@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: a skos:ConceptScheme .
psr:-F7H3K8H1-0
  skos:prefLabel "coefficient binomial"@fr, "binomial coefficient"@en ;
  a skos:Concept ;
  skos:narrower psr:-QKQ1ZP6D-T .

psr:-C4XZKP2N-4
  skos:prefLabel "permutation"@en, "permutation"@fr ;
  a skos:Concept ;
  skos:narrower psr:-QKQ1ZP6D-T .

psr:-QKQ1ZP6D-T
  skos:broader psr:-C4XZKP2N-4, psr:-F7H3K8H1-0 ;
  skos:definition """In combinatorial mathematics and statistics, the <b>Fuss–Catalan</b> numbers are numbers of the form  <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 A_{m}(p,r)\\\\equiv {\\rac {r}{mp+r}}{\\inom {mp+r}{m}}={\\rac {r}{m!}}\\\\prod _{i=1}^{m-1}(mp+r-i)=r{\\rac {\\\\Gamma (mp+r)}{\\\\Gamma (1+m)\\\\Gamma (m(p-1)+r+1)}}.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>A</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>m</mi>           </mrow>         </msub>         <mo stretchy="false">(</mo>         <mi>p</mi>         <mo>,</mo>         <mi>r</mi>         <mo stretchy="false">)</mo>         <mo>≡<!-- ≡ --></mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mi>r</mi>             <mrow>               <mi>m</mi>               <mi>p</mi>               <mo>+</mo>               <mi>r</mi>             </mrow>           </mfrac>         </mrow>         <mrow class="MJX-TeXAtom-ORD">           <mrow>             <mrow class="MJX-TeXAtom-OPEN">               <mo maxsize="2.047em" minsize="2.047em">(</mo>             </mrow>             <mfrac linethickness="0">               <mrow>                 <mi>m</mi>                 <mi>p</mi>                 <mo>+</mo>                 <mi>r</mi>               </mrow>               <mi>m</mi>             </mfrac>             <mrow class="MJX-TeXAtom-CLOSE">               <mo maxsize="2.047em" minsize="2.047em">)</mo>             </mrow>           </mrow>         </mrow>         <mo>=</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mi>r</mi>             <mrow>               <mi>m</mi>               <mo>!</mo>             </mrow>           </mfrac>         </mrow>         <munderover>           <mo>∏<!-- ∏ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>i</mi>             <mo>=</mo>             <mn>1</mn>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi>m</mi>             <mo>−<!-- − --></mo>             <mn>1</mn>           </mrow>         </munderover>         <mo stretchy="false">(</mo>         <mi>m</mi>         <mi>p</mi>         <mo>+</mo>         <mi>r</mi>         <mo>−<!-- − --></mo>         <mi>i</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mi>r</mi>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mrow>               <mi mathvariant="normal">Γ<!-- Γ --></mi>               <mo stretchy="false">(</mo>               <mi>m</mi>               <mi>p</mi>               <mo>+</mo>               <mi>r</mi>               <mo stretchy="false">)</mo>             </mrow>             <mrow>               <mi mathvariant="normal">Γ<!-- Γ --></mi>               <mo stretchy="false">(</mo>               <mn>1</mn>               <mo>+</mo>               <mi>m</mi>               <mo stretchy="false">)</mo>               <mi mathvariant="normal">Γ<!-- Γ --></mi>               <mo stretchy="false">(</mo>               <mi>m</mi>               <mo stretchy="false">(</mo>               <mi>p</mi>               <mo>−<!-- − --></mo>               <mn>1</mn>               <mo stretchy="false">)</mo>               <mo>+</mo>               <mi>r</mi>               <mo>+</mo>               <mn>1</mn>               <mo stretchy="false">)</mo>             </mrow>           </mfrac>         </mrow>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle A_{m}(p,r)\\\\equiv {\\rac {r}{mp+r}}{\\inom {mp+r}{m}}={\\rac {r}{m!}}\\\\prod _{i=1}^{m-1}(mp+r-i)=r{\\rac {\\\\Gamma (mp+r)}{\\\\Gamma (1+m)\\\\Gamma (m(p-1)+r+1)}}.}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1032236901060178c759799e9e6c40e5cfaa85b3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:88.248ex; height:7.343ex;" alt="{\\\\displaystyle A_{m}(p,r)\\\\equiv {\\rac {r}{mp+r}}{\\inom {mp+r}{m}}={\\rac {r}{m!}}\\\\prod _{i=1}^{m-1}(mp+r-i)=r{\\rac {\\\\Gamma (mp+r)}{\\\\Gamma (1+m)\\\\Gamma (m(p-1)+r+1)}}.}"></span></dd></dl> They are named after N. I. Fuss and Eugène Charles Catalan. In some publications this equation is sometimes referred to as <b>Two-parameter Fuss–Catalan numbers</b> or <b>Raney numbers</b>.  The implication is the <i>single-parameter Fuss-Catalan numbers</i> are when <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 \\\\,r=1\\\\,}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mspace width="thinmathspace"></mspace>         <mi>r</mi>         <mo>=</mo>         <mn>1</mn>         <mspace width="thinmathspace"></mspace>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\,r=1\\\\,}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ce3978a71bdcbfdd463ac2164c22234d7d3f274" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.084ex; height:2.176ex;" alt="{\\\\displaystyle \\\\,r=1\\\\,}"></span> and <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 \\\\,p=2\\\\,}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mspace width="thinmathspace"></mspace>         <mi>p</mi>         <mo>=</mo>         <mn>2</mn>         <mspace width="thinmathspace"></mspace>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\,p=2\\\\,}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fc8d6cc8a94250dd58f23393b54cca540a09117" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.205ex; height:2.509ex;" alt="{\\\\displaystyle \\\\,p=2\\\\,}"></span>. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Fuss%E2%80%93Catalan_number">https://en.wikipedia.org/wiki/Fuss%E2%80%93Catalan_number</a>)"""@en ;
  dc:created "2023-08-24"^^xsd:date ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Fuss%E2%80%93Catalan_number> ;
  dc:modified "2024-10-18"^^xsd:date ;
  skos:prefLabel "nombre de Fuss-Catalan"@fr, "Fuss-Catalan number"@en ;
  skos:inScheme psr: ;
  a skos:Concept .