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Recently, Fici, Restivo, Silva, and Zamboni introduced the notion of a$k$ -anti-power, which is defined as a word of the form$w^{(1)} w^{(2)} \\cdots w^{(k)}$ , where$w^{(1)}, w^{(2)}, \\ldots, w^{(k)}$are distinct words of the same length. For an infinite word$w$and a positive integer$k$ , define$AP_j(w,k)$to be the set of all integers$m$such that$w_{j+1} w_{j+2} \\cdots w_{j+km}$is a$k$ -anti-power, where$w_i$denotes the$i$ -th letter of$w$ . Define also$\\mathcal{F}_j(k) = (2 \\mathbb{Z}^+ - 1) \\cap AP_j(\\mathbf{t},k)$ , where$\\mathbf{t}$denotes the Thue-Morse word. For all$k \\in \\mathbb{Z}^+$ ,$\\gamma_j(k) = \\min (AP_j(\\mathbf{t},k))$is a well-defined positive integer, and for$k \\in \\mathbb{Z}^+$sufficiently large,$\\Gamma_j(k) = \\sup ((2 \\mathbb{Z}^+ -1) \\setminus \\mathcal{F}_j(k))$is a well-defined odd positive integer. In his 2018 paper, Defant shows that$\\gamma_0(k)$and$\\Gamma_0(k)$grow linearly in$k$ . We generalize Defant's methods to prove that$\\gamma_j(k)$and$\\Gamma_j(k)$grow linearly in$k$for any nonnegative integer$j$ . In particular, we show that$\\displaystyle 1/10 \\leq \\liminf_{k \\rightarrow \\infty} (\\gamma_j(k)/k) \\leq 9/10$and$\\displaystyle 1/5 \\leq \\limsup_{k \\rightarrow \\infty} (\\gamma_j(k)/k) \\leq 3/2$ . Additionally, we show that$\\displaystyle \\liminf_{k \\rightarrow \\infty} (\\Gamma_j(k)/k) = 3/2$and$\\displaystyle \\limsup_{k \\rightarrow \\infty} (\\Gamma_j(k)/k) = 3$ .

Interval parking functions are a generalization of parking functions in which cars have an interval preference for their parking. We generalize this definition to parking functions with$n$cars and$m\\geq n$parking spots, which we call interval rational parking functions and provide a formula for their enumeration. By specifying an integer parameter$\\ell\\geq 0$ , we then consider the subset of interval rational parking functions in which each car parks at most$\\ell$spots away from their initial preference. We call these$\\ell$ -interval rational parking functions and provide recursive formulas to enumerate this set for all positive integers$m\\geq n$and$\\ell$ . We also establish formulas for the number of nondecreasing$\\ell$ -interval rational parking functions via the outcome map on rational parking functions. We also consider the intersection between$\\ell$ -interval parking functions and Fubini rankings and show the enumeration of these sets is given by generalized Fibonacci numbers. We conclude by specializing$\\ell=1$ , and establish that the set of$1$ -interval rational parking functions with$n$cars and$m$spots are in bijection with the set of barred preferential arrangements of$[n]$with$m-n$bars. This readily implies enumerative formulas. Further, in the case where$\\ell=1$ , we recover the results of Hadaway and Harris that unit interval parking functions are in bijection with the set of Fubini rankings, which are enumerated by the Fubini numbers.

This paper studies the relationship between the modified Foata $\\unicode{x2013}$ Strehl action (a.k.a. valley-hopping) $\\unicode{x2014}$ a group action on permutations used to demonstrate the$\\gamma$ -positivity of the Eulerian polynomials $\\unicode{x2014}$ and the number of rixed points$\\operatorname{rix}$ $\\unicode{x2014}$ a recursively-defined permutation statistic introduced by Lin in the context of an equidistribution problem. We give a linear-time iterative algorithm for computing the set of rixed points, and prove that the$\\operatorname{rix}$statistic is homomesic under valley-hopping. We also demonstrate that a bijection$\\Phi$introduced by Lin and Zeng in the study of the$\\operatorname{rix}$statistic sends orbits of the valley-hopping action to orbits of a cyclic version of valley-hopping, which implies that the number of fixed points$\\operatorname{fix}$is homomesic under cyclic valley-hopping.

This article explores the relationship between Hessenberg varieties associated with semisimple operators with two eigenvalues and orbit closures of a spherical subgroup of the general linear group. We establish the specific conditions under which these semisimple Hessenberg varieties are irreducible. We determine the dimension of each irreducible Hessenberg variety under consideration and show that the number of such varieties is a Catalan number. We then apply a theorem of Brion to compute a polynomial representative for the cohomology class of each such variety. Additionally, we calculate the intersections of a standard (Schubert) hyperplane section of the flag variety with each of our Hessenberg varieties and prove that this intersection possesses a cohomological multiplicity-free property.

In this paper, we consider pattern avoidance in a subset of words on $\\{1,1,2,2,\\dots,n,n\\}$ called reverse double lists. In particular a reverse double list is a word formed by concatenating a permutation with its reversal. We enumerate reverse double lists avoiding any permutation pattern of length at most 4 and completely determine the corresponding Wilf classes. For permutation patterns $\\rho$ of length 5 or more, we characterize when the number of $\\rho$-avoiding reverse double lists on $n$ letters has polynomial growth. We also determine the number of $1\\cdots k$-avoiders of maximum length for any positive integer $k$. Comment: 24 pages, 5 figures, 4 tables

Two mesh patterns are coincident if they are avoided by the same set of permutations, and are Wilf-equivalent if they have the same number of avoiders of each length. We provide sufficient conditions for coincidence of mesh patterns, when only permutations also avoiding a longer classical pattern are considered. Using these conditions we completely classify coincidences between families containing a mesh pattern of length 2 and a classical pattern of length 3. Furthermore, we completely Wilf-classify mesh patterns of length 2 inside the class of 231-avoiding permutations.

A permutation class $C$ is splittable if it is contained in a merge of two of its proper subclasses, and it is 1-amalgamable if given two permutations $\\sigma$ and $\\tau$ in $C$, each with a marked element, we can find a permutation $\\pi$ in $C$ containing both $\\sigma$ and $\\tau$ such that the two marked elements coincide. It was previously shown that unsplittability implies 1-amalgamability. We prove that unsplittability and 1-amalgamability are not equivalent properties of permutation classes by showing that the class $Av(1423, 1342)$ is both splittable and 1-amalgamable. Our construction is based on the concept of LR-inflations, which we introduce here and which may be of independent interest. Comment: 17 pages, 7 figures

The integer sequence = ‧‧‧ is said to be an if 0 ≤ ≤ – 1 for all . Let denote the set of inversion sequences of length , represented using positive instead of non-negative integers. We consider here two new statistics defined on the bargraph representation ) of an inversion sequence which record the number of unit squares touching the boundary of ) and that are either exterior or interior to ). We denote these statistics on recording the number of outer and inner perimeter squares respectively by oper and iper. In this paper, we study the distribution of oper and iper on and also on members of that end in a particular letter. We find explicit formulas for the maximum and minimum values of oper and iper achieved by a member of as well as for the average value of these parameters. We make use of both algebraic and combinatorial arguments in establishing our results.

In this paper, we focus on shape partitions. We show that for any fixed , one can symbolically characterize the shape partition on a × rectangular grid by a context-free grammar. We explicitly give this grammar for = 2 and = 3 (for = 1, this corresponds to compositions of integers). From these grammars, we deduce the number of shape partitions for the × rectangular grids for ∈ {1, 2, 3} and every , as well as the limiting Gaussian distribution of the number of connected components. This also enables us to randomly and uniformly generate shape partitions of large size.