IA Scholar Query: Generalized Stirling and Lah numbers.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgThu, 25 Aug 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Totally non-negativity of a family of change-of-basis matrices
https://scholar.archive.org/work/clwovlpkqfh43ovoqx7he5nup4
Let a=(a_1, a_2, ..., a_n) and e=(e_1, e_2, ..., e_n) be real sequences. Denote by M_ e→ a the (n+1)×(n+1) matrix whose (m,k) entry (m, k ∈{0,..., n}) is the coefficient of the polynomial (x-a_1)⋯(x-a_k) in the expansion of (x-e_1)⋯(x-e_m) as a linear combination of the polynomials 1, x-a_1, ..., (x-a_1)⋯(x-a_m). By appropriate choice of a and e the matrix M_ e→ a can encode many familiar doubly-indexed combinatorial sequences, such as binomial coefficients, Stirling numbers of both kinds, Lah numbers and central factorial numbers. In all four of these examples, M_ e→ a enjoys the property of total non-negativity – the determinants of all its square submatrices are non-negative. This leads to a natural question: when, in general, is M_ e→ a totally non-negative? Galvin and Pacurar found a simple condition on e that characterizes total non-negativity of M_ e→ a when a is non-decreasing. Here we fully extend this result. For arbitrary real sequences a and e, we give a condition that can be checked in O(n^2) time that determines whether M_ e→ a is totally non-negative. When M_ e→ a is totally non-negative, we witness this with a planar network whose weights are non-negative and whose path matrix is M_ e→ a. When it is not, we witness this with an explicit negative minor.David Galvin, Yufei Zhangwork_clwovlpkqfh43ovoqx7he5nup4Thu, 25 Aug 2022 00:00:00 GMTOn the homotopy type of multipath complexes
https://scholar.archive.org/work/k5vgp6x3y5e5rmecizdb2mo4ay
A multipath in a directed graph is a disjoint union of paths. The multipath complex of a directed graph G is the simplicial complex whose faces are the multipaths of G. We compute the Euler characteristic, and associated generating function, of the multipath complex for some families of graphs, including transitive tournaments and complete bipartite graphs. Then, we compute the homotopy type of multipath complexes of linear graphs, polygons, small grids and transitive tournaments. We show that they are all contractible or wedges of spheres. We introduce a new technique for decomposing directed graphs into dynamical regions, which allows us to simplify the homotopy computations.Luigi Caputi, Carlo Collari, Sabino Di Trani, Jason P. Smithwork_k5vgp6x3y5e5rmecizdb2mo4ayTue, 09 Aug 2022 00:00:00 GMTTriangular Recurrences, Generalized Eulerian Numbers, and Related Number Triangles
https://scholar.archive.org/work/wssk7qlyfze3rjp3uhkgq2lm2y
Many combinatorial and other number triangles are solutions of recurrences of the Graham-Knuth-Patashnik (GKP) type. Such triangles and their defining recurrences are investigated analytically. They are acted on by a transformation group generated by two involutions: a left-right reflection and an upper binomial transformation, acting row-wise. The group also acts on the bivariate exponential generating function (EGF) of the triangle. By the method of characteristics, the EGF of any GKP triangle has an implicit representation in terms of the Gauss hypergeometric function. There are several parametric cases when this EGF can be obtained in closed form. One is when the triangle elements are the generalized Stirling numbers of Hsu and Shiue. Another is when they are generalized Eulerian numbers of a newly defined kind. These numbers are related to the Hsu-Shiue ones by an upper binomial transformation, and can be viewed as coefficients of connection between polynomial bases, in a manner that generalizes the classical Worpitzky identity. Many identities involving these generalized Eulerian numbers and related generalized Narayana numbers are derived, including closed-form evaluations in combinatorially significant cases.Robert S. Maierwork_wssk7qlyfze3rjp3uhkgq2lm2yWed, 20 Jul 2022 00:00:00 GMTIdentities of Degenerate Poly-Changhee Polynomials Arising from λ -Sheffer Sequences
https://scholar.archive.org/work/qcfiw6wnprfofl2bnri6sbm7ka
In the 1970s, Gian-Carlo Rota constructed the umbral calculus for investigating the properties of special functions, and by Kim-Kim, umbral calculus is generalized called λ -umbral calculus. In this paper, we find some important relationships between degenerate Changhee polynomials and some important special polynomials by expressing the Changhee polynomial as a linear combination of some special polynomials. In addition, we derive some interesting identities related to degenerate poly-Changhee polynomials and some important special functions by using λ -umbral calculus.Sang Jo Yun, Jin-Woo Park, Barbara Martinucciwork_qcfiw6wnprfofl2bnri6sbm7kaThu, 14 Jul 2022 00:00:00 GMTkStatistics: Unbiased Estimates of Joint Cumulant Products from the Multivariate Faà Di Bruno's Formula
https://scholar.archive.org/work/xttr3c4h2bhyxdbobczkgucpbi
kStatistics is a package in R that serves as a unified framework for estimating univariate and multivariate cumulants as well as products of univariate and multivariate cumulants of a random sample, using unbiased estimators with minimum variance. The main computational machinery of kStatistics is an algorithm for computing multi-index partitions. The same algorithm underlies the general-purpose multivariate Fa\'a di Bruno's formula, which has been therefore included in the last release of the package. This formula gives the coefficients of formal power series compositions as well as the partial derivatives of multivariable function compositions. One of the most significant applications of this formula is the possibility to generate many well-known polynomial families as special cases. So, in the package, there are special functions for generating very popular polynomial families, such as the Bell polynomials. However further families can be obtained, for suitable choices of the formal power series involved in the composition or when suitable symbolic strategies are employed. In both cases, we give examples on how to modify the R codes of the package to accomplish this task. Future developments are addressed at the end of the paper.E. Di Nardo, G. Guarinowork_xttr3c4h2bhyxdbobczkgucpbiThu, 30 Jun 2022 00:00:00 GMTExtensions and Limits of the Specker-Blatter Theorem
https://scholar.archive.org/work/vmg3jvzxzne45ca2lpelb2hocu
The original Specker-Blatter Theorem (1983) was formulated for classes of structures 𝒞 of one or several binary relations definable in Monadic Second Order Logic MSOL. It states that the number of such structures on the set [n] is modularly C-finite (MC-finite). In previous work we extended this to structures definable in CMSOL, MSOL extended with modular counting quantifiers. The first author also showed that the Specker-Blatter Theorem does not hold for one quaternary relation (2003). If the vocabulary allows a constant symbol c, there are n possible interpretations on [n] for c. We say that a constant c is hard-wired if c is always interpreted by the same element j ∈ [n]. In this paper we show: 1. The Specker-Blatter Theorem also holds for CMSOL when hard-wired constants are allowed. The proof method of Specker and Blatter does not work in this case. 2. The Specker-Blatter Theorem does not hold already for 𝒞 with one ternary relation definable in First Order Logic FOL. This was left open since 1983. Using hard-wired constants allows us to show MC-finiteness of counting functions of various restricted partition functions which were not known to be MC-finite till now. Among them we have the restricted Bell numbers B_r,A, restricted Stirling numbers of the second kind S_r,A or restricted Lah-numbers L_r,A. Here r is an non-negative integer and A is an ultimately periodic set of non-negative integers.Eldar Fischer, Johann A. Makowskywork_vmg3jvzxzne45ca2lpelb2hocuMon, 27 Jun 2022 00:00:00 GMTContext-Free Grammars for Several Triangular Arrays
https://scholar.archive.org/work/etti42qllffttmrpxq67flsduq
In this paper, we present a unified grammatical interpretation of the numbers that satisfy a kind of four-term recurrence relation, including the Bell triangle, the coefficients of modified Hermite polynomials, and the Bessel polynomials. Additionally, as an application, a criterion for real zeros of row-generating polynomials is also presented.Roberta Rui Zhou, Jean Yeh, Fuquan Renwork_etti42qllffttmrpxq67flsduqMon, 20 Jun 2022 00:00:00 GMTMoment Generating Stirling Numbers and Applications
https://scholar.archive.org/work/ukcfq227onbbngod652eswhh2m
In this paper, we investigate certain combinatorial numbers, the moment generating Stirling numbers. They are a special case of Hsu's generalized Stirling numbers and satisfy many more properties and combinatorial identities than are known in the general case. As application, we provide the computation of the moments and central moments of the phase type distribution, the recurrence time in Markov chains, the geometric distribution, the negative binomial distribution and of a class of distributions generalizing the negative binomial distribution. All computations can be performed in closed form without recursion. We also present the relationship to the Markov renewal process.Ludwig Frankwork_ukcfq227onbbngod652eswhh2mFri, 17 Jun 2022 00:00:00 GMTLah distribution: Stirling numbers, records on compositions, and convex hulls of high-dimensional random walks
https://scholar.archive.org/work/atmuuenl5fe7tc57nz7iwlq5ia
Let ξ_1,ξ_2,... be a sequence of independent copies of a random vector in ℝ^d having an absolutely continuous distribution. Consider a random walk S_i:=ξ_1+⋯+ξ_i, and let C_n,d:=conv(0,S_1,S_2,...,S_n) be the convex hull of the first n+1 points it has visited. The polytope C_n,d is called k-neighborly if for every indices 0≤ i_0 <⋯ < i_k≤ n the convex hull of the k+1 points S_i_0,..., S_i_k is a k-dimensional face of C_n,d. We study the probability that C_n,d is k-neighborly in various high-dimensional asymptotic regimes, i.e. when n, d, and possibly also k diverge to ∞. There is an explicit formula for the expected number of k-dimensional faces of C_n,d which involves Stirling numbers of both kinds. Motivated by this formula, we introduce a distribution, called the Lah distribution, and study its properties. In particular, we provide a combinatorial interpretation of the Lah distribution in terms of random compositions and records, and explicitly compute its factorial moments. Limit theorems which we prove for the Lah distribution imply neighborliness properties of C_n,d. This yields a new class of random polytopes exhibiting phase transitions parallel to those discovered by Vershik and Sporyshev, Donoho and Tanner for random projections of regular simplices and crosspolytopes.Zakhar Kabluchko, Alexander Marynychwork_atmuuenl5fe7tc57nz7iwlq5iaSat, 21 May 2022 00:00:00 GMTEVOLUTIONARY MATHEMATICS AND SCIENCE FOR COMBINATORIAL ALGEBRA III
https://scholar.archive.org/work/loxx2hfhfzbhjarq5bjzy7i74e
Tsao, Hung-ping (2022). Evolutionary mathematics and science for Combinatorial Algebra III. In: "Evolutionary Progress in Science, Technology, Engineering, Arts, and Mathematics (STEAM)", Wang, Lawrence K. and Tsao, Hung-ping (editors). Volume 4, Number 8C, August 2022; 100 pages. Lenox Institute Press, MA, USA. No. STEAM-VOL4-NUM8C-AUG2022; ISBN 978-0-9890870-3-2. ................ABSTRACT: This is the third of three sequential chapters of Combinatorial Algebra prepared for high school education in Taiwan, ROC or any other countries of the world for that matter despite of being written in Chinese language.Hung-ping Tsao, Lawrence K Wangwork_loxx2hfhfzbhjarq5bjzy7i74eFri, 01 Apr 2022 00:00:00 GMTNext speaker selection in Indonesian: A study of typical and atypical interactions
https://scholar.archive.org/work/lpncyx3dyrelhi6dmvkxytfq4y
This study explores the turn-taking system in conversations involving speakers of Indonesian, focusing on explicit next speaker selection. This study draws on "typical" and "atypical" datasets. The typical dataset comes from nine and a half hours of recordings of everyday conversations between 64 people. The atypical dataset comes from two and a half hours of recordings of conversation between four people with aphasia and 11 of their conversation partners. Using conversation analysis, this study examines how typical and atypical Indonesian speakers use two explicit practices for next speaker selection - address terms and touch - in questions. Specifically, it focuses on 238 questions including an address term, and 71 questions including a touch. This study demonstrates that address terms are used to commence courses of action and deal with problems of mutual orientation, deal with problems that emerge in a turn or sequence, address a person-specific action, or carry out fine aspects of action formation. It also demonstrates that touch can similarly deal with problems of mutual orientation, pursue a response from a recipient, or add a specific quality or salience to a question. These practices operate similarly in interactions involving people with aphasia, but people with aphasia experience difficult using maximally explicit practices, and problems with participation may arise despite successful next speaker selection. These findings offer an important basis for describing diversity and commonality in conversation across languages and cultures, and for characterising the disruptions to participation caused by aphasia.Fakry Hamdaniwork_lpncyx3dyrelhi6dmvkxytfq4yMon, 28 Mar 2022 00:00:00 GMTEVOLUTIONARY MATHEMATICS AND SCIENCE FOR COMBINATORIAL ALGEBRA II
https://scholar.archive.org/work/swcon32ytjbxpn3evjh47xd4eq
Tsao, Hung-ping (2022). Evolutionary mathematics and science for Combinatorial Algebra II. In: "Evolutionary Progress in Science, Technology, Engineering, Arts, and Mathematics (STEAM)", Wang, Lawrence K. and Tsao, Hung-ping (editors). Volume 4, Number 8B, August 2022; 100 pages. Lenox Institute Press, MA, USA. No. STEAM-VOL4-NUM8B-AUG2022; ISBN 978-0-9890870-3-2 ................ ABSTRACT: This is the second of three sequential chapters of Combinatorial Algebra prepared for high school education in Taiwan, ROC or any other countries of the world for that matter despite of being written in Chinese language.Hung-ping Tsao, Lawrence K Wangwork_swcon32ytjbxpn3evjh47xd4eqSat, 26 Mar 2022 00:00:00 GMTCentral Limit Theorems for Combinatorial Numbers Associated with Laguerre Polynomials
https://scholar.archive.org/work/zliliblnivfnji7jcghtfqrcqi
In this paper, we study limit theorems for numbers satisfying a class of triangular arrays, which are defined by a bivariate linear recurrence with bivariate linear coefficients. We obtain analytical expressions for the semi-exponential generating function of several classes of the numbers, including combinatorial numbers associated with Laguerre polynomials. We apply these results to prove the numbers' asymptotic normality and specify the convergence rate to the limiting distribution.Igoris Belovaswork_zliliblnivfnji7jcghtfqrcqiWed, 09 Mar 2022 00:00:00 GMTMatroids are not Ehrhart positive
https://scholar.archive.org/work/sp6kfep2rjgankmbwrk644yqlm
In this article we disprove the conjectures asserting the positivity of the coefficients of the Ehrhart polynomial of matroid polytopes by De Loera, Haws and Köppe (2007) and of generalized permutohedra by Castillo and Liu (2015). We prove constructively that for every n≥ 19 there exist connected matroids on n elements that are not Ehrhart positive. Also, we prove that for every k≥ 3 there exist connected matroids of rank k that are not Ehrhart positive. Our proofs rely on our previous results on the geometric interpretation of the operation of circuit-hyperplane relaxation and our formulas for the Ehrhart polynomials of hypersimplices and minimal matroids. This allows us to give a precise expression for the Ehrhart polynomials of all sparse paving matroids, a class of matroids which is conjectured to be predominant and which contains the counterexamples arising from our construction.Luis Ferroniwork_sp6kfep2rjgankmbwrk644yqlmMon, 07 Mar 2022 00:00:00 GMTOn the Connection Between Stirling Numbers and Bessel Numbers
https://scholar.archive.org/work/qbcbal6k4ba75f76eki3lme6sm
We present new proofs for some summation identities involving Stirling numbers of both first and second kind. The two main identities show a connection between Stirling numbers and Bessel numbers. Our method is based on solving a particular recurrence relation in two different ways and comparing the coefficients in the resulting polynomial expressions. We also briefly discuss a probabilistic setting where this recurrence relation occurs.David Stenlundwork_qbcbal6k4ba75f76eki3lme6smFri, 25 Feb 2022 00:00:00 GMTThe Daily Texan
https://scholar.archive.org/work/k5smfhxdn5h2jog6h2mnz4dek4
B y BILL BARNES Texan Staff Writer A vvhitc-shirted APO slithered through th e crowd in the Main Ballroom and worked his w ay in and around the chairs set up in the co m ers w here the election re turns w ere being tabulated on a blackboard. In the center of that crowd a chubby, tall youth, his f a c e burnt from cam paigning in the Min, hurriedly ground out his cigaret to and raced to the board; he stood tip to e d , neck craned, breath held, w hile th® figures w ere put in the rectan gles. HU boy's lead increased. His expression bloomed triumpliantly. " Whoooo . w hoooooo," he shouted, slapping shoulders with each hand. Af Ihe tab les in the n orth end of th e ro o m O ra n g e Ja c k e ts , Spooks, and A PO 's w e re still slu m p ed o v e r sh eets of p a p e r, p en cils scrib b lin g , lips m oving silently. N u m ero u s w rite-in c a n d id a te sm o re th a n 150 in one box slow ed the counting. V oters had w rit ten in the nam es o f personal friends. Jack Kennedy, B arry Goldwater, M ickey M ouse, and injured racing car driver Stirling M oss. " It's p ath etic," said an Election Com m ission m em ber about the vote for M oss, " but that's what som e people will d o." A funnier write-in w as for Marc E, Gronsberg, a first-year law student whom cla ssm a tes nominated for the Board of R e gents. Tho cro w d co n tin u ally filte re d th ro u g h th e d o o rs in q u e st of Cokes and coffee. Coffee from a larg e u rn w as c irc u la te d a m o n g the ta b le s to re fre sh th e c o u n ters. In th e tw o-hour p erio d from 8 in ~ ,u o cro w d doubled • to 250 p erso n s, d w e re th e m odes nkled sh irt tails ; dark fra te rn ity s ta rc h e d w hite b ro g an s, sh o w er mg c risp cotton ight-w eig h t sw eatflat s. t ? Cd * P o P 5 g g Cl lf CO a H-O Ct S o m e stu d e n ts tried to ignore th e e x c ite m e n t a c co m p a n y in g th e retu rn s so th e y could stu d y. A girl wa® read in g a n o v e l; a boy plotted a p a ra b o la on a grap h . A s t h e ev en in g pro g r e sse d and trends e v o lv e d , h ow ev er. (took*, p a p er s, and p u rses w e r e c o n sp ic u o u sly d e serted on folding c h a irs a s the crow d su rged around the c o r ner w h ere the b lackb oard of m a g ic fig u r es stood.University Of Texas At Austin, Austin, The University Of Texas Atwork_k5smfhxdn5h2jog6h2mnz4dek4Fri, 25 Feb 2022 00:00:00 GMTLattice points in slices of prisms
https://scholar.archive.org/work/ldjempkswzecjez3rfjqx6mdx4
We study the Ehrhart polynomials of certain slices of rectangular prisms. These polytopes are generalizations of the hypersimplex and are contained in the larger class of polypositroids introduced by Lam and Postnikov. We give a combinatorial formula for all the Ehrhart coefficients in terms of the number of weighted permutations satisfying certain compatibility properties. This result proves that all these polytopes are Ehrhart positive. Additionally, inspired by a result due to Early and Kim, we give a combinatorial interpretation for all the coefficients of the h^*-polynomial; this solves the problem of providing an interpretation for the numerator of the Hilbert series, also known as the h-vector of all algebras of Veronese type. As corollaries of our results, we obtain an expression for the volumes of these slices of prisms as weighted combinations of Eulerian numbers; we use this to provide some generalizations of Laplace's result on the combinatorial interpretation of the volume of the hypersimplex. We discuss an application regarding a generalization of the flag Eulerian numbers and certain refinements, and give a short proof of the Ehrhart positivity of the independence polytope of all uniform matroids.Luis Ferroni, Daniel McGinniswork_ldjempkswzecjez3rfjqx6mdx4Wed, 23 Feb 2022 00:00:00 GMTStirling numbers associated with sequences of polynomials
https://scholar.archive.org/work/yy4fossjljannhat6vsrtyn7zu
Let P be the set of the sequence of polynomials of degree n. The aim of this paper is to study the Stirling numbers of the second kind associated with P and of the first kind associated with P, in a unified and systematic way with the help of umbral calculus technique. Our results are illustrated with many examples which give rise to interesting inverse relations in each case.Dae san Kim, taekyun Kimwork_yy4fossjljannhat6vsrtyn7zuWed, 23 Feb 2022 00:00:00 GMTCoefficientwise Hankel-total positivity of row-generating polynomials for the m-Jacobi-Rogers triangle
https://scholar.archive.org/work/xx4rzuokbfanjb6kbe73jfcxci
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. 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. This immediately implies that the corresponding m-Jacobi-Rogers triangular convolution preserves the Stieltjes moment property of sequences and its zeroth column sequence is coefficientwise Hankel-totally positive and log-convex of higher order in all the indeterminates. In consequence, for m=1, we immediately obtain some results on coefficientwise Hankel-total positivity for the Catalan-Stieltjes matrices. For the general m, combining our criterion and a function satisfying an autonomous differential equation, we present different criteria for coefficientwise Hankel-total positivity of the row-generating polynomial sequence for exponential Rirodan arrays. In addition, we also derive some results for the coefficientwise Hankel-total positivity in terms of compositional functions and m-branched Stieltjes continued fractions. We apply our results to many combinatorial polynomials in a unified manner. In particular, we also solve some conjcetures proposed by Sokal.Bao-Xuan Zhuwork_xx4rzuokbfanjb6kbe73jfcxciTue, 08 Feb 2022 00:00:00 GMTPositive hulls of random walks and bridges
https://scholar.archive.org/work/lisotlpxdrdeliz4ro3ndow7qa
We study random convex cones defned as positive hulls of d-dimensional random walks and bridges. We compute expectations of various geometric functionals of these cones such as the number of k-dimensional faces and the sums of conic quermassintegrals of their k-dimensional faces. These expectations are expressed in terms of Stirling numbers of both kinds and their B-analogues.Thomas Godland, Zakhar Kabluchkowork_lisotlpxdrdeliz4ro3ndow7qaFri, 28 Jan 2022 00:00:00 GMT