2 - Andrews-Gordon恒等式和柱面划分

Transactions of the American Mathematical Society, Series B Pub Date : 2021-11-15 DOI:10.1090/btran/147

S. Warnaar

{"title":"2 - Andrews-Gordon恒等式和柱面划分","authors":"S. Warnaar","doi":"10.1090/btran/147","DOIUrl":null,"url":null,"abstract":"<p>Inspired by a number of recent papers by Corteel, Dousse, Foda, Uncu and Welsh on cylindric partitions and Rogers–Ramanujan-type identities, we obtain the <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper A 2\">\n <mml:semantics>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">A</mml:mi>\n </mml:mrow>\n <mml:mn>2</mml:mn>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\mathrm {A}_2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> (or <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper A 2 Superscript left-parenthesis 1 right-parenthesis\">\n <mml:semantics>\n <mml:msubsup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">A</mml:mi>\n </mml:mrow>\n <mml:mn>2</mml:mn>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n </mml:msubsup>\n <mml:annotation encoding=\"application/x-tex\">\\mathrm {A}_2^{(1)}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>) analogues of the celebrated Andrews–Gordon identities. We further prove <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"q\">\n <mml:semantics>\n <mml:mi>q</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">q</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-series identities that correspond to the infinite-level limit of the Andrews–Gordon identities for <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper A Subscript r minus 1\">\n <mml:semantics>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">A</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>r</mml:mi>\n <mml:mo>−</mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\mathrm {A}_{r-1}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> (or <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper A Subscript r minus 1 Superscript left-parenthesis 1 right-parenthesis\">\n <mml:semantics>\n <mml:msubsup>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">A</mml:mi>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi>r</mml:mi>\n <mml:mo>−</mml:mo>\n <mml:mn>1</mml:mn>\n </mml:mrow>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>1</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n </mml:msubsup>\n <mml:annotation encoding=\"application/x-tex\">\\mathrm {A}_{r-1}^{(1)}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>) for arbitrary rank <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"r\">\n <mml:semantics>\n <mml:mi>r</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">r</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. Our results for <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper A 2\">\n <mml:semantics>\n <mml:msub>\n <mml:mrow class=\"MJX-TeXAtom-ORD\">\n <mml:mi mathvariant=\"normal\">A</mml:mi>\n </mml:mrow>\n <mml:mn>2</mml:mn>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">\\mathrm {A}_2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> also lead to conjectural, manifestly positive, combinatorial formulas for the <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"2\">\n <mml:semantics>\n <mml:mn>2</mml:mn>\n <mml:annotation encoding=\"application/x-tex\">2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-variable generating function of cylindric partitions of rank <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"3\">\n <mml:semantics>\n <mml:mn>3</mml:mn>\n <mml:annotation encoding=\"application/x-tex\">3</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and level <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d\">\n <mml:semantics>\n <mml:mi>d</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">d</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, such that <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d\">\n <mml:semantics>\n <mml:mi>d</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">d</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is not a multiple of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"3\">\n <mml:semantics>\n <mml:mn>3</mml:mn>\n <mml:annotation encoding=\"application/x-tex\">3</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>.</p>","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"68 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2021-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":"{\"title\":\"The 𝐴₂ Andrews–Gordon identities and cylindric partitions\",\"authors\":\"S. Warnaar\",\"doi\":\"10.1090/btran/147\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Inspired by a number of recent papers by Corteel, Dousse, Foda, Uncu and Welsh on cylindric partitions and Rogers–Ramanujan-type identities, we obtain the <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal upper A 2\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"normal\\\">A</mml:mi>\\n </mml:mrow>\\n <mml:mn>2</mml:mn>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathrm {A}_2</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> (or <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal upper A 2 Superscript left-parenthesis 1 right-parenthesis\\\">\\n <mml:semantics>\\n <mml:msubsup>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"normal\\\">A</mml:mi>\\n </mml:mrow>\\n <mml:mn>2</mml:mn>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mn>1</mml:mn>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n </mml:msubsup>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathrm {A}_2^{(1)}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>) analogues of the celebrated Andrews–Gordon identities. We further prove <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"q\\\">\\n <mml:semantics>\\n <mml:mi>q</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">q</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-series identities that correspond to the infinite-level limit of the Andrews–Gordon identities for <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal upper A Subscript r minus 1\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"normal\\\">A</mml:mi>\\n </mml:mrow>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>r</mml:mi>\\n <mml:mo>−</mml:mo>\\n <mml:mn>1</mml:mn>\\n </mml:mrow>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathrm {A}_{r-1}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> (or <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal upper A Subscript r minus 1 Superscript left-parenthesis 1 right-parenthesis\\\">\\n <mml:semantics>\\n <mml:msubsup>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"normal\\\">A</mml:mi>\\n </mml:mrow>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi>r</mml:mi>\\n <mml:mo>−</mml:mo>\\n <mml:mn>1</mml:mn>\\n </mml:mrow>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mo stretchy=\\\"false\\\">(</mml:mo>\\n <mml:mn>1</mml:mn>\\n <mml:mo stretchy=\\\"false\\\">)</mml:mo>\\n </mml:mrow>\\n </mml:msubsup>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathrm {A}_{r-1}^{(1)}</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>) for arbitrary rank <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"r\\\">\\n <mml:semantics>\\n <mml:mi>r</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">r</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. Our results for <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"normal upper A 2\\\">\\n <mml:semantics>\\n <mml:msub>\\n <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\">\\n <mml:mi mathvariant=\\\"normal\\\">A</mml:mi>\\n </mml:mrow>\\n <mml:mn>2</mml:mn>\\n </mml:msub>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathrm {A}_2</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> also lead to conjectural, manifestly positive, combinatorial formulas for the <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"2\\\">\\n <mml:semantics>\\n <mml:mn>2</mml:mn>\\n <mml:annotation encoding=\\\"application/x-tex\\\">2</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-variable generating function of cylindric partitions of rank <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"3\\\">\\n <mml:semantics>\\n <mml:mn>3</mml:mn>\\n <mml:annotation encoding=\\\"application/x-tex\\\">3</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> and level <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"d\\\">\\n <mml:semantics>\\n <mml:mi>d</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">d</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>, such that <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"d\\\">\\n <mml:semantics>\\n <mml:mi>d</mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">d</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> is not a multiple of <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"3\\\">\\n <mml:semantics>\\n <mml:mn>3</mml:mn>\\n <mml:annotation encoding=\\\"application/x-tex\\\">3</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>.</p>\",\"PeriodicalId\":377306,\"journal\":{\"name\":\"Transactions of the American Mathematical Society, Series B\",\"volume\":\"68 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-11-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"5\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Transactions of the American Mathematical Society, Series B\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/btran/147\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions of the American Mathematical Society, Series B","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/btran/147","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 5

摘要

受Corteel, Dousse, Foda, Uncu和Welsh最近关于圆柱划分和rogers - ramanujan型恒等式的一些论文的启发，我们得到了与著名的Andrews-Gordon恒等式类似的a2 \ mathm {a}_2(或a2 (1) \ mathm {a}_2^{(1)})。我们进一步证明了q q级数恒等式对应于任意秩rr的Andrews-Gordon恒等式的无穷级极限，即A r−1 \ mathm {A}_{r} -1}(或A r−1 (1)\ mathm {A}_{r-1}^{(1)})。我们对a2 \ maththrm {A}_2的结果也得出了对秩为33和阶为d d的圆柱形分区的22变量生成函数的推测性的、明显正的组合公式，使得d d不是33的倍数。

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The 𝐴₂ Andrews–Gordon identities and cylindric partitions

Inspired by a number of recent papers by Corteel, Dousse, Foda, Uncu and Welsh on cylindric partitions and Rogers–Ramanujan-type identities, we obtain the A 2 \mathrm {A}_2 (or A 2 ( 1 ) \mathrm {A}_2^{(1)} ) analogues of the celebrated Andrews–Gordon identities. We further prove q q -series identities that correspond to the infinite-level limit of the Andrews–Gordon identities for A r − 1 \mathrm {A}_{r-1} (or A r − 1 ( 1 ) \mathrm {A}_{r-1}^{(1)} ) for arbitrary rank r r . Our results for A 2 \mathrm {A}_2 also lead to conjectural, manifestly positive, combinatorial formulas for the 2 2 -variable generating function of cylindric partitions of rank 3 3 and level d d , such that d d is not a multiple of 3 3 .

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