Cyclic sieving and dihedral sieving on noncrossing (1,2)-configurations

IF 0.7 3区 数学 Q2 MATHEMATICS
{"title":"Cyclic sieving and dihedral sieving on noncrossing (1,2)-configurations","authors":"","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":null,"pages":null},"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 (1,2)-configurations (denoted by Xn), which is a class of set partitions of [n1]. More precisely, Thiel proved that, with a natural action of the cyclic group Cn1 on Xn, the triple (Xn,Cn1,Catn(q)) exhibits the CSP, where Catn(q)1[n+1]q[2nn]q is MacMahon's q-Catalan number. Recently, in a study of the fermionic diagonal coinvariant ring FDRn, Jesse Kim found a combinatorial basis for FDRn indexed by Xn. In this paper, we continue to study Xn and obtain the following results:

  • (1)

    We define a statistic on Xn whose generating function is Catn(q), which answers a problem of Thiel.

  • (2)

    We show that Catn(q) is equivalent tok,x,y2k+x+y=n1[n12k,x,y]qCatk(q)qk+(x2)+(y2)+(n2) modulo qn11, 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 I2(n1) (for even n), we prove a dihedral sieving result on Xn.

非交叉 (1,2) 构型上的循环筛分和二面筛分
为了验证普罗普和赖纳关于循环筛分现象(CSP)的猜想,蒂尔(M. Thiel)引入了一个称为非交叉(1,2)配置(用 Xn 表示)的加泰罗尼亚对象,它是 [n-1] 的一类集合分区。更确切地说,蒂尔证明了在循环群 Cn-1 对 Xn 的自然作用下,三重(Xn,Cn-1,Catn(q))展示了 CSP,其中 Catn(q)≔1[n+1]q[2nn]q 是麦克马洪的 q 加泰罗尼亚数。最近,在对费米对角共变环 FDRn 的研究中,Jesse Kim 发现了以 Xn 为索引的 FDRn 组合基。(2)我们证明 Catn(q) 等价于∑k,x,y2k+x+y=n-1[n-12k,x,y]qCatk(q)qk+(x2)+(y2)+(n2) modulo qn-1-1,这回答了 Jesse Kim 的一个问题。(3)我们考虑二面筛分,它是 CSP 的广义化,最近由 Rao 和 Suk 提出。在二面群 I2(n-1)(偶数 n)的自然作用下,我们证明了 Xn 上的二面筛分结果。
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来源期刊
Discrete Mathematics
Discrete Mathematics 数学-数学
CiteScore
1.50
自引率
12.50%
发文量
424
审稿时长
6 months
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. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.
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