Lyndon Words and Transition Matrices between Elementary, Homogeneous and Monomial Symmetric Functions

Extending the elementary and complete homogeneous symmetric functions, we introduce the truncated homogeneous symmetric function in (1) for any integer partition λ, and show that the transition matrix from to the power sum symmetric functions is given by where and z are nonsingular diagonal matrices. Consequently, forms a basis of the ring Λ of symmetric functions. In addition, we show that the generating function satisfies where ω is the involution of Λ sending each elementary symmetric function to the complete homogeneous symmetric function .

We present the basic elements of a generalization of symmetric function theory involving functions of commuting and anticommuting (Grassmannian) variables. These new functions, called symmetric functions in superspace, are invariant under the diagonal action of the symmetric group on the sets of commuting and anticommuting variables. In this work, we present the superspace extension of the classical bases, namely, the monomial symmetric functions, the elementary symmetric functions, the completely symmetric functions, and the power sums. Various basic results, such as the generating functions for the multiplicative bases, Cauchy formulas, involution operations as well as the combinatorial scalar product are also generalized.

In this note we prove with a bijective argument that every monomial symmetric function is a polynomial with integer coefficients in the elementary symmetric functions.

Two important classes of symmetric Boolean functions are the equal-weight Boolean functions and the elementary (or homogeneous) symmetric Boolean functions. In this paper we studied the equal-weight symmetric Boolean functions. First the Walsh spectra of the equal-weight symmetric Boolean functions are given. Second the sufficient and necessary condition on correlation-immunity of the equal-weight symmetric Boolean function is derived and other cryptology properties such as the nonlinearity, balance and propagation criterion are taken into account. In particular, the nonlinearity of the equal-weight symmetric Boolean functions with n (n ≥ 10) variables is determined by their Hamming weight. Considering these properties will be helpful in further investigations of symmetric Boolean functions.

The elementary, complete homogeneous, and power sum symmetric functions

In 1992, Loeb (Adv Math, 91:64–74, 1992) considered a natural extension of the binomial coefficients to negative entries and gave a combinatorial interpretation in terms of hybrid sets. He showed that many of the fundamental properties of binomial coefficients continue to hold in this extended setting. Recently, Formichella and Straub (Ann Comb, 23:725–748, 2019) showed that these results can be extended to the q-binomial coefficients with arbitrary integer values and extended the work of Loeb further by examining the arithmetic properties of the q-binomial coefficients. In this paper, we give an alternative combinatorial interpretation in terms of lattice paths and consider an extension of the more general weight-dependent binomial coefficients, first defined by Schlosser (Sém Lothar Combin, 81:24, 2020), to arbitrary integer values. Remarkably, many of the results of Loeb, Formichella and Straub continue to hold in the general weighted setting. We also examine important special cases of the weight-dependent binomial coefficients, including ordinary, q- and elliptic binomial coefficients as well as elementary and complete homogeneous symmetric functions.

In [24] the authors introduced inhomogeneous bases of the ring of symmetric functions. The elements in these bases have the property that they evaluate to characters of symmetric groups. In this article we develop further properties of these bases by proving product and coproduct formulae. In addition, we give the transition coefficients between the elementary symmetric functions and the irreducible character basis.

Recursive relations for objects of statistical interest have long been important for computation, and they remain so even with hugely improved computing power. Such recursions are frequently derived by exploiting relations between generating functions. For example, the top-order zonal polynomials that occur in much distribution theory under normality can be recursively related to other (easily computed) symmetric functions (power-sum and elementary symmetric functions; Ruben, 1962,Annals of Mathematical Statistics33, 542–570; Hillier, Kan, and Wang, 2009,Econometric Theory25, 211–242). Typically, in a recursion of this type thekth object of interest,dk, say, is expressed in terms of all lower orderdj’s. In Hillier et al. (2009) we pointed out that, in the case of top-order zonal polynomials and other invariant polynomials of multiple matrix argument, afixed lengthrecursion can be deduced. We refer to this as a short recursion. The present paper shows that the main results in Hillier et al. (2009) can be generalized and that short recursions can be obtained for a much larger class of objects/generating functions. As applications, we show that short recursions can be obtained for various problems involving quadratic forms in noncentral normal vectors, including moments, product moments, and expectations of ratios of powers of quadratic forms. For this class of problems, we also show that the length of the recursion can be further reduced by an application of a generalization of Horner’s method (cf. Brown, 1986,SIAM Journal on Scientific and Statistical Computing7, 689–695), producing a super-short recursion that is significantly more efficient than even the short recursion.

Description of Minimum Weight Codewords of Cyclic Codes by Algebraic Systems

We study families of multivariate orthogonal polynomials with respect to the symmetric weight function in d variables Bγ(x)=∏i=1dω(xi)∏i<j|xi-xj|2γ+1,x∈(a,b)d,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} B_{\\gamma }(\\mathtt {x}) = \\prod \\limits _{i=1}^{d} \\omega (x_{i}) \\prod \\limits _{i<j} |x_{i}-x_{j}|^{2\\gamma +1}, \\quad \\mathtt {x}\\in (a,b)^{d}, \\end{aligned}$$\\end{document}for gamma >-1, where omega (t) is an univariate weight function in t in (a,b) and mathtt {x} = (x_{1},x_{2}, ldots , x_{d}) with x_{i} in (a,b). Applying the change of variables x_{i},i=1,2,ldots ,d, into u_{r},r=1,2,ldots ,d, where u_{r} is the r-th elementary symmetric function, we obtain the domain region in terms of the discriminant of the polynomials having x_{i},i=1,2,ldots ,d, as its zeros and in terms of the corresponding Sturm sequence. Choosing the univariate weight function as the Hermite, Laguerre, and Jacobi weight functions, we obtain the representation in terms of the variables u_{r} for the partial differential operators such that the respective Hermite, Laguerre, and Jacobi generalized multivariate orthogonal polynomials are the eigenfunctions. Finally, we present explicitly the partial differential operators for Hermite, Laguerre, and Jacobi generalized polynomials, for d=2 and d=3 variables.

The Jack polynomials J (x; ) form a remarkable class of symmetric polynomials. They are indexed by a partition and depend on a parameter . One of their properties is that several classical families of symmetric functions can be obtained by specializing , e.g., the monomial symmetric functions m ( =∞), the elementary functions e ′ ( = 0), the Schur functions s ( = 1) and nally the two classes of zonal polynomials ( = 2, = 1=2). The Jack polynomials can be de ned in various ways, e.g.: a) as an orthogonal family of functions which is compatible with the canonical ltration of the ring of symmetric functions or b) as simultaneous eigenfunctions of certain di erential operators (the Sekiguchi–Debiard operators). Recently Opdam, [O], constructed a similar family F (x; ) of nonsymmetric polynomials. The index runs now through all compositions ∈ Nn. They are de ned in a completely similar fashion, e.g., the Sekiguchi–Debiard operators are being replaced by the Cherednik di erential-re ection operators (see Sect. 3). It is becoming more and more clear that these polynomials are as important as their symmetric counterparts. The purpose of this paper is to add to the existing characterizations of Jack polynomials two further ones: c) a recursion formula among the F together with two formulas to obtain J from them. d) combinatorial formulas of both J and F in terms of certain generalized tableaux.

The q-Stirling numbers of both kinds are specializations of the complete or elementary symmetric functions. In this note, we use this fact to prove that the q-Stirling numbers can be expressed in terms of the q-binomial coefficients and vice versa.

The r-Whitney numbers of both kinds are specializations of complete and elementary symmetric functions. In this paper, we use this fact to express a finite discrete convolution involving r-Whitney numbers of both kinds in terms of Bernoulli polynomials.

The computation of elementary symmetric functions and their derivatives is an integral part of conditional maximum likelihood estimation of item parameters un der the Rasch model. The conditional approach has the advantages of parameter estimates that are consistent (assuming the model is correct) and statistically rigorous goodness-of-fit tests. Despite these characteristics, the conditional approach has been limited by problems in computing the elementary symmetric functions. The in troduction of recursive formulas for computing these functions and the availability of modem computers has largely mediated these problems; however, detailed documentation of how these formulas work is lacking. This paper describes how various recursion formulas work and how they are used to compute elementary symmetric functions and their derivatives. The availabil ity of this information should promote a more thorough understanding of item parameter estimation in the Rasch model among both measurement specialists and practitioners.

Rotation symmetric functions have been used as components of different cryptosystems. Functions in this class are invariant under circular translation of indices. In this paper, we will do some enumeration on homogeneous rotation symmetric functions over F p . A formula for counting homogeneous rotation symmetric functions over F p is presented when gcd(n ,d ) is a prime power, where n is the number of input variables and d is the algebraic degree of the function, which demonstrates that we solved one of the problems in [7].