dass jeder Summand nur einmal vorkommen darf (strikte Zahlpartitionen), diese Variante wird ebenfalls Partitionsfunktion, manchmal auch strikte Partitionsfunktion genannt, als oder notiert und ist Folge in OEIS. Mit einer aus der Partitionsfunktion abgeleiteten zahlentheoretischen Funktion kann die Anzahl der Isomorphietypen für die endlichen abelschen Gruppen angegeben werden. (de) In number theory, 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 the five partitions 1 + 1 + 1 + 1, 1 + 1 + 2, 1 + 3, 2 + 2, and 4. No closed-form expression for the partition function is known, but it has both asymptotic expansions that accurately approximate it and recurrence relations by which it can be calculated exactly. It grows as an exponential function of the square root of its argument. The multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal number theorem this function is an alternating sum of pentagonal number powers of its argument. Srinivasa Ramanujan first discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences. For instance, whenever the decimal representation of n ends in the digit 4 or 9, the number of partitions of n will be divisible by 5. (en) 数論における分割数（ぶんかつすう、英: partition function） p(n) は自然数 n の分割（n をその順番の違いを除いて自然数の和として表す方法）の総数を表す数論的函数である。ただし、規約として p(0) = 1 および負の整数に対して p(n) = 0 と定める。 (ja) Em teoria dos números, a partição de um inteiro positivo n é uma forma de decomposição de n como soma de inteiros positivos. Duas somas são consideradas iguais somente se possuírem o mesmo número de parcelas e as mesmas parcelas, mesmo que em ordem diferente. Rigorosamente, uma partição de um inteiro positivo n é uma sequência de inteiros positivos , tais que: . As possíveis partições de um inteiro n podem ser melhor visualizadas com o uso dos chamados diagramas de Ferrers ou diagramas de Young. (pt) |
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| dbp:authorlink | Paul Erdős (en) Ken Ono (en) |
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| rdfs:comment | 数論における分割数（ぶんかつすう、英: partition function） p(n) は自然数 n の分割（n をその順番の違いを除いて自然数の和として表す方法）の総数を表す数論的函数である。ただし、規約として p(0) = 1 および負の整数に対して p(n) = 0 と定める。 (ja) Em teoria dos números, a partição de um inteiro positivo n é uma forma de decomposição de n como soma de inteiros positivos. Duas somas são consideradas iguais somente se possuírem o mesmo número de parcelas e as mesmas parcelas, mesmo que em ordem diferente. Rigorosamente, uma partição de um inteiro positivo n é uma sequência de inteiros positivos , tais que: . As possíveis partições de um inteiro n podem ser melhor visualizadas com o uso dos chamados diagramas de Ferrers ou diagramas de Young. (pt) Die Partitionsfunktionen geben die Anzahl der Möglichkeiten an, positive, ganze Zahlen in positive, ganze Summanden zu zerlegen. Üblicherweise betrachtet man die Zerlegungen ohne Berücksichtigung der Reihenfolge. Jede solche Zerlegung wird in der Kombinatorik als (ungeordnete) Zahlpartition oder manchmal kurz Partition bezeichnet. Die Bestimmung aller Zahlpartitionen für eine bestimmte (große) natürliche Zahl ist ein wichtiges Problem sowohl in der theoretischen als auch der praktischen Informatik. Siehe dazu den Artikel Partitionierungsproblem. (de) In number theory, 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 the five partitions 1 + 1 + 1 + 1, 1 + 1 + 2, 1 + 3, 2 + 2, and 4. Srinivasa Ramanujan first discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences. For instance, whenever the decimal representation of n ends in the digit 4 or 9, the number of partitions of n will be divisible by 5. (en) |
| rdfs:label | Partitionsfunktion (de) 分割数 (ja) Partition function (number theory) (en) Função de partição (matemática) (pt) |
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