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Consider k-binomial equivalence: two finite words are equivalent if they share the same subwords of length at most k with the same multiplicities. With this relation, the k-binomial complexity of an infinite word x maps the integer n to the number of pairwise non-equivalent factors of length n occurring in x. In this paper based on the notion of template introduced by Currie et al., we show that, for all k ≥ 2 , the k-binomial complexity of the Tribonacci word coincides with its usual factor complexity p ( n ) = 2 n + 1 . A similar result was already known for Sturmian words, but the proof relies on completely different techniques that seemingly could not be applied for Tribonacci.

This paper studies the mathematical properties of collectively canalizing Boolean functions, a class of functions that has arisen from applications in systems biology. Boolean networks are an increasingly popular modeling framework for regulatory networks, and the class of functions studied here captures a key feature of biological network dynamics, namely that a subset of one or more variables, under certain conditions, can dominate the value of a Boolean function, to the exclusion of all others. These functions have rich mathematical properties to be explored. The paper shows how the number and type of such sets influence a function's behavior and define a new measure for the canalizing strength of any Boolean function. We further connect the concept of collective canalization with the well-studied concept of the average sensitivity of a Boolean function. The relationship between Boolean functions and the dynamics of the networks they form is important in a wide range of applications beyond biology, such as computer science, and has been studied with statistical and simulation-based methods. But the rich relationship between structure and dynamics remains largely unexplored, and this paper is intended as a contribution to its mathematical foundation., Comment: 17 pages, 3 figures

A systematic study of avoidance of mesh patterns of length 2 was conducted by Hilmarsson et al., where 25 out of 65 non-equivalent cases were solved. In this paper, we give 27 distribution results for these patterns including 14 distributions for which avoidance was not known. Moreover, for the unsolved cases, we prove an equidistribution result (out of 6 equidistribution results we prove in total), and conjecture 6 more equidistributions. Finally, we find seemingly unknown distribution of the well known permutation statistic ``strict fixed point'', which plays a key role in many of our enumerative results. This paper is the first systematic study of distributions of mesh patterns. Our techniques to obtain the results include, but are not limited to, obtaining functional relations for generating functions, and finding recurrence relations and bijections., Comment: Advances in Applied Mathematics 110 (2019) 1-32; there is a mistake in the proof of Theorem 5.1 in version 1 of the paper; its statement is now a conjecture

In this paper, we study the distribution of consecutive patterns in the set of 123-avoiding permutations and the set of 132-avoiding permutations, that is, in S n ( 123 ) and S n ( 132 ) . We first study the distribution of consecutive pattern γ-matches in S n ( 123 ) and S n ( 132 ) for each length 3 consecutive pattern γ. Then we extend our methods to study the joint distributions of multiple consecutive patterns. Some more general cases are discussed in this paper as well.

In 2007, Andrews introduced the odd rank of odd Durfee symbols. Let N 0 ( m , n ) denote the number of odd Durfee symbols of n with odd rank m, and N 0 ( r , m ; n ) be the number of odd Durfee symbols of n with odd rank congruent to r modulo m. In this paper, we give the generating functions for N 0 ( r , 12 ; n ) by utilizing some identities involving Appell-Lerch sums m ( x , q , z ) and a universal mock theta function g ( x , q ) . Based on these formulas, we determine the signs of N 0 ( r , 12 ; 4 n + t ) − N 0 ( s , 12 ; 4 n + t ) for all 0 ≤ r , s ≤ 6 and 0 ≤ t ≤ 3 . In particular, we prove that N 0 ( 2 , 12 ; 4 n + 1 ) = N 0 ( 4 , 12 ; 4 n + 1 ) . Moreover, let D k 0 ( n ) denote the number of k-marked odd Durfee symbols of n. Andrews conjectured that D 2 0 ( 8 n + s ) and D 3 0 ( 16 n + t ) are even with s ∈ { 4 , 6 } and t ∈ { 1 , 9 , 11 , 13 } which were confirmed by Wang. In this paper, we found new congruences for D k 0 ( n ) . In particular, for k = 2 or 3, we give characterizations of n such that D k 0 ( n ) is odd and prove that D k 0 ( n ) take even values with probability 1 for n ≥ 0 .

This paper addresses the question of the impact of the boundary on the dynamical behaviour of finite Boolean automata networks on Z2. The evolution over discrete time of such networks is governed by a specific stochastic threshold non-linear transition rule derived from the classical rule of formal neural networks. More precisely, the networks considered in this paper are finite but the study is done for arbitrarily large sizes. Moreover, the boundary impact is viewed as a classical definition of a phase transition in probability theory, characterising in our context the fact that a network admits distinct asymptotic behaviours when different boundary instances are assumed. The main contribution of this paper is the highlight of a formula for a necessary condition for boundary sensitivity, whose sufficiency and necessity are entirely proven with natural constraints on interaction potentials.

It is a classical result that the parity-balance of the number of weak excedances of all permutations (derangements, respectively) of length n is the Euler number E n , alternating in sign, if n is odd (even, respectively). Josuat-Verges obtained a q-analog of the results respecting the number of crossings of a permutation. One of the goals in this paper is to extend the results to the permutations (derangements, respectively) of types B and D, on the basis of the joint distribution in statistics excedances, crossings and the number of negative entries obtained by Corteel, Josuat-Verges and Kim. Springer numbers are analogous Euler numbers that count the alternating permutations of type B, called snakes. Josuat-Verges derived bivariate polynomials Q n ( t , q ) and R n ( t , q ) as generalized Euler numbers via successive q-derivatives and multiplications by t on polynomials in t. The other goal in this paper is to give a combinatorial interpretation of Q n ( t , q ) and R n ( t , q ) as the enumerators of the snakes with restrictions.

Phylogenetic networks are used to represent evolutionary relationships between species in biology. Such networks are often categorized into classes by their topological features, which stem from both biological and computational motivations. We study two network classes in this paper: tree-based networks and orchard networks. Tree-based networks are those that can be obtained by inserting edges between the edges of an underlying tree. Orchard networks are a recently introduced generalization of the class of tree-child networks. Structural characterizations have already been discovered for tree-based networks; this is not the case for orchard networks. In this paper, we introduce cherry covers—a unifying characterization of both network classes—in which we decompose the edges of the networks into so-called cherry shapes and reticulated cherry shapes. We show that cherry covers can be used to characterize the class of tree-based networks as well as the class of orchard networks. Moreover, we also generalize these results to non-binary networks.

本文在一类具有时滞效应和收获效应的浮游植物–浮游动物模型基础上，考虑了浮游动物的年龄结构特征，从而得到了一类具有时滞效应，收获效应和年龄结构的浮游植物和浮游动物模型。文章的主要目的是研究年龄结构对该模型动力学性质的影响。通过模型分析，将其转化成一类抽象的柯西问题。在此基础上，本文主要研究了该模型平衡点的动力学性质，包括存在性和唯一性。 In this paper, we study an age-structured phytoplankton-zooplankton model to describe the dy-namics between the two populations. The model is described by considering age-structured zoop-lankton, time delay and harvesting effect. The aim of this paper is to understand the effect of age structure on this model. Sufficient conditions are obtained for the existence of the equilibrium. It is shown that the equilibrium state of the model is related to the age of zooplankton.

本文讨论Fredholm积分微分方程边值问题的数值方法。通过建立满足边界条件的再生核空间，获得简单易行的再生核数值逼近方法。给出方程精确解的级数表达式，通过截断级数获得方程的近似解，并给出了误差估计。数值模拟结果说明本文方法的有效性。 This paper discusses the numerical method for a class of Fredholm integro-differential boundary value problems. By constructing the reproducing kernel space which satisfies the boundary condi-tions, the simple reproducing kernel numerical approximate method is established. The paper describes both the exact solution obtained in the form of series and the approximate solution ob-tained by truncating the series representation of the exact solution. Error estimation of the method was discussed. The results of numerical simulation demonstrate the validity of the method in the paper.

本文从研究二元多项式插值的适定性问题着手，在构造二元分次插值适定结点组的“添加横直线法”和“添加竖直线法”的基础上，对二元分次插值适定性问题进一步研究和探讨，给出了二元分次插值适定结点组的几何结构和基本特征，构造了二元分次插值适定结点组的“添加抛物线”方法，推广了已有的研究结果，最后给出算例对所得研究结果进行了验证。 The paper studies from the posedness of bivariate polynomial interpolation. Based on the methods of “adding a horizontal line” and “adding vertical line” in constructing well-posed node group of binary interpolation, this paper further researches and discusses the posedness of binary graded interpolation, and gives geometrical structure and basic characteristics of well-posed node group of binary graded interpolation, then constructs the method that adds parabola for the well-posed node group of binary graded interpolation and generalizes the existing research results. Finally, the numerical examples are given to verify the research results.

In 2012, Sagan and Savage introduced the notion of st-Wilf equivalence for a statistic st and for sets of permutations that avoid particular permutation patterns. In this paper we consider inv-Wilf equivalence on sets of permutations that avoid two or more consecutive permutation patterns. We say that two sets of generalized permutation patterns Π and Π ′ are inv-Wilf equivalent if the generating function for the inversion statistic on the permutations that simultaneously avoid all elements of Π is equal to the generating function for the inversion statistic on the permutations that simultaneously avoid all elements of Π ′ . In 2013, Cameron and Killpatrick gave the inversion generating function for Fibonacci tableaux which are in one-to-one correspondence with the set of permutations that simultaneously avoid the consecutive patterns 321 and 312. In this paper, we use the language of Fibonacci tableaux to study the inversion generating functions for permutations that avoid Π where Π is a set of three or more consecutive permutation patterns. In addition, we introduce the more general notion of strip tableaux which are a useful combinatorial object for studying consecutive pattern avoidance. We give the inversion generating functions for Π a subset of 4 or 5 consecutive permutation patterns and for all but one of the cases where Π is a subset of three consecutive permutation patterns.

This paper develops the theory of Dirac reduction by symmetry for nonholonomic systems on Lie groups with broken symmetry. The reduction is carried out for the Dirac structures, as well as for the associated Lagrange-Dirac and Hamilton-Dirac dynamical systems. This reduction procedure is accompanied by reduction of the associated variational structures on both Lagrangian and Hamiltonian sides. The reduced dynamical systems obtained are called the implicit Euler-Poincar\'e-Suslov equations with advected parameters and the implicit Lie-Poisson-Suslov equations with advected parameters. The theory is illustrated with the help of finite and infinite dimensional examples. It is shown that equations of motion for second order Rivlin-Ericksen fluids can be formulated as an infinite dimensional nonholonomic system in the framework of the present paper.

The cogirth, $g^\ast(M)$, of a matroid $M$ is the size of a smallest cocircuit of $M$. Finding the cogirth of a graphic matroid can be done in polynomial time, but Vardy showed in 1997 that it is NP-hard to find the cogirth of a binary matroid. In this paper, we show that $g^\ast(M)\leq \frac{1}{2}\vert E(M)\vert$ when $M$ is binary, unless $M$ simplifies to a projective geometry. We also show that, when equality holds, $M$ simplifies to a Bose-Burton geometry, that is, a matroid of the form $PG(r-1,2)-PG(k-1,2)$. These results extend to matroids representable over arbitrary finite fields., 8 pages

Given an infinite family ${\mathcal G}$ of graphs and a monotone property ${\mathcal P}$, an (upper) threshold for ${\mathcal G}$ and ${\mathcal P}$ is a "fastest growing" function $p: \mathbb{N} \to [0,1]$ such that $\lim_{n \to \infty} \Pr(G_n(p(n)) \in {\mathcal P})= 1$ for any sequence $(G_n)_{n \in \mathbb{N}}$ over ${\mathcal G}$ with $\lim_{n \to \infty}\lvert V(G_n) \rvert = \infty$, where $G_n(p(n))$ is the random subgraph of $G_n$ such that each edge remains independently with probability $p(n)$. In this paper we study the upper threshold for the family of $H$-minor free graphs and for the graph property of being $(r-1)$-degenerate, which is one fundamental graph property with many applications. Even a constant factor approximation for the upper threshold for all pairs $(r,H)$ is expected to be very difficult by its close connection to a major open question in extremal graph theory. We determine asymptotically the thresholds (up to a constant factor) for being $(r-1)$-degenerate for a large class of pairs $(r,H)$, including all graphs $H$ of minimum degree at least $r$ and all graphs $H$ with no vertex-cover of size at most $r$, and provide lower bounds for the rest of the pairs of $(r,H)$. The results generalize to arbitrary proper minor-closed families and the properties of being $r$-colorable, being $r$-choosable, or containing an $r$-regular subgraph, respectively.

In 1977 Smith characterized graphs with exactly one positive eigenvalue. Since then, many particular results related to graphs with exactly two positive eigenvalues have emerged. In this paper we conclude this investigation by giving a full characterization of these graphs., 19 pages, 7 figures

This paper presents a geometric model of the Auslander-Reiten quiver of a type A quiver together with a stability function for which all indecomposable modules are stable. We also introduce a new Catalan object which we call a maximal almost rigid representation. We show that its endomorphism algebra is a tilted algebra of type A. We define a partial order on the set of maximal almost rigid representations and use our new geometric model to show that this partial order is a Cambrian lattice., 28 pages, 16 figures. Comments are welcome

A bipartite graph is subcubic if it is an irregular bipartite graph with maximum degree three. In this paper, we prove that the asymptotic value of maximum spectral radius over subcubic bipartite graphs of order $n$ is $3-\varTheta(\frac{\pi^{2}}{n^{2}})$. Our key approach is taking full advantage of the eigenvalues of certain tridiagonal matrices, due to Willms [SIAM J. Matrix Anal. Appl. 30 (2008) 639--656]. Moreover, for large maximum degree, i.e., the maximum degree is at least $\lfloor n/2 \rfloor$, we characterize irregular bipartite graphs with maximum spectral radius. For general maximum degree, we present an upper bound on the spectral radius of irregular bipartite graphs in terms of the order and maximum degree., Comment: 15 pages, 1 figure

In this paper we study a class of convex sets which are called closed pseudo-cones and study a new duality of this class. It turns out that the duality characterizes closed pseudo-cones and is essentially the only possible abstract duality of them.The characterization of the duality is corresponding to the classification of endomorphisms closed pseudo-cones., Comment: 26 pages

This is a computational study of bottlenecks on algebraic varieties. The bottlenecks of a smooth variety $X \subseteq \mathbb{C}^n$ are the lines in $\mathbb{C}^n$ which are normal to $X$ at two distinct points. The main result is a numerical homotopy that can be used to approximate all isolated bottlenecks. This homotopy has the optimal number of paths under certain genericity assumptions. In the process we prove bounds on the number of bottlenecks in terms of the Euclidean distance degree. Applications include the optimization problem of computing the distance between two real varieties. Also, computing bottlenecks may be seen as part of the problem of computing the reach of a smooth real variety and efficient methods to compute the reach are still to be developed. Relations to triangulation of real varieties and meshing algorithms used in computer graphics are discussed in the paper. The resulting algorithms have been implemented with Bertini and Macaulay2., Changed title, added refs

Frieze patterns form a nexus between algebra, combinatorics, and geometry. T-paths with respect to triangulations of surfaces have been used to obtain expansion formulae for cluster variables. This paper will introduce the concepts of weak friezes and T-paths with respect to dissections of polygons. Our main result is that weak friezes are characterised by satisfying an expansion formula which we call the T-path formula. We also show that weak friezes can be glued together, and that the resulting weak frieze is a frieze if and only if so was each of the weak friezes being glued., 17 pages

In this paper, we revisit the application of generating trees to the pattern avoidance problem for permutations. In particular, we study this problem for certain general sets of patterns and propose a new procedure leveraging the FinLabel algorithm and exploiting the subregularities in the associated generating trees. We consider some general kinds of generating trees for which the FinLabel algorithm fails to determine in a finite number of iterations the generating function that enumerates the underlying class of permutations. Our procedure provides a unified approach in these cases leading to a system of equations satisfied by a certain finite set of generating functions which can be readily solved with the aid of programming.

The Turan inequality and its higher order analog arise in the study of Maclaurin coefficients of an entire function in the Laguerre-Polya class. It is well known that if a real entire function ψ ( x ) is in the LP class, the Maclaurin coefficients satisfy both the Turan inequality and the higher order Turan inequality. Chen, Jia and Wang proved that for n ≥ 95 , the higher order Turan inequality holds for the partition function p ( n ) and the 3-rd associated Jensen polynomials p ( n ) + 3 p ( n + 1 ) x + 3 p ( n + 2 ) x 2 + p ( n + 3 ) x 3 have only real zeros. Recently, Griffin, Ono, Rolen and Zagier showed that Jensen polynomials for a large family of functions, including those associated to ξ ( s ) and the partition function, are hyperbolic for sufficiently large n. This result gave evidence for Riemann hypothesis. In this paper, we give a unified approach to investigate the higher order Turan inequality for the sequences { a n / n ! } n ≥ 0 , where a n satisfy a three-term recurrence relation. In particular, we prove higher order Turan inequality for the sequences { a n / n ! } n ≥ 0 , where a n are the Motzkin numbers, the Fine number, the Franel numbers of order 3 and the Domb numbers. As a consequence, for these combinatorial sequences, the 3-rd associated Jensen polynomials a n n ! + 3 a n + 1 ( n + 1 ) ! x + 3 a n + 2 ( n + 2 ) ! x 2 + a n + 3 ( n + 3 ) ! x 3 have only real zeros. Furthermore, for these combinatorial sequences we conjecture that for any given integer m ≥ 4 , there exists an integer N ( m ) such that for n > N ( m ) , the m-th associated Jensen polynomials ∑ i = 0 m ( m i ) a n + i ( n + i ) ! x i have only real zeros.

We consider two extensions of free probability that have been studied in the research literature, and are based on the notions of c-freeness and respectively of infinitesimal freeness for noncommutative random variables. In a 2012 paper, Belinschi and Shlyakhtenko pointed out a connection between these two frameworks, at the level of their operations of 1-dimensional free additive convolution. Motivated by that, we propose a construction which produces a multi-variate version of the Belinschi-Shlyakhtenko result, together with a result concerning free products of multi-variate noncommutative distributions. Our arguments are based on the combinatorics of the specific types of cumulants used in c-free and in infinitesimal free probability. They work in a rather general setting, where the initial data consists of a vector space $V$ given together with a linear map $\Delta : V \to V \otimes V$. In this setting, all the needed brands of cumulants live in the guise of families of multilinear functionals on $V$, and our main result concerns a certain transformation $\Delta^{*}$ on such families of multilinear functionals., Comment: Version 2: Minor revision, added references

In the paper ‘A probabilistic analysis of a discrete-time evolution in recombination’ [4] the evolution of the recombination transformation Ξ = ∑ δ ρ δ ⨂ J ∈ δ μ J was described by a Markov chain ( Y n ) on a set of partitions, which converges to the finest partition. Our main results were the description of the geometric decay rate to the limit and the quasi-stationary behavior of the Markov chain when conditioned on the event that the chain does not hit the limit. All these results continue to be true, but the Markov chain ( Y n ) that was claimed to satisfy Ξ n = E ( ⨂ J ∈ Y n μ J ) required to be modified. This is done in this Corrigendum.

Using the equivariant cohomology theory, Naruse obtained a hook length formula for the number of standard Young tableaux of skew shape λ / μ . Morales, Pak and Panova found two q-analogues of Naruse's formula respectively by counting semistandard Young tableaux of shape λ / μ and reverse plane partitions of shape λ / μ . When λ and μ are both staircase shape partitions, Morales, Pak and Panova conjectured that the generating function of reverse plane partitions of shape λ / μ can be expressed as a determinant whose entries are related to q-analogues of the Euler numbers. The objective of this paper is to settle this conjecture.

This paper presents a new approach to the recently established Orlicz Brunn-Minkowski inequality which is due to Xi, Jin and Leng and is stronger than the classical and L p Brunn-Minkowski inequalities. Our approach is based on the shadow system and does not rely on the Steiner symmetrization.

Simplicial arrangements are classical objects in discrete geometry. Their classification remains an open problem but there is a list conjectured to be complete at least for rank three. A further important class in the theory of hyperplane arrangements with particularly nice geometric, algebraic, topological, and combinatorial properties are the supersolvable arrangements. In this paper we give a complete classification of supersolvable simplicial arrangements (in all ranks). For each fixed rank, our classification already includes almost all known simplicial arrangements. Surprisingly, for irreducible simplicial arrangements of rank greater than three, our result shows that supersolvability imposes a strong integrality property; such an arrangement is called crystallographic. Furthermore we introduce Coxeter graphs for simplicial arrangements which serve as our main tool of investigation., 37 pages, 20 figures

Consequent to the introduction of the concept of Parikh matrix of a word which is based on the notion of subwords of a word, there has been an extensive research and study based on subwords. Parikh word representable graph is one such notion which has been introduced in the recent times. On the other hand connections of partitions of a number with counts of certain subword in a binary word are known. In this paper we introduce the notion of dual of a word and investigate its relationship with conjugate partition. As a result of this study, an expression for the number of nonisomorphic Parikh word representable graphs with a given number of edges, is obtained. Several other properties of Parikh word representable graphs are also derived.

The size ‖ σ ‖ of a split σ (a bipartition) of the leaf set of a tree is the cardinality of the smaller part. There exist many studies of properties of the phylogenetic tree shapes (trees with no vertex of degree 2) and general trees related to the split sizes. A general function Φ f ( T ) = ∑ σ ∈ Σ ⁎ ( T ) f ( ‖ σ ‖ ) was proposed for an (strictly) increasing function f, where the sum is over all non-trivial splits induced by internal edges. There are many interesting applications of this important concept. In particular, Φ f ( ⋅ ) was used to measure the balance of phylogenetic tree shapes. Extremal problems for Φ f ( ⋅ ) have been considered and partially answered for binary trees. In this paper, we study the extremal trees with a given degree sequence that maximize or minimize Φ f ( ⋅ ) . These results can also be directly applied to phylogenetic tree shapes and their degree sequences. When the number of leaves is fixed, we can also compare the extremal trees of different degree sequences. This comparison is conducted for degree sequences of phylogenetic tree shapes with given number of leaves. Immediate consequences are shown. Related problems are also proposed for potential future work.

For any sequences u = { u ( n ) } n ≥ 0 , v = { v ( n ) } n ≥ 0 , we define u v : = { u ( n ) v ( n ) } n ≥ 0 and u + v : = { u ( n ) + v ( n ) } n ≥ 0 . We say an integer sequence u = { u ( n ) } n ≥ 0 is a polynomial generated sequence if there exists an integer k ≥ 1 and polynomials f i ( x ) ( 0 ≤ i k ) whose coefficients are integer sequences such that { u ( k n + i ) } n ≥ 0 = f i ( u ) , ( 0 ≤ i k ) . In this paper, we study the polynomial generated sequences. Assume k ≥ 2 and f i ( x ) = a i x + b i ( 0 ≤ i k ) . If a i are k-automatic and b i are k-regular for 0 ≤ i k , then we prove that the corresponding polynomial generated sequences are k-regular. As a application, we prove that the Hankel determinant sequence { det ⁡ ( p ( i + j ) ) i , j = 0 n − 1 } n ≥ 0 is 2-regular, where { p ( n ) } n ≥ 0 = 0110100010000 ⋯ is the characteristic sequence of powers of 2. Moreover, we give a proof of a conjecture of Cigler about the Hankel determinants.

Let G be a connected graph on n vertices. For a vertex u∈G, the eccentricity of u is defined as ε(u)=max⁡{d(u,v)|v∈V(G)}, where d(u,v) denotes the distance between u and v. The eccentricity matrix E(G)=(ϵuv), where ϵuv:={d(u,v)if d(u,v)=min⁡{ε(u),ε(v)},0otherwise, has been firstly introduced in Chemical Graph Theory. In literature, it is also known as the DMAX-matrix. Graphs with the diameter equal to the radius are called self-centered graphs. Two non-isomorphic graphs are said to be M-cospectral with respect to a given matrix M if they have the same M-eigenvalues. In this paper, we show that, when n→∞, the fractions of non-isomorphic cospectral graphs with respect to the adjacency and the eccentricity matrix behave like those only concerning the self-centered graphs with diameter two. Secondly, we prove that a graph G has just two distinct E-eigenvalues if and only if G is an r-antipodal graph. Thirdly, we obtain many pairs of E-cospectral graphs by using strong and lexicographic products. Finally we formulate some problems waiting to be solved in order to build up a spectral theory based on the eccentricity matrix.

Define the $n$-th fibotomic polynomial to be the product of the monic irredicible factors of the $n$-th Fibonacci polynomial which are not factors of any Fibonacci polynomial of smaller degree. In this paper, we prove a number of properties of the fibotomic polynomials. This includes determining the discriminant of the fibotomic polynomials and the resultant of pairs of fibotomic polynomials. Furthermore, we completely determine the factorization form of the fibotomic polynomials in prime fields. Results are also generalized for the bivariate homogenous fibotomic polynomials.

In this paper, we study the Wilf-type equivalence relations among multiset permutations. We identify all multiset equivalences among pairs of patterns consisting of a pattern of length three and another pattern of length at most four. To establish our results, we make use of a variety of techniques, including Ferrers-equivalence arguments, sorting by minimal/maximal letters, analysis of active sites and direct bijections. In several cases, our arguments may be extended to prove multiset equivalences for infinite families of pattern pairs. Our results apply equally well to the Wilf-type classification of compositions, and as a consequence, we obtain a complete description of the Wilf-equivalence classes for pairs of patterns of type (3,3) and (3,4) on compositions, with the possible exception of two classes of type (3,4)., 26 pages

We study the interpolation Macdonald functions, remarkable inhomogeneous generalizations of Macdonald functions, and a sequence $A^1, A^2, \ldots$ of commuting operators that are diagonalized by them. Such a sequence of operators arises in the projective limit of finite families of commuting q-difference operators studied by Okounkov, Knop and Sahi. The main theorem is an explicit formula for the operators $A^k$. Our formula involves the family of Hall-Littlewood functions and a new family of inhomogeneous Hall-Littlewood functions, for which we give an explicit construction and identify as a degeneration of the interpolation Macdonald functions in the regime $q \rightarrow 0$. This article is inspired by the recent papers of Nazarov-Sklyanin on Macdonald and Sekiguchi-Debiard operators, and our main theorem is an extension of their results., 34 pages, 1 figure

In this paper, we study the generating function for the number of set partitions of [ n ] represented as bargraphs according to the number of interior vertices. In particular, we find an explicit formula for the total number of interior vertices over set partitions of [ n ] .

The paper provides a description of the large deviation behavior for the Euclidean norm of projections of l p n -balls to high-dimensional random subspaces. More precisely, for each integer n ≥ 1 , let k n ∈ { 1 , … , n − 1 } , E ( n ) be a uniform random k n -dimensional subspace of R n and X ( n ) be a random point that is uniformly distributed in the l p n -ball of R n for some p ∈ [ 1 , ∞ ] . Then the Euclidean norms ‖ P E ( n ) X ( n ) ‖ 2 of the orthogonal projections are shown to satisfy a large deviation principle as the space dimension n tends to infinity. Its speed and rate function are identified, making thereby visible how they depend on p and the growth of the sequence of subspace dimensions k n . As a key tool we prove a probabilistic representation of ‖ P E ( n ) X ( n ) ‖ 2 which allows us to separate the influence of the parameter p and the subspace dimension k n .

A natural question for matroids follows: Which matroids N guarantee that, when present as minors in a larger matroid M, their duals are present as minors? The main result of this paper focuses on this question with the additional constraints that N and M are 3-connected and graphic. We also solve the problem with other connectivity constraints.

In this paper, we discuss three different approaches to attack the problem of determining all rational solutions for a first-order algebraic ordinary differential equation (AODE). We first give a sufficient condition for first-order AODEs to have the property that poles of rational solutions can only occur at the zeros of the leading coefficient. A combinatorial approach is presented to determine all rational solutions, if there are any, of the family of first-order AODEs satisfying this condition. Algebraic considerations based on algebraic function theory yield another algorithm for quasi-linear first-order AODEs. And finally ideas from algebraic geometry combine these results to an algorithm for finding all rational solutions of parametrizable first-order AODEs.

In this paper we set up a bivariate representation of partial theta functions which not only unifies some famous identities for partial theta functions due to Andrews and Warnaar but also unveils a new characteristic of such identities. As further applications, we establish a general form of Warnaar's identity and a general q-series transformation associated with Bailey pairs via the use of the power series expansion of partial theta functions.

A numerical monoid is an additive submonoid of the non-negative integers. Given a numerical monoid S, consider the family of “shifted” monoids M n obtained by adding n to each generator of S. In this paper, we characterize the Apery set of M n in terms of the Apery set of the base monoid S when n is sufficiently large. We give a highly efficient algorithm for computing the Apery set of M n in this case, and prove that several numerical monoid invariants, such as the genus and Frobenius number, are eventually quasipolynomial as a function of n.

本文研究分数阶Bagley-Torvik方程不确定边值条件下的解。基于Caputo分数阶导数定义和广义的Hukuhara可微性，引进模糊Laplace变换，不确定边界条件为模糊数，给出了问题的级数解。数值结果分析了解的性态。 This paper investigates the problem of the fractional Bagley-Torvik equation with uncertainty boundary-value conditions. Under the Caputo’s H-differentiability, the fuzzy Laplace transform is introduced. The uncertainty boundary-value conditions are assumed to be fuzzy numbers. The se-ries solution of fractional Bagley-Torvik equation is given. Numerical results are shown to illustrate the obtained solution.

本文主要研究几何稳定过程的性质。首先，我们得到了由𝛼-𝑠𝑡𝑎𝑏𝑙𝑒过程驱动的几何稳定过程并给出几何稳定过程的解。其次，证明了在随机噪声较大时，几何稳定过程的解几乎处处以指数速率趋近于零。 The main purpose of this paper is to investigate the properties of geometric stable process. First, a model driven by 𝛼-𝑠𝑡𝑎𝑏𝑙𝑒 process is present. We obtain the solution of such model. Then, we prove that if the noise is sufficiently large, the solution of the geometric stable process will tend to zero at an exponential rate with probability one.

本文研究了稳定过程的平方过程的性质。基于标准对称稳定过程的性质，我们证明了稳定过程的平方过程也是一个稳定过程，并计算了稳定过程平方的尾分布；其次，我们证明了稳定过程平方的二次变差是一个非负的稳定过程；最后，我们简单介绍了对参数α进行估计的方法。 This paper studies on the properties of square processes of stable processes. Based on the proper-ties of the classical symmetric stable processes, we prove that the square process of stable process is also a stable process and we calculate the tail distribution of it; furthermore, we prove that the quadratic variation of it tends to be a nonnegative stable process. Finally, we briefly introduce the methods to estimate the parameters in the square processes of stable processes.

Grinberg defined Nyldon words as those words which cannot be factorized into a sequence of lexicographically nondecreasing smaller Nyldon words. He was inspired by Lyndon words, defined the same way except with "nondecreasing" replaced by "nonincreasing." Charlier, Philibert, and Stipulanti proved that, like Lyndon words, any word has a unique nondecreasing factorization into Nyldon words. They also show that the Nyldon words form a right Lazard set, and equivalently, a right Hall set. In this paper, we provide a new proof of unique factorization into Nyldon words related to Hall set theory and resolve several questions of Charlier et al. In particular, we prove that Nyldon words of a fixed length form a circular code, we prove a result on factorizing powers of words into Nyldon words, and we investigate the Lazard procedure for generating Nyldon words. We show that these results generalize to a new class of Hall sets, of which Nyldon words are an example, that we name "Nyldon-like sets.", Comment: 27 pages; incorporated reviewer comments