@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:-JJRPZSZ2-M
  skos:prefLabel "combinatoire algébrique"@fr, "algebraic combinatorics"@en ;
  a skos:Concept ;
  skos:narrower psr:-WDH948HN-H .

psr:-CVDPQB0Q-M
  skos:prefLabel "natural numbers"@en, "entier naturel"@fr ;
  a skos:Concept ;
  skos:narrower psr:-WDH948HN-H .

psr: a skos:ConceptScheme .
psr:-WDH948HN-H
  skos:broader psr:-CVDPQB0Q-M, psr:-FM1M1PDT-5, psr:-JJRPZSZ2-M ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Nombre_de_Kostka>, <https://en.wikipedia.org/wiki/Kostka_number> ;
  a skos:Concept ;
  skos:prefLabel "nombre de Kostka"@fr, "Kostka number"@en ;
  skos:related psr:-KCRM7MC2-6 ;
  dc:modified "2024-10-18"^^xsd:date ;
  dc:created "2023-08-18"^^xsd:date ;
  skos:definition """En mathématiques, le <b>nombre de Kostka</b> <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 K_{\\\\lambda \\\\mu }}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>K</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>λ<!-- λ --></mi>             <mi>μ<!-- μ --></mi>           </mrow>         </msub>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle K_{\\\\lambda \\\\mu }}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7edb0fc88f8b52729c1bae08cea098ef40812efd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.155ex; height:2.843ex;" alt="{\\\\displaystyle K_{\\\\lambda \\\\mu }}"></span>, paramétré par deux partition d'un entier <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 \\\\lambda }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>λ<!-- λ --></mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\lambda }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="\\\\lambda "></span> et <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 \\\\mu }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>μ<!-- μ --></mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mu }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\\\\mu "></span>, est un entier naturel qui est égal au nombre de tableaux de Young semi-standard de forme <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 \\\\lambda }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>λ<!-- λ --></mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\lambda }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="\\\\lambda "></span> et de poids <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 \\\\mu }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>μ<!-- μ --></mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mu }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\\\\mu "></span>.  Ils ont été introduits par le mathématicien Carl Kostka dans ses études des fonctions symétriques</span>,</span>. Par exemple, si <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 \\\\lambda =(3,2)}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>λ<!-- λ --></mi>         <mo>=</mo>         <mo stretchy="false">(</mo>         <mn>3</mn>         <mo>,</mo>         <mn>2</mn>         <mo stretchy="false">)</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\lambda =(3,2)}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/846ab75299f2d404510f28f4394a76a10d0effbb" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.622ex; height:2.843ex;" alt="{\\\\displaystyle \\\\lambda =(3,2)}"></span> et <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 \\\\mu =(1,1,2,1)}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>μ<!-- μ --></mi>         <mo>=</mo>         <mo stretchy="false">(</mo>         <mn>1</mn>         <mo>,</mo>         <mn>1</mn>         <mo>,</mo>         <mn>2</mn>         <mo>,</mo>         <mn>1</mn>         <mo stretchy="false">)</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mu =(1,1,2,1)}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea5fc32f31a26647a2271813622da8ded1d27f8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.061ex; height:2.843ex;" alt="{\\\\displaystyle \\\\mu =(1,1,2,1)}"></span>, le nombre de Kostka <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 K_{\\\\lambda \\\\mu }}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>K</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>λ<!-- λ --></mi>             <mi>μ<!-- μ --></mi>           </mrow>         </msub>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle K_{\\\\lambda \\\\mu }}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7edb0fc88f8b52729c1bae08cea098ef40812efd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.155ex; height:2.843ex;" alt="{\\\\displaystyle K_{\\\\lambda \\\\mu }}"></span> compte le nombre de manières de remplir une collection de 5 cellules alignée à gauche, avec 3 cellules dans la première ligne et 2 dans la seconde, et contenant une fois les entiers 1 et 2, deux fois l'entier 3 et une fois l'entier 4. De plus, les entiers doivent être strictement croissants en colonne, et faiblement croissants en ligne. Les trois tableaux possibles sont montrés sur la figure, et on a donc <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 K_{(3,2)(1,1,2,1)}=3}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>K</mi>           <mrow class="MJX-TeXAtom-ORD">             <mo stretchy="false">(</mo>             <mn>3</mn>             <mo>,</mo>             <mn>2</mn>             <mo stretchy="false">)</mo>             <mo stretchy="false">(</mo>             <mn>1</mn>             <mo>,</mo>             <mn>1</mn>             <mo>,</mo>             <mn>2</mn>             <mo>,</mo>             <mn>1</mn>             <mo stretchy="false">)</mo>           </mrow>         </msub>         <mo>=</mo>         <mn>3</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle K_{(3,2)(1,1,2,1)}=3}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/605db86850b1cad408de5979181d4199148d2cb1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.786ex; height:3.009ex;" alt="{\\\\displaystyle K_{(3,2)(1,1,2,1)}=3}"></span>. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Nombre_de_Kostka">https://fr.wikipedia.org/wiki/Nombre_de_Kostka</a>)"""@fr, """In mathematics, the <b>Kostka number</b> <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 K_{\\\\lambda \\\\mu }}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>K</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>λ<!-- λ --></mi>             <mi>μ<!-- μ --></mi>           </mrow>         </msub>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle K_{\\\\lambda \\\\mu }}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7edb0fc88f8b52729c1bae08cea098ef40812efd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.155ex; height:2.843ex;" alt="K_{{\\\\lambda \\\\mu }}"></span> (depending on two integer partitions <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 \\\\lambda }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>λ<!-- λ --></mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\lambda }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="\\\\lambda "></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 \\\\mu }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>μ<!-- μ --></mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mu }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\\\\mu "></span>) is a non-negative integer that is equal to the number of semistandard Young tableaux of shape <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 \\\\lambda }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>λ<!-- λ --></mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\lambda }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="\\\\lambda "></span> and weight <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 \\\\mu }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>μ<!-- μ --></mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mu }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="\\\\mu "></span>.  They were introduced by the mathematician Carl Kostka in his study of symmetric functions (Kostka (1882)). For example, if <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 \\\\lambda =(3,2)}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>λ<!-- λ --></mi>         <mo>=</mo>         <mo stretchy="false">(</mo>         <mn>3</mn>         <mo>,</mo>         <mn>2</mn>         <mo stretchy="false">)</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\lambda =(3,2)}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/846ab75299f2d404510f28f4394a76a10d0effbb" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.622ex; height:2.843ex;" alt="{\\\\displaystyle \\\\lambda =(3,2)}"></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 \\\\mu =(1,1,2,1)}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>μ<!-- μ --></mi>         <mo>=</mo>         <mo stretchy="false">(</mo>         <mn>1</mn>         <mo>,</mo>         <mn>1</mn>         <mo>,</mo>         <mn>2</mn>         <mo>,</mo>         <mn>1</mn>         <mo stretchy="false">)</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mu =(1,1,2,1)}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea5fc32f31a26647a2271813622da8ded1d27f8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.061ex; height:2.843ex;" alt="{\\\\displaystyle \\\\mu =(1,1,2,1)}"></span>, the Kostka number <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 K_{\\\\lambda \\\\mu }}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>K</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>λ<!-- λ --></mi>             <mi>μ<!-- μ --></mi>           </mrow>         </msub>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle K_{\\\\lambda \\\\mu }}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7edb0fc88f8b52729c1bae08cea098ef40812efd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.155ex; height:2.843ex;" alt="K_{{\\\\lambda \\\\mu }}"></span> counts the number of ways to fill a left-aligned collection of boxes with 3 in the first row and 2 in the second row with 1 copy of the number 1, 1 copy of the number 2, 2 copies of the number 3 and 1 copy of the number 4 such that the entries increase along columns and do not decrease along rows.  The three such tableaux are shown at right, 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 K_{(3,2)(1,1,2,1)}=3}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>K</mi>           <mrow class="MJX-TeXAtom-ORD">             <mo stretchy="false">(</mo>             <mn>3</mn>             <mo>,</mo>             <mn>2</mn>             <mo stretchy="false">)</mo>             <mo stretchy="false">(</mo>             <mn>1</mn>             <mo>,</mo>             <mn>1</mn>             <mo>,</mo>             <mn>2</mn>             <mo>,</mo>             <mn>1</mn>             <mo stretchy="false">)</mo>           </mrow>         </msub>         <mo>=</mo>         <mn>3</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle K_{(3,2)(1,1,2,1)}=3}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/605db86850b1cad408de5979181d4199148d2cb1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.786ex; height:3.009ex;" alt="{\\\\displaystyle K_{(3,2)(1,1,2,1)}=3}"></span>. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Kostka_number">https://en.wikipedia.org/wiki/Kostka_number</a>)"""@en ;
  skos:inScheme psr: .

psr:-KCRM7MC2-6
  skos:prefLabel "Kostka polynomial"@en, "polynôme de Kostka"@fr ;
  a skos:Concept ;
  skos:related psr:-WDH948HN-H .

psr:-FM1M1PDT-5
  skos:prefLabel "suite d'entiers"@fr, "integer sequence"@en ;
  a skos:Concept ;
  skos:narrower psr:-WDH948HN-H .