Search

Symmetry is often regarded as an integral aspect about aesthetics. This motivates the pursuit of interdisciplinary studies based on the use of subjects in mathematics concerned with symmetry in conjunction with aesthetics. What is referred to as a symmetric function in the field of algebraic combinatorics is an abstraction based on polynomials that exhibit a symmetric property, and this leads us to pursue an algebraic combinatorics-inspired exploration based on aesthetics. In particular, we use different bases and transitions between them to create aesthetically pleasing visualizations of symmetric functions. We see that these visualizations in turn raise new and interesting questions. [ABSTRACT FROM AUTHOR]

We prove that there are no 6-cycles of the form xn+3 = xif(xj, xk), with i, j, k ∈ {n, n+1, n+2} pairwise distinct, whenever f : (0,∞) x (0,∞) → (0,∞) is a continuous symmetric function, that is, f(x, y) = f(y, x), for all x, y > 0. Moreover, we obtain all the 6-cycles of potential form and present some open questions relative to the search of p-cycles whenever symmetry does not hold. [ABSTRACT FROM AUTHOR]

We describe an algebraic technique for operating with power series whose coefficients are represented by integrals of symmetric functions fn defined on the Cartesian powers Ωn of a set Ω with a measure μ. Moreover, each of the coefficient functions fn is obtained by means of a special mapping from graphs with n labeled vertices belonging to a fixed class. This technique has application to equilibrium statistical mechanics and to problems of enumeration of graphs. [ABSTRACT FROM AUTHOR]

In the plane of a triangle ABC, let L 1 as L 2 be a pair of nonparallel lines. A coordinate system having L 1 as x-axis and L 2 as y-axis is defined, and transformation formulas for representations of points and lines in (x, y) coordinates and barycentric coordinates are derived. The set of central lines consists of two types: even and odd, with consequences for the cases that L 1 and L 2 are both even, or both odd, or one even and the other odd. A mapping T(P, U) is defined for pairs (P, U) of points. This mapping maps lines to lines. A distinctive example of a bivariate coordinate system is introduced as the Steiner system, in which the axes are those of the Steiner circumellipse. In this system, the line y = x is the Euler line of ABC. [ABSTRACT FROM AUTHOR]

Two classes of determinants with their matrix entries being polynomial differences are evaluated in closed product expressions. They extend with many free parameters the Vandermonde-like determinant identities associated with the classical root systems. [ABSTRACT FROM AUTHOR]

We study functions of the roots of an integer polynomial p = p (x) with m ≥ 2 distinct roots ρ = (ρ 1 , ... , ρ m) of multiplicity μ = (μ 1 , ... , μ m) , μ 1 ≥ ⋯ ≥ μ m ≥ 1. Traditionally, root functions are studied via the theory of symmetric polynomials; we generalize this theory to μ -symmetric polynomials. We initiate the study of the vector space of μ -symmetric polynomials of a given degree δ via the concepts of μ -gist and μ -ideal. In particular, we are interested in the root D μ + (r 1 , ... , r m) := ∏ 1 ≤ i < j ≤ m (r i − r j) μ i + μ j . The D-plus discriminant of p is D + (p) := D μ + (ρ 1 , ... , ρ m). This quantity appears in the complexity analysis of the root clustering algorithm of Becker et al. (ISSAC 2016). We conjecture that D μ + (r 1 , ... , r m) is μ -symmetric, which implies D + (p) is rational. To explore this conjecture experimentally, we introduce algorithms for checking if a given root function is μ -symmetric. We design three such algorithms: one based on Gröbner bases, another based on canonical bases and reduction, and the third based on solving linear equations. Each of these algorithms has variants that depend on the choice of a basis for the μ -symmetric functions. We implement these algorithms (and their variants) in Maple and experiments show that the latter two algorithms are significantly faster than the first. [ABSTRACT FROM AUTHOR]

The enumeration of lattice paths is an important counting model in enumerative combinatorics. Because it can provide powerful methods and technical support in the study of discrete structural objects in different disciplines, it has attracted much attention and is a hot research field. In this paper, we summarize two kinds of the lattice path counting models that are single lattice paths and family of nonintersecting lattice paths and their applications in terms of the change of dimensions, steps, constrained conditions, the positions of starting and end points, and so on. (1) The progress of classical lattice path such as Dyck lattice is introduced. (2) A method to study the enumeration of lattice paths problem by generating function is introduced. (3) Some methods of studying the enumeration of lattice paths problem by matrix are introduced. (4) The family of lattice paths problem and some counting methods are introduced. (5) Some applications of family of lattice paths in symmetric function theory are introduced, and a related open problem is proposed. [ABSTRACT FROM AUTHOR]

In this paper, we discuss and introduce a new study on the connection between geometric function theory, especially sandwich theorems, and Viete's theorem in elementary algebra. We obtain some conclusions for differential subordination and superordination for a new formula of complete homogeneous symmetric functions class involving an ordered cyclic operator. In addition, certain sandwich theorems are found. [ABSTRACT FROM AUTHOR]

Symmetric functions play an important role in several subjects of mathematics, such as algebraic combinatorics, representation theory of finite groups and algebraic geometry. In this paper, we study the solutions of the following system of equations related to bases of symmetric functions: where , are natural numbers, and . Our main theorems generalize several known results. [ABSTRACT FROM AUTHOR]

Product fault diagnosis has always been the focus of quality and reliability research. However, a failure–rate curve of some products is a symmetrical function, the fault analysis result is not true because the failure period of the products cannot be judged accurately. In order to solve the problem of fault diagnosis, this paper proposes a new Takagi-Sugeno (T-S) dynamic fault tree analysis method based on a Bayesian network accompanying the Wiener process. Firstly, the top event, middle event, and bottom event of the product failure mode are determined, and the T-S dynamic fault tree is constructed. Secondly, in order to form the Bayesian network diagram of the T-S dynamic fault tree, the events in the fault tree are transformed into nodes, and the T-S dynamic gate is also transformed into directed edges. Then, the Wiener process is used to model the performance degradation process of the stationary independent increment of the symmetric function distribution, and the maximum likelihood estimation method is applied to estimate the unknown parameters of the degradation model. Next, the product residual life prediction model is established based on the concept of first arrival time, and a symmetric function of failure–rate curve is obtained by using the product failure probability density function. According to the fault density function derived from the Wiener process, the reverse reasoning algorithm of the Bayesian network is established. Combined with the prior probability of the bottom event, the posterior probability of the root node is calculated and sorted as well. Finally, taking the insufficient braking force of electromagnetic brakes as an example, the practicability and objectivity of the new method are proved. [ABSTRACT FROM AUTHOR]

We examine the coefficients in front of Chern numbers for complex genera, and pay special attention to the Td 1 2 -genus, the Γ-genus as well as the Todd genus. Some related geometric applications to hyper-Kähler and Calabi-Yau manifolds are discussed. Along this line and building on the work of Doubilet in 1970s, various Hoffman-type formulas for multiple-(star) zeta values and transition matrices among canonical bases of the ring of symmetric functions can be uniformly treated in a more general framework. [ABSTRACT FROM AUTHOR]

In this paper, we extend the chromatic symmetric function X to a chromatic k-multisymmetric function X k , defined for graphs equipped with a partition of their vertex set into k parts. We demonstrate that this new function retains the basic properties and basis expansions of X , and we give a method for systematically deriving new linear relationships for X from previous ones by passing them through X k. In particular, we show how to take advantage of homogeneous sets of G (those S ⊆ V (G) such that each vertex of V (G) ﹨ S is either adjacent to all of S or is nonadjacent to all of S) to relate the chromatic symmetric function of G to those of simpler graphs. Furthermore, we show how extending this idea to homogeneous pairs S 1 ⊔ S 2 ⊆ V (G) generalizes the process used by Guay-Paquet to reduce the Stanley-Stembridge conjecture to unit interval graphs. [ABSTRACT FROM AUTHOR]

We introduce a family of varieties Y n , λ , s , which we call the Δ -Springer varieties , that generalize the type A Springer fibers. We give an explicit presentation of the cohomology ring H ⁎ (Y n , λ , s) and show that there is a symmetric group action on this ring generalizing the Springer action on the cohomology of a Springer fiber. In particular, the top cohomology groups are induction products of Specht modules with trivial modules. The λ = (1 k) case of this construction gives a compact geometric realization for the expression in the Delta Conjecture at t = 0. Finally, we generalize results of De Concini and Procesi on the scheme of diagonal nilpotent matrices by constructing an ind-variety Y n , λ whose cohomology ring is isomorphic to the coordinate ring of the scheme-theoretic intersection of an Eisenbud–Saltman rank variety and diagonal matrices. [ABSTRACT FROM AUTHOR]

Research in the area of secure multi-party computation using a deck of playing cards, often called card-based cryptography, started from the introduction of the five-card trick protocol to compute the logical AND function by den Boer in 1989. Since then, many card-based protocols to compute various functions have been developed. In this paper, we propose two new protocols that securely compute the n -variable equality function (determining whether all inputs are equal) E : { 0 , 1 } n → { 0 , 1 } using 2 n cards. The first protocol can be generalized to compute any doubly symmetric function f : { 0 , 1 } n → Z using 2 n cards, and any symmetric function f : { 0 , 1 } n → Z using 2 n + 2 cards. The second protocol can be generalized to compute the k -candidate n -variable equality function E : (Z / k Z) n → { 0 , 1 } using 2 ⌈ lg ⁡ k ⌉ n cards. [ABSTRACT FROM AUTHOR]

It is well known that Heron's equality provides an explicit formula for the area of a triangle, as a symmetric function of the lengths of its edges. It has been extended by Brahmagupta to quadrilaterals inscribed in a circle (cyclic quadrilaterals). A natural problem is trying to further generalize the result to cyclic polygons with a larger number of edges. Surprisingly, this has proved to be far from simple, and no explicit solutions exist for cyclic polygons having n > 4 edges. In this paper we investigate such a problem by following a new and elementary approach, based on the idea that the simple geometry underlying Heron's and Brahmagupta's equalities hides the real players of the game. In details, we propose to focus on the dissection of the edges determined by the incircles of a suitable triangulation of the cyclic polygon, showing that this approach leads to an explicit formula for the area as a symmetric function of the lengths of these segments. We also show that such a symmetry can be rediscovered in Heron's and Brahmagupta's results, which consequently represent special cases of the provided general equality. [ABSTRACT FROM AUTHOR]

Motivated by a question in Schubert calculus, we study the interplay of quasisymmetric polynomials with the divided symmetrization operator, which was introduced by Postnikov in the context of volume polynomials of permutahedra. Divided symmetrization is a linear form which acts on the space of polynomials in n indeterminates of degree n - 1 . We first show that divided symmetrization applied to a quasisymmetric polynomial in m indeterminates can be easily determined. Several examples with a strong combinatorial flavor are given. Then, we prove that the divided symmetrization of any polynomial can be naturally computed with respect to a direct sum decomposition due to Aval–Bergeron–Bergeron, involving the ideal generated by positive degree quasisymmetric polynomials in n indeterminates. [ABSTRACT FROM AUTHOR]

VLSI technology makes the integration of multiple electronic components into a single electronic chip possible but Moore has already predicted that further miniaturization of VLSI chips will cause excess heat dissipation. Because of this impediment faced by this technology, the scientists have turned their focus to nano-technological alternatives to keep up the continuous device improvement and continue with the miniaturization procedure without facing any hurdle from the aspect of heat dissipation and space and time complexity. Memristor can be such a nano-alternative, which has the potential of very rapid execution, miniaturization and very low power consumption compared to transistor-based technology. The main objective of this paper is to show an application of symmetric function in cryptography. In order to do so, first, three-input and five-input symmetric functions are implemented in memristor-based nanocrossbar circuits. Then a dynamic circuit of symmetric functions is implemented where the input voltages convert the circuit to three-input or five-input symmetric function circuit as and when required. The simulation results show the satisfactory behavior of our proposed design. [ABSTRACT FROM AUTHOR]

In 2000, Kadell gave an orthogonality conjecture for a symmetric function generalization of the Zeilberger-Bressoud q-Dyson theorem or the q-Dyson constant term identity. This conjecture was proved by Károlyi, Lascoux and Warnaar in 2015. In this paper, by slightly changing the variables of Kadell's conjecture, we obtain another symmetric function generalization of the q-Dyson constant term identity. This new generalized constant term admits a simple product-form expression. [ABSTRACT FROM AUTHOR]

Two open problems recently proposed by Xi and Luo (Adv. Differ. Equ. 2021:38, 2021) are resolved by evaluating explicitly three binomial sums involving harmonic numbers, that are realized mainly by utilizing the generating function method and symmetric functions. [ABSTRACT FROM AUTHOR]

We introduce a family of ideals In,λ,s in Q[x1, . . . , xn] for λ a partition of k ≤ n and an integer s ≥ l (λ). This family contains both the Tanisaki ideals Iλ and the ideals In,k of Haglund-Rhoades-Shimozono as special cases. We study the corresponding quotient rings Rn,λ,s as symmetric group modules. When n = k and s is arbitrary, we recover the Garsia-Procesi modules, and when λ = (1k) and s = k, we recover the generalized coinvariant algebras of Haglund-Rhoades-Shimozono. We give a monomial basis for Rn,λ, s in terms of (n,λ,s)-staircases, unifying the monomial bases studied by Garsia-Procesi and Haglund-Rhoades-Shimozono. We realize the Sn-module structure of Rn,λ,s in terms of an action on (n, λ, s)-ordered set partitions. We find a formula for the Hilbert series of Rn,λ,s in terms of inversion and diagonal inversion statistics on a set of fillings in bijection with (n, λ, s)-ordered set partitions. Furthermore, we prove an expansion of the graded Frobenius characteristic of our rings into Gessel's fundamental quasisymmetric basis. We connect our work with Eisenbud-Saltman rank varieties using results of Weyman. As an application of our results on Rn,λ,s, we give a monomial basis, Hilbert series formula, and graded Frobenius characteristic formula for the coordinate ring of the scheme-theoretic intersection of a rank variety with diagonal matrices. [ABSTRACT FROM AUTHOR]

In the paper, the authors obtain some properties, including an explicit formula in symmetric function form, a Hankel determinant, the orthogonality and inverse relations, and a Qi type formula of (p, q)-analogues of the r-Whitney numbers. [ABSTRACT FROM AUTHOR]

Sum of Power had gathered interest of many classical mathematicians for more than two thousand years ago. The quests of finding sum of power or discrete sum of numerical power can be traced back from the time of Archimedes in third BC then to Faulhaber in the sixteen centuries. Until today there is no closed form sums of power formulation for an arithmetic progression has been found. Many mathematicians were involved in this research and many approaches have been introduced but none is found to be conclusive. The generalized equation for sums of power discovered in this research has been compared to Faulhaber's sums of power for integers and it is found that this new generalized equation can be used for both integers and arithmetic progression, thus offering a new frontier in studying symmetric function, Fermat's last theorem, Riemman's Zeta function etc. [ABSTRACT FROM AUTHOR]

Using the description of multiline queues as functions on words, we introduce the notion of a spectral weight of a word by defining a new weighting on multiline queues. We show that the spectral weight of a word is invariant under a natural action of the symmetric group, giving a proof of the commutativity conjecture of Arita, Ayyer, Mallick, and Prolhac. We give a determinant formula for the spectral weight of a word, which gives a proof of a conjecture of the first author and Linusson. [ABSTRACT FROM AUTHOR]

A Boolean function f : { 0 , 1 } n → { 0 , 1 } is weighted symmetric if there exist a function g : Z → { 0 , 1 } and integers w 0 , w 1 , ... , w n such that f (x 1 , ... , x n) = g (w 0 + ∑ i = 1 n w i x i) holds. In this paper, we present algorithms for the circuit satisfiability problem of bounded depth circuits with AND, OR, NOT gates and a limited number of weighted symmetric gates. Our algorithms run in time super-polynomially faster than 2 n even when the number of gates is super-polynomial and the maximum weight of symmetric gates is nearly exponential. As a special case, we obtain an algorithm for the maximum satisfiability problem that runs in time poly (n t) ⋅ 2 n − n 1 / O (t) for instances with n variables and O (n t) clauses. Through the analysis of our algorithms, we show average-case lower bounds and compression algorithms for such circuits and worst-case lower bounds for majority votes of such circuits. [ABSTRACT FROM AUTHOR]

According to [5] , the arithmetic mean is a function characterized by four features: it is non-decreasing, idempotent, symmetric and additive. The first three of them can be naturally converted to the theory of posets but the last one generally can not. Due to this problem, we will replace it with another suitable property, which is called transfer-stability. However, we do not get the exact arithmetic mean but some approximation. These functions will be called transfer-stable means. The first aim of the paper is to show that transfer-stable means on a finite chain form a lattice which is isomorphic to the direct power of a finite chain. The second goal is to create a generating set of transfer-stable means, i.e., means that can generate all other transfer-stable means of the same arity by classical composition of functions. The last goal deals with question of how to generate all transfer-stable means of any arity by binary transfer-stable means only. For this problem we define special transfer-stable means composition. [ABSTRACT FROM AUTHOR]

Let (x 1 , ... , x n , y 1 , ... , y n) be a list of 2 n commuting variables, (θ 1 , ... , θ n , ξ 1 , ... , ξ n) be a list of 2 n anticommuting variables, and C [ x n , y n ] ⊗ ∧ { θ n , ξ n } be the algebra generated by these variables. D'Adderio, Iraci, and Vanden Wyngaerd introduced the Theta operators on the ring of symmetric functions and used them to conjecture a formula for the quadruply-graded S n -isomorphism type of C [ x n , y n ] ⊗ ∧ { θ n , ξ n } / I where I is the ideal generated by S n -invariants with vanishing constant term. We prove their conjecture in the 'purely fermionic setting' obtained by setting the commuting variables x i , y i equal to zero. [ABSTRACT FROM AUTHOR]

We construct an action of the Hecke algebra Hn(q) on a quotient of the polynomial ring F[x1, . . . , xn], where F = Q(q). The dimension of our quotient ring is the number of k-block ordered set partitions of {1, 2, . . . , n}. This gives a quantum analog of a construction of Haglund-Rhoades-Shimozono and interpolates between their result at q = 1 and work of Huang-Rhoades at q = 0. [ABSTRACT FROM AUTHOR]

Abstract We consider a certain quotient of a polynomial ring categorified by both the isomorphic Green rings of the symmetric groups and Schur algebras generated by the signed Young permutation modules and mixed powers respectively. They have bases parametrised by pairs of partitions whose second partitions are multiples of the odd prime p the characteristic of the underlying field. We provide an explicit formula rewriting a signed Young permutation module (respectively, mixed power) in terms of signed Young permutation modules (respectively, mixed powers) labelled by those pairs of partitions. As a result, for each partition λ , we discovered the number of compositions δ such that δ can be rearranged to λ and whose partial sums of δ are not divisible by p. [ABSTRACT FROM AUTHOR]

The focus of this work is a family of maps from the space of symmetric matrices, , into the space for any , invariant under the conjugate action of the orthogonal group . This family, called the generated k-isotropic functions, generalizes known types of maps with a similar invariance property, such as the spectral, isotropic, primary matrix functions, multiplicative compound, and additive compound matrices on . We give necessary and sufficient conditions for the analyticity of these maps. [ABSTRACT FROM AUTHOR]

The localized equivariant homology of the quiver variety of type A(1) N-1 can be identified with the level one Fock space by assigning a normalized torus fixed point basis to certain symmetric functions: Jack(glN) symmetric functions introduced by Uglov. We show that this correspondence is compatible with actions of two algebras, the Yangian for slN and the affine Lie algebra s^lN, on both sides. Consequently we obtain the affine Yangian action on the Fock space. [ABSTRACT FROM AUTHOR]

We review recent results obtained at the intersection of the theory of quantum deformed Calogero-Moser-Sutherland systems and the theory of Lie superalgebras. We begin with a definition of admissible deformations of root systems of basic classical Lie superalgebras. For classical series, we prove the existence of Lax pairs. Connections between infinite-dimensional Calogero-Moser-Sutherland systems, deformed quantum CMS systems, and representation theory of Lie superalgebras are discussed. [ABSTRACT FROM AUTHOR]

An ( r , k ) -parking function of length n may be defined as a sequence ( a 1 , … , a n ) of positive integers whose increasing rearrangement b 1 ≤ ⋯ ≤ b n satisfies b i ≤ k + ( i − 1 ) r . The case r = k = 1 corresponds to ordinary parking functions. We develop numerous properties of ( r , k ) -parking functions. In particular, if F n ( r , k ) denotes the Frobenius characteristic of the action of the symmetric group S n on the set of all ( r , k ) -parking functions of length n , then we find a combinatorial interpretation of the coefficients of the power series ( ∑ n ≥ 0 F n ( r , 1 ) t n ) k for any k ∈ Z . When k > 0 , this power series is just ∑ n ≥ 0 F n ( r , k ) t n ; when k < 0 , we obtain a dual to ( r , k ) -parking functions. We also give a q -analogue of this result. For fixed r , we can use the symmetric functions F n ( r , 1 ) to define a multiplicative basis for the ring Λ of symmetric functions. We investigate some of the properties of this basis. [ABSTRACT FROM AUTHOR]

In this paper we propose the mathematical model for a vane with stochastic rotation. By backward stochastic differential equation and the Feynman–Kac formula, we obtain the representation of solution to the proposed convection–diffusion equation with symmetric initial condition through the transition probability density of stochastic differential equation. Moreover the explicit probability density function of solution with t = 1 and x = 0 is obtained, which shows that the distribution of k < 0 is bimodal distribution with double maximum. By comparing to the experiment, it is found that the angular velocity with stochastic bidirectional is the binormal distribution with k < 0 , thus our model can describe the rotation of a vane with stochastic rotation very well. • We propose the mathematical model for a vane with stochastic rotation. • The solution to the proposed equation with symmetric initial condition is obtained. • The model can describe the rotation of a vane with stochastic rotation. [ABSTRACT FROM AUTHOR]

We propose and investigate a family of maps from the space of n × n symmetric matrices, Sn, into the space Snk for any k = 1, . . . ,n which are invariant under the conjugate action of the orthogonal group On. This family, called k-isotropic functions, not only generalizes all known types of maps with similar invariance property, such as the spectral, isotropic, primary matrix functions, multiplicative compound, and additive compound matrices on Sn, but also contains many previously unknown and potentially interesting subclasses of functions. We give necessary and sufficient conditions for these maps to be r-times differentiable and exhibit a recursive formula for the rth derivative. Moreover, we provide a full description of the linear k-isotropic functions which provides a generalization to constitutive equation in classical linear elasticity also know as Hooke law. Finally, we exhibit a duality motivated by the Hodge star operator between different classes of k-isotropic functions. The study of isotropic maps is motivated by constitutive relations in continuum mechanics, where the properties of materials and their symmetry groups are related. [ABSTRACT FROM AUTHOR]

The symmetric group S n acts on the polynomial ring Q [ x n ] = Q [ x 1 , … , x n ] by variable permutation. The invariant ideal I n is the ideal generated by all S n -invariant polynomials with vanishing constant term. The quotient R n = Q [ x n ] I n is called the coinvariant algebra . The coinvariant algebra R n has received a great deal of study in algebraic and geometric combinatorics. We introduce a generalization I n , k ⊆ Q [ x n ] of the ideal I n indexed by two positive integers k ≤ n . The corresponding quotient R n , k : = Q [ x n ] I n , k carries a graded action of S n and specializes to R n when k = n . We generalize many of the nice properties of R n to R n , k . In particular, we describe the Hilbert series of R n , k , give extensions of the Artin and Garsia–Stanton monomial bases of R n to R n , k , determine the reduced Gröbner basis for I n , k with respect to the lexicographic monomial order, and describe the graded Frobenius series of R n , k . Just as the combinatorics of R n are controlled by permutations in S n , we will show that the combinatorics of R n , k are controlled by ordered set partitions of { 1 , 2 , … , n } with k blocks. The Delta Conjecture of Haglund, Remmel, and Wilson is a generalization of the Shuffle Conjecture in the theory of diagonal coinvariants. We will show that the graded Frobenius series of R n , k is (up to a minor twist) the t = 0 specialization of the combinatorial side of the Delta Conjecture. It remains an open problem to give a bigraded S n -module V n , k whose Frobenius image is even conjecturally equal to any of the expressions in the Delta Conjecture; our module R n , k solves this problem in the specialization t = 0 . [ABSTRACT FROM AUTHOR]

The notion of operator monotonicity dates back to a work by Löwner in 1934. A map F : S n → S m is called operator monotone , if A ⪰ B implies F ( A ) ⪰ F ( B ) . (Here, S n is the space of symmetric matrices with the semidefinite partial order ⪰.) Often, the function F is defined in terms of an underlying simpler function f . Of main interest is to find the properties of f that characterize operator monotonicity of F . In that case, it is said that f is also operator monotone. Classical examples are the Löwner operators and the spectral (scalar-valued isotropic) functions. Operator monotonicity for these two classes of functions is characterized in seemingly very different ways. This work extends the notion of operator monotonicity to symmetric functions f on k arguments. The latter is used to define (generated) k-isotropic maps F : S n → S ( n k ) for any n ≥ k . Necessary and sufficient conditions are given for f to generate an operator monotone k -isotropic map F . When k = 1 , the k -isotropic map becomes a Löwner operator and when k = n it becomes a spectral function. This allows us to reconcile and explain the differences between the conditions for monotonicity for the Löwner operators and the spectral functions. [ABSTRACT FROM AUTHOR]

The Delta Conjecture of Haglund, Remmel, and Wilson is a recent generalization of the Shuffle Conjecture in the field of diagonal harmonics. In this paper we give evidence for the Delta Conjecture by proving a pair of conjectures of Wilson and Haglund–Remmel–Wilson which give equidistribution results for statistics related to inversion count and major index on objects related to ordered set partitions. Our results generalize the famous result of MacMahon that major index and inversion number share the same distribution on permutations. [ABSTRACT FROM AUTHOR]

The main aim of the article is to prove that the symmetric function Φ n ( x , r ) = ∏ i 1 + i 2 + ⋯ + i n = r ( x 1 i 1 + x 2 i 2 + ⋯ + x n i n ) is Schur geometrically convex for x ∈ R + + n and fixed r ∈ N + = { 1 , 2 , ⋯ } , where i 1 , i 2 , ⋯ , i n are non-negative integers. Further, we obtain Φ n ( x , r ) is also Schur m -power convex for m ≤ 0 . As applications, a Klamkin–Newman type inequality is derived. Finally, we list a counter example to illustrate Φ n ( x , r ) is neither Schur convex nor Schur concave. [ABSTRACT FROM AUTHOR]

We enumerate interlaced pairs of parking functions whose underlying Dyck path has a bounded height. We obtain an explicit formula for this enumeration in the form of a quotient of analogs of Chebyshev polynomials having coefficients in the ring of symmetric functions. [ABSTRACT FROM AUTHOR]

The decision theorem of the Schur harmonic convexity for the compositions involving Schur harmonically convex functions is established and used to determine the Schur harmonic convexity of some symmetric functions. [ABSTRACT FROM AUTHOR]

We relate the m -truncated Kronecker products of symmetric functions to the semi-invariant rings of a family of quiver representations. We find cluster algebra structures for these semi-invariant rings when m = 2 . Each g -vector cone G ◇ l of these cluster algebras controls the 2-truncated Kronecker products for all symmetric functions of degree no greater than l . As a consequence, each relevant Kronecker coefficient is the difference of the number of the lattice points inside two rational polytopes. We also give explicit description of all G ◇ l 's. As an application, we compute some invariant rings. [ABSTRACT FROM AUTHOR]

The decision theorem of the Schur harmonic convexity for the compositions involving Schur harmonically convex functions is established and used to determine the Schur harmonic convexity of some symmetric functions. [ABSTRACT FROM AUTHOR]

Let n ∈ N and [ A j k ] j , k = 1 , 2 be a Hermitian 2 n × 2 n matrix partitioned into four quadratic matrices A j k of order n . Marcus and Watkins proved in 1971 that [ tr A j k 2 ] is positive semidefinite whenever [ A j k ] is positive semidefinite. Let A be an n × n matrix. If f is a C -valued function on C , denote by f ( A ) the value of the primary matrix function associated with f on the matrix A . If F is a symmetric C -valued function on C n , let F ( A ) be the value of F on the eigenvalues of A . Generalizing Marcus and Watkins' result we describe those functions f and F , for which [ F ( f ( A j k ) ) ] is positive semidefinite whenever [ A j k ] is positive semidefinite. We extend an example by Choudhury, which gives a negative result for 3 × 3 block matrices. [ABSTRACT FROM AUTHOR]

In 2010, the first author introduced a combinatorial model for Schur polynomials based on labeled abaci. We generalize this construction to give analogous models for the Hall–Littlewood symmetric polynomials P λ , Q λ , and R λ using objects called abacus-tournaments. We introduce various cancellation mechanisms on abacus-tournaments to obtain simpler combinatorial formulas and explain why these polynomials are divisible by certain products of t -factorials. These tools are then applied to give bijective proofs of several identities involving Hall–Littlewood polynomials, including the Pieri rule that expands the product P μ e k into a linear combination of Hall–Littlewood polynomials. [ABSTRACT FROM AUTHOR]

This article gives a new matrix function named ‘twisted immanant,’ which can be regarded as an analogue of the immanant. This is defined for each self-conjugate partition through a ‘twisted’ analogue of the irreducible character of the symmetric group. This twisted immanant has some interesting properties. For example, it satisfies Cauchy–Binet type formulas. Moreover it is closely related to the following results for matrices whose entries anticommute with each other: (i) the description of the invariants under the conjugations, and (ii) an analogue of the Cauchy identities for symmetric polynomials. [ABSTRACT FROM AUTHOR]

The Thoma simplex Ω is an infinite-dimensional space, a kind of dual object to the infinite symmetric group. The z-measures are probability measures on Ω depending on three continuous parameters. One of them is the parameter of the Jack symmetric functions, and in the limit as it goes to 0, the z-measures turn into the Poisson-Dirichlet distributions. The definition of the z-measures is somewhat implicit. We show that the topological support of any nondegenerate z-measure is the whole space Ω. [ABSTRACT FROM AUTHOR]

In 2004 Rosas and Sagan asked whether there was a way to define a basis in the algebra of symmetric functions in noncommuting variables, NCSym, having properties analogous to the classical Schur functions. This was because they had constructed a partial such set that was not a basis. We answer their question by defining Schur functions in noncommuting variables using a noncommutative analogue of the Jacobi-Trudi determinant. Our Schur functions in NCSym map to classical Schur functions under commutation, and a subset of them indexed by set partitions forms a basis for NCSym. Amongst other properties, Schur functions in NCSym also satisfy a noncommutative analogue of the product rule for classical Schur functions in terms of skew Schur functions. We also show how Schur functions in NCSym are related to Specht modules, and naturally refine the Rosas-Sagan Schur functions. Moreover, by generalizing Rosas-Sagan Schur functions to skew Schur functions in the natural way, we prove noncommutative analogues of the Littlewood-Richardson rule and coproduct rule for them. Finally, we relate our functions to noncommutative symmetric functions by proving a subset of our functions are natural extensions of noncommutative ribbon Schur functions, and immaculate functions indexed by integer partitions. [ABSTRACT FROM AUTHOR]

The coinvariant algebra is a quotient of the polynomial ring Q [ x 1 , ... , x n ] whose algebraic properties are governed by the combinatorics of permutations of length n. A word w = w 1 ... w n over the positive integers is packed if whenever i > 2 appears as a letter of w , so does i − 1. We introduce a quotient S n of Q [ x 1 , ... , x n ] which is governed by the combinatorics of packed words. We relate our quotient S n to the generalized coinvariant rings of Haglund, Rhoades, and Shimozono as well as the superspace coinvariant ring. [ABSTRACT FROM AUTHOR]

A subtraction-free definition of the big Witt vector construction was recently given by the first author. This allows one to define the big Witt vectors of any semiring. Here we give an explicit combinatorial description of the big Witt vectors of the Boolean semiring. We do the same for two variants of the big Witt vector construction: the Schur Witt vectors and the p-typical Witt vectors. We use this to give an explicit description of the Schur Witt vectors of the natural numbers, which can be viewed as the classification of totally positive power series with integral coefficients, first obtained by Davydov. We also determine the cardinalities of some Witt vector algebras with entries in various concrete arithmetic semirings. [ABSTRACT FROM AUTHOR]