{"title":"B 型和 D 型出差多项式的一些行列式表示法","authors":"Chak-On Chow","doi":"10.1016/j.disc.2024.114155","DOIUrl":null,"url":null,"abstract":"<div><p>Chow (2024) recently computed expressions of the types <em>B</em> and <em>D</em> derangement polynomials <span><math><msubsup><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>B</mi></mrow></msubsup><mo>(</mo><mi>q</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>B</mi></mrow></msubsup></mrow></msub><msup><mrow><mi>q</mi></mrow><mrow><mi>fmaj</mi><mo>(</mo><mi>σ</mi><mo>)</mo></mrow></msup></math></span> and <span><math><msubsup><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>D</mi></mrow></msubsup><mo>(</mo><mi>q</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>D</mi></mrow></msubsup></mrow></msub><msup><mrow><mi>q</mi></mrow><mrow><mi>maj</mi><mo>(</mo><mi>σ</mi><mo>)</mo></mrow></msup></math></span> as tridiagonal and lower Hessenberg determinants of order <em>n</em>. Qi, Wang, and Guo (2016), based on a determinantal formula for the <em>n</em>th derivative of a quotient of two functions, derived an expression of the classical derangement number <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>n</mi><mo>!</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi></mrow></msubsup><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>k</mi></mrow></msup><mo>/</mo><mi>k</mi><mo>!</mo></math></span> as a tridiagonal determinant of order <span><math><mi>n</mi><mo>+</mo><mn>1</mn></math></span>. By <em>q</em>-extending the approach of Qi et al., we present in this work yet another determinantal expressions of <span><math><msubsup><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>B</mi></mrow></msubsup><mo>(</mo><mi>q</mi><mo>)</mo></math></span> and <span><math><msubsup><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>D</mi></mrow></msubsup><mo>(</mo><mi>q</mi><mo>)</mo></math></span> as determinants of order <span><math><mi>n</mi><mo>+</mo><mn>1</mn></math></span>.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Some determinantal representations of derangement polynomials of types B and D\",\"authors\":\"Chak-On Chow\",\"doi\":\"10.1016/j.disc.2024.114155\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Chow (2024) recently computed expressions of the types <em>B</em> and <em>D</em> derangement polynomials <span><math><msubsup><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>B</mi></mrow></msubsup><mo>(</mo><mi>q</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>B</mi></mrow></msubsup></mrow></msub><msup><mrow><mi>q</mi></mrow><mrow><mi>fmaj</mi><mo>(</mo><mi>σ</mi><mo>)</mo></mrow></msup></math></span> and <span><math><msubsup><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>D</mi></mrow></msubsup><mo>(</mo><mi>q</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>D</mi></mrow></msubsup></mrow></msub><msup><mrow><mi>q</mi></mrow><mrow><mi>maj</mi><mo>(</mo><mi>σ</mi><mo>)</mo></mrow></msup></math></span> as tridiagonal and lower Hessenberg determinants of order <em>n</em>. Qi, Wang, and Guo (2016), based on a determinantal formula for the <em>n</em>th derivative of a quotient of two functions, derived an expression of the classical derangement number <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>n</mi><mo>!</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi></mrow></msubsup><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>k</mi></mrow></msup><mo>/</mo><mi>k</mi><mo>!</mo></math></span> as a tridiagonal determinant of order <span><math><mi>n</mi><mo>+</mo><mn>1</mn></math></span>. By <em>q</em>-extending the approach of Qi et al., we present in this work yet another determinantal expressions of <span><math><msubsup><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>B</mi></mrow></msubsup><mo>(</mo><mi>q</mi><mo>)</mo></math></span> and <span><math><msubsup><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>D</mi></mrow></msubsup><mo>(</mo><mi>q</mi><mo>)</mo></math></span> as determinants of order <span><math><mi>n</mi><mo>+</mo><mn>1</mn></math></span>.</p></div>\",\"PeriodicalId\":50572,\"journal\":{\"name\":\"Discrete Mathematics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-07-14\",\"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/S0012365X24002863\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24002863","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
摘要
Chow (2024) 最近计算了 B 和 D 型失真多项式 dnB(q)=∑σ∈DnBqfmaj(σ) 和 dnD(q)=∑σ∈DnDqmaj(σ) 的表达式,它们是 n 阶的三对角和下海森伯行列式。Qi、Wang和Guo(2016)基于两个函数商的n次导数的行列式公式,推导出了经典失真数dn=n!∑k=0n(-1)k/k!作为n+1阶的三对角行列式的表达式。通过对齐等人的方法进行 q 扩展,我们在本文中提出了 dnB(q) 和 dnD(q) 作为 n+1 阶行列式的另一种行列式表达式。
Some determinantal representations of derangement polynomials of types B and D
Chow (2024) recently computed expressions of the types B and D derangement polynomials and as tridiagonal and lower Hessenberg determinants of order n. Qi, Wang, and Guo (2016), based on a determinantal formula for the nth derivative of a quotient of two functions, derived an expression of the classical derangement number as a tridiagonal determinant of order . By q-extending the approach of Qi et al., we present in this work yet another determinantal expressions of and as determinants of order .
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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