@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-B373Q2P1-V
  skos:prefLabel "combinatorics"@en, "combinatoire"@fr ;
  a skos:Concept ;
  skos:narrower psr:-QLWQNVHX-2 .

psr: a skos:ConceptScheme .
psr:-QLWQNVHX-2
  skos:prefLabel "nombre de Motzkin"@fr, "Motzkin number"@en ;
  dc:modified "2023-07-26"^^xsd:date ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Motzkin_number>, <https://fr.wikipedia.org/wiki/Nombre_de_Motzkin> ;
  skos:broader psr:-B373Q2P1-V, psr:-FM1M1PDT-5 ;
  skos:definition """In mathematics, the nth Motzkin number is the number of different ways of drawing non-intersecting chords between n points on a circle (not necessarily touching every point by a chord). The Motzkin numbers are named after Theodore Motzkin and have diverse applications in geometry, combinatorics and number theory. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Motzkin_number">https://en.wikipedia.org/wiki/Motzkin_number</a>)"""@en, """En mathématiques, et plus particulièrement en combinatoire, les <b>nombres de Motzkin</b> forment une suite d'entiers naturels utilisée dans divers problèmes de dénombrement. Ils sont nommés ainsi d'après le mathématicien  Théodore Motzkin (1908-1970). Les nombres de Motzkin ont de nombreuses applications en géométrie, combinatoire et théorie des nombres.
<br/>Le <dfn>nombre de Motzkin</dfn> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle M_{n}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>M</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle M_{n}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8656f32ad5c50e679b491b361a423727491496a0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:3.473ex; height:2.509ex;" alt="M_{n}"></span> d'indice <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle n}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>n</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle n}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="n"></span> est le nombre de façons de choisir des cordes ne se coupant pas, parmi les cordes reliant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle n}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>n</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle n}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="n"></span> points disposés sur un cercle. Les nombres de Motzkin satisfont la relation de récurrence suivante&nbsp;:
<br/>
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle M_{n+1}=M_{n}+\\\\sum _{i=0}^{n-1}M_{i}M_{n-1-i}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>M</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>            <mo>+</mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>=</mo>
<br/>        <msub>
<br/>          <mi>M</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>+</mo>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>            <mo>=</mo>
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mn>1</mn>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <msub>
<br/>          <mi>M</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <msub>
<br/>          <mi>M</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mn>1</mn>
<br/>            <mo>−<!-- − --></mo>
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle M_{n+1}=M_{n}+\\\\sum _{i=0}^{n-1}M_{i}M_{n-1-i}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a7cff3dde04680d5e69bc93696977fd14d0f7f0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.005ex; width:29.199ex; height:7.343ex;" alt="{\\\\displaystyle M_{n+1}=M_{n}+\\\\sum _{i=0}^{n-1}M_{i}M_{n-1-i}}"> 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Nombre_de_Motzkin">https://fr.wikipedia.org/wiki/Nombre_de_Motzkin</a>)"""@fr ;
  dc:created "2023-07-26"^^xsd:date ;
  skos:inScheme psr: ;
  a skos:Concept .

psr:-FM1M1PDT-5
  skos:prefLabel "suite d'entiers"@fr, "integer sequence"@en ;
  a skos:Concept ;
  skos:narrower psr:-QLWQNVHX-2 .