Advances in Applied Mathematics最新文献

{"title":"Some interlacing properties related to the Eulerian and derangement polynomials","authors":"Lily Li Liu,&nbsp;Xue Yan","doi":"10.1016/j.aam.2024.102776","DOIUrl":"10.1016/j.aam.2024.102776","url":null,"abstract":"<div><p>In this paper, we consider two matrices of polynomials <span><math><msub><mrow><mo>[</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>]</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span> and <span><math><msub><mrow><mo>[</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>]</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span>, which are defined by a recurrence relation from a sequence of polynomials with real coefficients. These matrices play an important role in the study of the Eulerian and derangement transformations by Athanasiadis, who asked when their rows form interlacing sequences of real-rooted polynomials. In this paper, we give an answer to this question. In our setup, all columns of these matrices are shown to be generalized Sturm sequences. As applications, we show that the derangement transformation, its type <em>B</em> analogue and the <em>r</em>-colored derangement transformation of a class of polynomials with nonnegative coefficients, and all roots in the interval <span><math><mo>[</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>]</mo></math></span>, have only real roots in a unified manner. The question about the type <em>B</em> derangement transformation was also raised by Athanasiadis. Furthermore, we show that the diagonal line of <span><math><msub><mrow><mo>[</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>]</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span> and <span><math><msub><mrow><mo>[</mo><msub><mrow><mi>l</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>]</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span> forms a generalized Sturm sequence respectively, i.e., we give sufficient conditions for the binomial transformation to preserve the interlacing property.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"162 ","pages":"Article 102776"},"PeriodicalIF":1.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0196885824001088/pdfft?md5=4128db605d5a508bd857a5c53a34cfc3&pid=1-s2.0-S0196885824001088-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142163848","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Corrigendum to “Exact solutions in log-concave maximum likelihood estimation” [Adv. Appl. Math. 143 (2023) 102448]","authors":"Alexandros Grosdos ,&nbsp;Alexander Heaton ,&nbsp;Kaie Kubjas ,&nbsp;Olga Kuznetsova ,&nbsp;Georgy Scholten ,&nbsp;Miruna-Ştefana Sorea","doi":"10.1016/j.aam.2024.102777","DOIUrl":"10.1016/j.aam.2024.102777","url":null,"abstract":"","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"162 ","pages":"Article 102777"},"PeriodicalIF":1.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S019688582400109X/pdfft?md5=53ccd91901fe78ecb2557a95b23a8851&pid=1-s2.0-S019688582400109X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142163963","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Newton polytopes of dual k-Schur polynomials","authors":"Bo Wang,&nbsp;Candice X.T. Zhang,&nbsp;Zhong-Xue Zhang","doi":"10.1016/j.aam.2024.102773","DOIUrl":"10.1016/j.aam.2024.102773","url":null,"abstract":"<div><p>Rado's theorem about permutahedra and dominance order on partitions reveals that each Schur polynomial is M-convex, or equivalently, it has a saturated Newton polytope and this polytope is a generalized permutahedron as well. In this paper we show that the support of each dual <em>k</em>-Schur polynomial indexed by a <em>k</em>-bounded partition coincides with that of the Schur polynomial indexed by the same partition, and hence the two polynomials share the same saturated Newton polytope. The main result is based on our recursive algorithm to generate a semistandard <em>k</em>-tableau for a given shape and <em>k</em>-weight. As consequences, we obtain the M-convexity of dual <em>k</em>-Schur polynomials, affine Stanley symmetric polynomials and cylindric skew Schur polynomials.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"162 ","pages":"Article 102773"},"PeriodicalIF":1.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0196885824001052/pdfft?md5=3973fc564b253c4bed216b16fcf4396e&pid=1-s2.0-S0196885824001052-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142163846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Classical continued fractions for some multivariate polynomials generalizing the Genocchi and median Genocchi numbers","authors":"Bishal Deb ,&nbsp;Alan D. Sokal","doi":"10.1016/j.aam.2024.102756","DOIUrl":"10.1016/j.aam.2024.102756","url":null,"abstract":"<div><p>A D-permutation is a permutation of <span><math><mo>[</mo><mn>2</mn><mi>n</mi><mo>]</mo></math></span> satisfying <span><math><mn>2</mn><mi>k</mi><mo>−</mo><mn>1</mn><mo>≤</mo><mi>σ</mi><mo>(</mo><mn>2</mn><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> and <span><math><mn>2</mn><mi>k</mi><mo>≥</mo><mi>σ</mi><mo>(</mo><mn>2</mn><mi>k</mi><mo>)</mo></math></span> for all <em>k</em>; they provide a combinatorial model for the Genocchi and median Genocchi numbers. We find Stieltjes-type and Thron-type continued fractions for some multivariate polynomials that enumerate D-permutations with respect to a very large (sometimes infinite) number of simultaneous statistics that measure cycle status, record status, crossings and nestings.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"161 ","pages":"Article 102756"},"PeriodicalIF":1.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0196885824000885/pdfft?md5=3810ddd45e6b3ed90210b39501d14be8&pid=1-s2.0-S0196885824000885-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142151475","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Sorting signed permutations by tandem duplication random loss and inverse tandem duplication random loss","authors":"Bruno J. Schmidt ,&nbsp;Tom Hartmann ,&nbsp;Peter F. Stadler","doi":"10.1016/j.aam.2024.102757","DOIUrl":"10.1016/j.aam.2024.102757","url":null,"abstract":"<div><p>Tandem duplication random loss (TDRL) and inverse tandem duplication random loss (iTDRL) are mechanisms of mitochondrial genome rearrangement that can be modeled as simple operations on signed permutations. Informally, they comprise the duplication of a subsequence of a permutation, where in the case of iTDRL the copy is inserted with inverted order and signs. In the second step, one copy of each duplicate element is removed, such that the result is again a signed permutation. The TDRL/iTDRL sorting problem consists in finding the minimal number of TDRL or iTDRL operations necessary to convert the identity permutation <em>ι</em> into a given permutation <em>π</em>. We introduce a simple signature, called the misc-encoding, of permutation <em>π</em>. This construction is used to design an <span><math><mi>O</mi><mo>(</mo><mi>n</mi><mi>log</mi><mo>⁡</mo><mi>n</mi><mo>)</mo></math></span> algorithm to solve the TDRL/iTDRL sorting problem.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"161 ","pages":"Article 102757"},"PeriodicalIF":1.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0196885824000897/pdfft?md5=8c4e38f63eb26de0ae762cdd3ebe2a78&pid=1-s2.0-S0196885824000897-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142151476","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Algebraic network reconstruction of discrete dynamical systems","authors":"Heather A. Harrington ,&nbsp;Mike Stillman ,&nbsp;Alan Veliz-Cuba","doi":"10.1016/j.aam.2024.102760","DOIUrl":"10.1016/j.aam.2024.102760","url":null,"abstract":"<div><p>We present a computational algebra solution to reverse engineering the network structure of discrete dynamical systems from data. We use pseudomonomial ideals to determine dependencies between variables that encode constraints on the possible wiring diagrams underlying the process generating the discrete-time, continuous-space data. Our work assumes that each variable is either monotone increasing or decreasing. We prove that with enough data, even in the presence of small noise, our method can reconstruct the correct unique wiring diagram.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"161 ","pages":"Article 102760"},"PeriodicalIF":1.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142136744","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Bijections on pattern avoiding inversion sequences and related objects","authors":"JiSun Huh ,&nbsp;Sangwook Kim ,&nbsp;Seunghyun Seo ,&nbsp;Heesung Shin","doi":"10.1016/j.aam.2024.102771","DOIUrl":"10.1016/j.aam.2024.102771","url":null,"abstract":"<div><p>The number of inversion sequences avoiding two patterns 101 and 102 is known to be the same as the number of permutations avoiding three patterns 2341, 2431, and 3241. This sequence also counts the number of Schröder paths without triple descents, restricted bicolored Dyck paths, <span><math><mo>(</mo><mn>101</mn><mo>,</mo><mn>021</mn><mo>)</mo></math></span>-avoiding inversion sequences, and weighted ordered trees. We provide bijections to integrate them together by introducing <em>F</em>-paths. Moreover, we define three kinds of statistics for each of the objects and count the number of each object with respect to these statistics. We also discuss direct sums of each object.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"161 ","pages":"Article 102771"},"PeriodicalIF":1.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142136743","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Extended Schur functions and bases related by involutions","authors":"Spencer Daugherty","doi":"10.1016/j.aam.2024.102770","DOIUrl":"10.1016/j.aam.2024.102770","url":null,"abstract":"<div><p>We introduce two new bases of <span><math><mi>Q</mi><mi>S</mi><mi>y</mi><mi>m</mi></math></span>, the reverse extended Schur functions and the row-strict reverse extended Schur functions, as well as their duals in <em>NSym</em>, the reverse shin functions and row-strict reverse shin functions. These bases are the images of the extended Schur basis and shin basis under the involutions <em>ρ</em> and <em>ω</em>, which generalize the classical involution <em>ω</em> on the symmetric functions. In addition, we prove a Jacobi-Trudi rule for certain shin functions using creation operators. We define skew extended Schur functions and skew-II extended Schur functions based on left and right actions of <em>NSym</em> and <span><math><mi>Q</mi><mi>S</mi><mi>y</mi><mi>m</mi></math></span> respectively. We then use the involutions <em>ρ</em> and <em>ω</em> to translate these and other known results to our reverse and row-strict reverse bases.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"161 ","pages":"Article 102770"},"PeriodicalIF":1.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142129395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Off-diagonally symmetric domino tilings of the Aztec diamond of odd order","authors":"Yi-Lin Lee","doi":"10.1016/j.aam.2024.102759","DOIUrl":"10.1016/j.aam.2024.102759","url":null,"abstract":"<div><p>We study the enumeration of off-diagonally symmetric domino tilings of odd-order Aztec diamonds in two directions: (1) with one boundary defect, and (2) with maximally-many zeroes on the diagonal. In the first direction, we prove a symmetry property which states that the numbers of off-diagonally symmetric domino tilings of the Aztec diamond of order <span><math><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></math></span> are equal when the boundary defect is at the <em>k</em>th position and the <span><math><mo>(</mo><mn>2</mn><mi>n</mi><mo>−</mo><mi>k</mi><mo>)</mo></math></span>th position on the boundary, respectively. This symmetry property proves a special case of a recent conjecture by Behrend, Fischer, and Koutschan.</p><p>In the second direction, a Pfaffian formula is obtained for the number of “nearly” off-diagonally symmetric domino tilings of odd-order Aztec diamonds, where the entries of the Pfaffian satisfy a simple recurrence relation. The numbers of domino tilings mentioned in the above two directions do not seem to have a simple product formula, but we show that these numbers satisfy simple matrix equations in which the entries of the matrix are given by Delannoy numbers. The proof of these results involves the method of non-intersecting lattice paths and a modification of Stembridge's Pfaffian formula for families of non-intersecting lattice paths. Finally, we propose conjectures concerning the log-concavity and asymptotic behavior of the number of off-diagonally symmetric domino tilings of odd-order Aztec diamonds.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"161 ","pages":"Article 102759"},"PeriodicalIF":1.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142087152","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stereologically adapted Crofton formulae for tensor valuations","authors":"Emil Dare","doi":"10.1016/j.aam.2024.102754","DOIUrl":"10.1016/j.aam.2024.102754","url":null,"abstract":"<div><p>The classical Crofton formula explains how intrinsic volumes of a convex body <em>K</em> in <em>n</em>-dimensional Euclidean space can be obtained from integrating a measurement function at sections of <em>K</em> with invariantly moved affine flats. We generalize this idea by constructing stereologically adapted Crofton formulae for translation invariant Minkowski tensors, expressing a prescribed tensor valuation as an invariant integral of a measurement function of section profiles with flats. The measurement functions are weighed sums of powers of the metric tensor times Minkowski valuations. The weights are determined explicitly from known Crofton formulae using Zeilberger's algorithm. The main result is an exhaustive set of measurement functions where the invariant integration is over flats.</p><p>With the main result at hand, a Blaschke-Petkantschin formula allows us to establish new measurement functions valid when the invariant integration over flats is replaced by an invariant integration over subspaces containing a fixed subspace of lower dimension. Likewise, a stereologically adapted Crofton formula valid in the scheme of vertical sections is constructed. Only some special cases of this result have been stated explicitly before, with even the three-dimensional case yielding a new stereological formula. Here, we obtain new vertical section formulae for the surface tensors of even rank.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"160 ","pages":"Article 102754"},"PeriodicalIF":1.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0196885824000861/pdfft?md5=7cfe89c5b6afba8f72d1292bc78aca12&pid=1-s2.0-S0196885824000861-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142083640","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}