Schröder–Hipparchus number (original) (raw)
In combinatorics, the Schröder–Hipparchus numbers form an integer sequence that can be used to count the number of plane trees with a given set of leaves, the number of ways of inserting parentheses into a sequence, and the number of ways of dissecting a convex polygon into smaller polygons by inserting diagonals. These numbers begin 1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, ... (sequence in the OEIS).
| Property | Value |
|---|---|
| dbo:abstract | En théorie des nombres, les nombres de Schröder-Hipparque forment une suite d'entiers qui servent à compter les arbres planaires avec un ensemble donné de feuilles, les insertions de parenthèses dans une suite, et le nombre de façons de découper un polygone convexe en polygones plus petits par l'insertion de diagonales. Cette suite de nombres commence par 1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859,... (c'est la suite de l'OEIS). Ils sont aussi appelés nombres super-Catalans, petits nombres de Schröder, ou nombres d'Hipparque, d'après Eugène Charles Catalan et ses nombres de Catalan, Ernst Schröder et ses (grands) nombres de Schröder très voisins, et d'après le mathématicien et astronome grec Hipparque qui, selon Plutarque, connaissait certainement ces nombres. (fr) In combinatorics, the Schröder–Hipparchus numbers form an integer sequence that can be used to count the number of plane trees with a given set of leaves, the number of ways of inserting parentheses into a sequence, and the number of ways of dissecting a convex polygon into smaller polygons by inserting diagonals. These numbers begin 1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, ... (sequence in the OEIS). They are also called the super-Catalan numbers, the little Schröder numbers, or the Hipparchus numbers, after Eugène Charles Catalan and his Catalan numbers, Ernst Schröder and the closely related Schröder numbers, and the ancient Greek mathematician Hipparchus who appears from evidence in Plutarch to have known of these numbers. (en) In matematica, data una griglia quadrata di dimensione nel 1° quadrante di un sistema di riferimento cartesiano, il numero di Schröder-Ipparco, , descrive il numero di cammini possibili per arrivare dal punto di coordinate al punto di coordinate , ammettendo di potersi muovere soltanto in verticale e in orizzontale o in diagonale verso destra e senza che il cammino oltrepassi mai la diagonale data dalla retta di equazione o abbia passi che corrono lungo di essa. I numeri di Schröder-Ipparco sono detti anche piccoli numeri di Schröder e differiscono dai "grandi numeri di Schröder" per il fatto che questi ultimi tengono anche conto dei cammini che non aderiscono all'ultima delle condizioni sopra elencate. La successione di tali numeri interi, che prendono il nome dal matematico tedesco Ernst Schröder e dall'antico matematico greco Ipparco di Nicea, che ci è noto conoscesse tali numeri, per , ha come primi elementi: 1, 1, 3, 11, 45, 197, 903, 4 279, 20 793, 103 049, ... (it) Числа Шрёдера — Гиппарха образуют , которую можно использовать для подсчёта числа плоских деревьев с заданным числом листьев, числа способов вставки скобок в последовательность и числа способов разбиения выпуклого многоугольника на меньшие многоугольники путём проведения диагоналей. Эта последовательность начинается с 1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, ... последовательность в OEIS. Эти числа называются также суперкаталановыми числами, малыми числами Шрёдера, или числами Гиппарха (Эжен Шарль Каталан и его числа Каталана, Эрнст Шрёдер и тесно связанные Числа Шрёдера, древнегреческий математик Гиппарх, который по свидетельству Плутарха знал эти числа). (ru) |
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| dbp:title | Super Catalan Number (en) |
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| rdfs:comment | In combinatorics, the Schröder–Hipparchus numbers form an integer sequence that can be used to count the number of plane trees with a given set of leaves, the number of ways of inserting parentheses into a sequence, and the number of ways of dissecting a convex polygon into smaller polygons by inserting diagonals. These numbers begin 1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, ... (sequence in the OEIS). (en) En théorie des nombres, les nombres de Schröder-Hipparque forment une suite d'entiers qui servent à compter les arbres planaires avec un ensemble donné de feuilles, les insertions de parenthèses dans une suite, et le nombre de façons de découper un polygone convexe en polygones plus petits par l'insertion de diagonales. Cette suite de nombres commence par 1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859,... (c'est la suite de l'OEIS). (fr) In matematica, data una griglia quadrata di dimensione nel 1° quadrante di un sistema di riferimento cartesiano, il numero di Schröder-Ipparco, , descrive il numero di cammini possibili per arrivare dal punto di coordinate al punto di coordinate , ammettendo di potersi muovere soltanto in verticale e in orizzontale o in diagonale verso destra e senza che il cammino oltrepassi mai la diagonale data dalla retta di equazione o abbia passi che corrono lungo di essa. I numeri di Schröder-Ipparco sono detti anche piccoli numeri di Schröder e differiscono dai "grandi numeri di Schröder" per il fatto che questi ultimi tengono anche conto dei cammini che non aderiscono all'ultima delle condizioni sopra elencate. (it) Числа Шрёдера — Гиппарха образуют , которую можно использовать для подсчёта числа плоских деревьев с заданным числом листьев, числа способов вставки скобок в последовательность и числа способов разбиения выпуклого многоугольника на меньшие многоугольники путём проведения диагоналей. Эта последовательность начинается с 1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, ... последовательность в OEIS. (ru) |
| rdfs:label | Nombre de Schröder-Hipparque (fr) Numero di Schröder-Ipparco (it) Schröder–Hipparchus number (en) Числа Шрёдера — Гиппарха (ru) |
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