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Mutual Interlacing and Eulerian-like Polynomials for Weyl Groups [article]

Arthur L.B. Yang, Philip B. Zhang
<span title="2014-01-24">2014</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We use the method of mutual interlacing to prove two conjectures on the real-rootedness of Eulerian-like polynomials: Brenti's conjecture on q-Eulerian polynomials for Weyl groups of type D, and Dilks,  ...  As a result, we get the real-rootedness of the affine Eulerian polynomials of type D.  ...  This work was supported by the 973 Project and the National Science Foundation of China.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1401.6273v1">arXiv:1401.6273v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dx5m2cnr45co3oc6vu2eomso6a">fatcat:dx5m2cnr45co3oc6vu2eomso6a</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20191018225316/https://arxiv.org/pdf/1401.6273v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/3d/9f/3d9f93c7168183c747606cd7eca69f9bd6bfec42.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1401.6273v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Stable multivariate W -Eulerian polynomials

Mirkó Visontai, Nathan Williams
<span title="">2013</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/z77xaqun7bcxjkh75wb7iseaty" style="color: black;">Journal of combinatorial theory. Series A</a> </i> &nbsp;
polynomials of types B and D, we indicate some methods of attack and pose some related open problems.  ...  We prove a multivariate strengthening of Brenti's result that every root of the Eulerian polynomial of type B is real.  ...  Acknowledgements We thank Jim Haglund for helpful suggestions and Dennis Stanton for showing us the connection between the the number of monomials and moments of Laguerre polynomials (and its connections  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jcta.2013.07.009">doi:10.1016/j.jcta.2013.07.009</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6prxb5oitjdcrim5g5cnieuvoq">fatcat:6prxb5oitjdcrim5g5cnieuvoq</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/download/arxiv-1203.0791/1203.0791.pdf" title="fulltext PDF download [not primary version]" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> File Archive [PDF] <span style="color: #f43e3e;">&#10033;</span> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jcta.2013.07.009"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Log-concavity of the Excedance Enumerators in positive elements of Type A and Type B Coxeter Groups [article]

Hiranya Kishore Dey
<span title="2020-10-20">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The classical Eulerian Numbers A_n,k are known to be log-concave. Let P_n,k and Q_n,k be the number of even and odd permutations with k excedances.  ...  We show similar results for Type B Coxeter Groups.  ...  Acknowledgements The author would like to thank his advisor Sivaramakrishnan Sivasubramanian for all the insightful discussions and comments during the preparation of the paper and also for going through  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2009.10655v2">arXiv:2009.10655v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tlc6xlbwnvel7apyfrwgh6yt5q">fatcat:tlc6xlbwnvel7apyfrwgh6yt5q</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200929101643/https://arxiv.org/pdf/2009.10655v1.pdf" title="fulltext PDF download [not primary version]" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <span style="color: #f43e3e;">&#10033;</span> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2009.10655v2" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Actions on permutations and unimodality of descent polynomials

Petter Brändén
<span title="">2008</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/54t3hgai4fhhthc74mj7z7tapu" style="color: black;">European journal of combinatorics (Print)</a> </i> &nbsp;
We prove that the generalized permutation patterns (13-2) and (2-31) are invariant under the action and use this to prove unimodality properties for a q-analog of the Eulerian numbers recently studied  ...  We also extend the action to linear extensions of sign-graded posets to give a new proof of the unimodality of the (P,ω)-Eulerian polynomials of sign-graded posets and a combinatorial interpretations (  ...  Acknowledgements The author would like to thank the anonymous referee for helpful advice on the presentation of this paper, for pointing out the correct credit for the actions and for showing how the bijection  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.ejc.2006.12.010">doi:10.1016/j.ejc.2006.12.010</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cxvftrfh5zbrtiej3m5rshvtiu">fatcat:cxvftrfh5zbrtiej3m5rshvtiu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170927054557/http://publisher-connector.core.ac.uk/resourcesync/data/elsevier/pdf/4ca/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS9zMDE5NTY2OTgwNzAwMDE5NA%3D%3D.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/64/3a/643aac352be6304761cff5a765c1d4bb11e5cf28.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.ejc.2006.12.010"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Coefficientwise Hankel-total positivity of row-generating polynomials for the m-Jacobi-Rogers triangle [article]

Bao-Xuan Zhu
<span title="2022-02-08">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Using the theory of production matrices, we gain a criterion for the coefficientwise Hankel-total positivity of the row-generating polynomial sequence of the m-Jacobi-Rogers triangle.  ...  The aim of this paper is to study the criteria for the row-generating polynomial sequence of the m-Jacobi-Rogers triangle being coefficientwise Hankel-totally positive and their applications.  ...  In classical analysis, one of important applications of the Pólya frequency is to characterize real rootedness of polynomials and entire functions.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2202.03793v1">arXiv:2202.03793v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vlcl67ceyrd4jpswirdc23qo6m">fatcat:vlcl67ceyrd4jpswirdc23qo6m</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220210225234/https://arxiv.org/pdf/2202.03793v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/c4/ca/c4ca9668152438eb2786d247740eb701223e0f6d.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2202.03793v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>