{"title":"Cyclic sieving and dihedral sieving on noncrossing (1,2)-configurations","authors":"Chuyi Zeng, Shiwen Zhang","doi":"10.1016/j.disc.2024.114262","DOIUrl":null,"url":null,"abstract":"<div><p>Verifying a suspicion of Propp and Reiner concerning the cyclic sieving phenomenon (CSP), M. Thiel introduced a Catalan object called noncrossing <span><math><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>-configurations (denoted by <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>), which is a class of set partitions of <span><math><mo>[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>]</mo></math></span>. More precisely, Thiel proved that, with a natural action of the cyclic group <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span> on <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, the triple <span><math><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mtext>Cat</mtext></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo><mo>)</mo></math></span> exhibits the CSP, where <span><math><msub><mrow><mtext>Cat</mtext></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo><mo>≔</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mo>[</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></mrow></mfrac><msub><mrow><mo>[</mo><mtable><mtr><mtd><mn>2</mn><mi>n</mi></mtd></mtr><mtr><mtd><mi>n</mi></mtd></mtr></mtable><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> is MacMahon's <em>q</em>-Catalan number. Recently, in a study of the fermionic diagonal coinvariant ring <span><math><mi>F</mi><mi>D</mi><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, Jesse Kim found a combinatorial basis for <span><math><mi>F</mi><mi>D</mi><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> indexed by <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. In this paper, we continue to study <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and obtain the following results:</p><ul><li><span>(1)</span><span><p>We define a statistic on <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> whose generating function is <span><math><msub><mrow><mtext>Cat</mtext></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span>, which answers a problem of Thiel.</p></span></li><li><span>(2)</span><span><p>We show that <span><math><msub><mrow><mtext>Cat</mtext></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span> is equivalent to<span><span><span><math><munder><mo>∑</mo><mrow><mtable><mtr><mtd><mi>k</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi></mtd></mtr><mtr><mtd><mn>2</mn><mi>k</mi><mo>+</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn></mtd></mtr></mtable></mrow></munder><msub><mrow><mo>[</mo><mtable><mtr><mtd><mi>n</mi><mo>−</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>2</mn><mi>k</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi></mtd></mtr></mtable><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub><msub><mrow><mtext>Cat</mtext></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>k</mi><mo>+</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>y</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></mrow></msup></math></span></span></span> modulo <span><math><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>1</mn></math></span>, which answers a problem of Jesse Kim. As mentioned by Jesse Kim, this result leads to a representation theoretic proof of the above cyclic sieving result of Thiel.</p></span></li><li><span>(3)</span><span><p>We consider the dihedral sieving, a generalization of the CSP, which was recently introduced by Rao and Suk. Under a natural action of the dihedral group <span><math><msub><mrow><mi>I</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> (for even <em>n</em>), we prove a dihedral sieving result on <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>.</p></span></li></ul></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114262"},"PeriodicalIF":0.7000,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24003935","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Verifying a suspicion of Propp and Reiner concerning the cyclic sieving phenomenon (CSP), M. Thiel introduced a Catalan object called noncrossing -configurations (denoted by ), which is a class of set partitions of . More precisely, Thiel proved that, with a natural action of the cyclic group on , the triple exhibits the CSP, where is MacMahon's q-Catalan number. Recently, in a study of the fermionic diagonal coinvariant ring , Jesse Kim found a combinatorial basis for indexed by . In this paper, we continue to study and obtain the following results:
(1)
We define a statistic on whose generating function is , which answers a problem of Thiel.
(2)
We show that is equivalent to modulo , which answers a problem of Jesse Kim. As mentioned by Jesse Kim, this result leads to a representation theoretic proof of the above cyclic sieving result of Thiel.
(3)
We consider the dihedral sieving, a generalization of the CSP, which was recently introduced by Rao and Suk. Under a natural action of the dihedral group (for even n), we prove a dihedral sieving result on .
期刊介绍:
Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory.
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