{"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}
引用次数: 0
Abstract
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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