In this paper, we consider the generating functions of the complete and elementary symmetric functions and provide a new generalization of these classical symmetric functions. Some classical relationships involving the complete and elementary symmetric functions are reformulated in a more general context. Combinatorial interpretations of these generalized symmetric functions are also introduced.
The $i^{th}$ elementary symmetric polynomial of the set of $n$ variables $\mathcal{R}=\{m_1,m_2,m_3,...,m_n\}$ is isolated from the expansion of the $i^{th}$ binomial product ${{m_1+...+m_n}\choose i}$ via an alternating sum.
The Jacobi-Stirling numbers and the Legendre-Stirling numbers of the first and second kind were first introduced in [6], [7]. In this paper we note that Jacobi-Stirling numbers and Legendre-Stirling numbers are specializations of elementary and complete symmetric functions. We then study combinatorial interpretations of this specialization and obtain new combinatorial interpretations of the Jacobi-Stirling and Legendre-Stirling numbers.
We bound the tensor ranks of elementary symmetric polynomials, and we give explicit decompositions into powers of linear forms. The bound is attained when the degree is odd.
We introduce the new combinatorial approach of plethystic type of tableaux, as a method to understand coefficients of Schur functions appearing in plethysms $s_ν[h_λ]$ and $s_ν[e_λ]$, for any partitions $λ$ and $ν$. We first give general results about this approach, then use results on tableaux, ribbon tableaux and integer points in polytopes to understand the case where $ν$ is a partition of $3$ and $λ$ has one part. We then use a \textit{Kronecker map} to extend these results to any partition $λ$.
We prove that the hyperbolicity cones of elementary symmetric polynomials are spectrahedral, i.e., they are slices of the cone of positive semidefinite matrices. The proof uses the matrix--tree theorem, an idea already present in Choe et al.
Previous work by Mora and Sala provides the reduced Groebner basis of the ideal formed by the elementary symmetric polynomials in $n$ variables of degrees $k=1,\dots,n$, $\langle e_{1,n}(x), \dots, e_{n,n}(x) \rangle$. Haglund, Rhoades, and Shimonozo expand upon this, finding the reduced Groebner basis of the ideal of elementary symmetric polynomials in $n$ variables of degree $d$ for $d=n-k+1,\dots,n$ for $k\leq n$. In this paper, we further generalize their findings by using symbolic computation and experimentation to construct the reduced Groebner basis for the ideal generated by the elemen…
We consider the expansion of the square of a complete homogeneous function $h_λ$, or of an elementary symmetric function $e_λ$, in the basis of Schur functions. This square also decomposes into two plethysms, $s_2[h_λ]$ and $s_{11}[h_λ]$ (resp. $s_2[h_λ]$ and $s_{11}[h_λ]$), which are called its symmetric and anti-symmetric parts, respectively. We define a sign statistic on the set of tableaux that index the Schur functions appearing in the square of those symmetric functions. This sign statistic allows to determine to which plethysm each Schur function contributes. We use mainly combinatorial…
In this article we establish the asymptotic behavior of generating functions related to the exponential sum over finite fields of elementary symmetric functions and their perturbations. This asymptotic behavior allows us to calculate the probability generating function of the probability that the the elementary symmetric polynomial of degree $k$ and its perturbations returns $β\in \mathbb{F}_q$ where $\mathbb{F}_q$ represents the field of $q$ elements. Our study extends many of the results known for perturbations over the binary field to any finite field. In particular, we establish when a par…
Major Percy A. MacMahon's first paper on plane partitions included a conjectured generating function for symmetric plane partitions. This conjecture was proven almost simultaneously by George Andrews and Ian Macdonald, Andrews using the machinery of basic hypergeometric series and Macdonald employing his knowledge of symmetric functions. The purpose of this paper is to simplify Macdonald's proof by providing a direct, inductive proof of his formula which expresses the sum of Schur functions whose partitions fit inside a rectangular box as a ratio of determinants.
Consideration of a classification of the number of partitions of a natural number according to the members of sub-partitions differing from unity leads to a non-recursive formula for the number of irreducible representations of the symmetric group Sn. This article was published, long ago, under the title A non-recursive expression for the number of irreducible representations of the Symmetric Group Sn, Physica 114A, 1982, 361-364, North-Holland Publishing Co. The Introduction has been, somewhat, improved, however, the handmade result remains unproved.
We conjecture an explicit positive combinatorial formula for the expansion of unicellular LLT polynomials in the elementary symmetric basis. This is an analogue of the Shareshian-Wachs conjecture and previously studied by Panova and the author in 2018. We show that the conjecture for unicellular LLT polynomials implies a similar formula for vertical-strip LLT polynomials. We prove positivity in the elementary basis in for the class of graphs called `melting lollipops' previously considered by Huh, Nam and Yoo. This is done by proving a curious relationship between a generalization of charge an…
In 1995 Stanley introduced a generalization of the chromatic polynomial of a graph $G$, called the chromatic symmetric function, $X_G$, which was generalized to noncommuting variables, $Y_G$, by Gebhard-Sagan in 2001. Recently there has been a renaissance in the study of $X_G$, in particular in classifying when $X_G$ is a positive linear combination of elementary symmetric or Schur functions. We extend this study from $X_G$ to $Y_G$, including establishing the multiplicativity of $Y_G$, and showing $Y_G$ satisfies the $k$-deletion property. Moreover, we completely classify when $Y_G$ is a posi…
We give an elementary symmetric function expansion for $MΔ_{m_γe_1}Πe_λ^{\ast}$ and $MΔ_{m_γe_1}Πs_λ^{\ast}$ when $t=1$ in terms of what we call $γ$-parking functions and lattice $γ$-parking functions. Here, $Δ_F$ and $Π$ are certain eigenoperators of the modified Macdonald basis and $M=(1-q)(1-t)$. Our main results in turn give an elementary basis expansion at $t=1$ for symmetric functions of the form $M Δ_{Fe_1} Θ_{G} J$ whenever $F$ is expanded in terms of monomials, $G$ is expanded in terms of the elementary basis, and $J$ is expanded in terms of the modified elementary basis $\{Πe_λ^\ast\…
Extending the elementary and complete homogeneous symmetric functions, we introduce the truncated homogeneous symmetric function $h_λ^{\dd}$ in $(\ref{THSF})$ for any integer partition $λ$, and show that the transition matrix from $h_λ^{\dd}$ to the power sum symmetric functions $p_λ$ is given by \[M(h^{\dd},p)=M'(p,m)z^{-1}D^{\dd},\] where $D^{\dd}$ and $z$ are nonsingular diagonal matrices. Consequently, $\{h_λ^{\dd}\}$ forms a basis of the ring $Λ$ of symmetric functions. In addition, we show that the generating function $H^{\dd}(t)=\ssum_{n\ge 0}h_n^{\dd}(x)t^n$ satisfies \[ω(H^{\dd}(t))=\…
I give an elementary proof that a quasi-symmetric design without repeated blocks on $v$ points has at most $v\choose2$ blocks, with equality if and only if it is a tight $4$-design.
Hypergeometric functions and zonal polynomials are the tools usually addressed in the literature to deal with the expected value of the elementary symmetric functions in non-central Wishart latent roots. The method here proposed recovers the expected value of these symmetric functions by using the umbral operator applied to the trace of suitable polynomial matrices and their cumulants. The employment of a suitable linear operator in place of hypergeometric functions and zonal polynomials was conjectured by de Waal in 1972. Here we show how the umbral operator accomplishes this task and consequ…
Let $P$ be a simple polytope of dimension $n$ with $m$ facets. In this paper we pay our attention on those elementary symmetric polynomials in the Stanley--Reisner face ring of $P$ and study how the decomposability of the $n$-th elementary symmetric polynomial influences on the combinatorics of $P$ and the topology and geometry of toric spaces over $P$. We give algebraic criterions of detecting the decomposability of $P$ and determining when $P$ is $n$-colorable in terms of the $n$-th elementary symmetric polynomial. In addition, we define the Stanley--Reisner {\em exterior} face ring $\mathca…
Often in mathematics it is useful to summarize a multivariate phenomenon with a single number and in fact, the determinant -- which is represented by det -- is one of the simplest cases. In fact, this number it is defined only for square matrices and a lot of its properties are very well-known. For instance, the determinant is a multiplicative function, i.e. det(AB)=detA detB, but it is not, in general, an additive function. Another interesting function in the matrix analysis is the characteristic polynomial -- in fact, given a matrix A, this function is defined by $p_A(t)=det(tI-A)$ where I i…
We present an elementary proof of a generalization of Kirchoff's matrix tree theorem to directed, weighted graphs. The proof is based on a specific factorization of the Laplacian matrices associated to the graphs, which only involves the two incidence matrices that capture the graph's topology. We also point out how this result can be used to calculate principal eigenvectors of the Laplacian matrices.
We take an elementary approach to derive a generalization of Kunth's formula using Lassalle's explicit formula. In particular, we give a formula for the Kostka numbers of a shape $μ\vdash n$ and weight $(m,1^{n-m})$ for $m=3,\;4$.
Let $P_n(y_1,\ldots,y_n):= \prod_{1\leq i<j\leq n}\left( 1 -\frac{y_i}{y_j}\right) $ and $P_n:= \sup_{(y_1,\ldots,y_n)}P_n(y_1,\ldots,y_n) $ where the supremum is taken over the $n$-ples $(y_1,\ldots,y_n)$ of real numbers satisfying $0 <|y_1| < |y_2|< \cdots < |y_n|$. We prove that $P_n \leq 2^{\lfloor n/2\rfloor}$ for every $n$, i.e., we extend to all $n$ the bound that Pohst proved for $n\leq 11$. As a consequence, the bound for the absolute discriminant of a totally real field in terms of its regulator is now proved for every degree of the field.
It is well-known that the famous Selberg integral is equivalent to the Morris constant term identity. In 1998, Baker and Forrester conjectured a generalization of the $q$-Morris constant term identity. This conjecture was proved and extended by Károlyi, Nagy, Petrov and Volkov in 2015. In this paper, we obtain two symmetric function generalizations of the $q$-Baker--Forrester ex-conjecture. These includes: (i) a $q$-Baker--Forrester type constant term identity for a product of a complete symmetric function and a Macdonald polynomial; (ii) a complete symmetric function generalization of KNPV's …
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.
In Chapter 2 we study the path-cycle symmetric function of a digraph, a symmetric function generalization of Chung and Graham's cover polynomial. Most of this material appears in either Advances in Math. 118 (1996), 71-98 or J. Algebraic Combin. 10 (1999), 227-240. Chapter 3 contains miscellaneous results about Stanley's symmetric function generalization X_G of the chromatic polynomial, e.g., we establish a connection with some of Tutte's work on the chromatic polynomial and use this to prove that X_G is reconstructible. Most of Chapter 3 does not appear elsewhere.
The symmetric function theorem states that a polynomial that is invariant under permutation of variables, is a polynomial in the elementary symmetric polynomials. We deduce this classical result, in the analytic setting, from the multivariate Faà di Bruno formula. In two variables, this allows us to completely determine all coefficients that occur in the inductive equations.
We prove that for any tree with a vertex of degree at least six, its chromatic symmetric function is not $e$-positive, that is, it cannot be written as a nonnegative linear combination of elementary symmetric functions. This makes significant progress towards a recent conjecture of Dahlberg, She, and van Willigenburg, who conjectured the result for all trees with a vertex of degree at least four. We also provide a series of conditions that can identify when the chromatic symmetric function of a spider, a tree consisting of multiple paths identified at an end, is not $e$-positive. These conditi…
A sequential dynamical system (SDS) consists of a graph $G$ with vertices $v_1,v_2,\ldots,v_n$, a state set $A$, a collection of "vertex functions" $\{f_{v_i}\}_{i=1}^n$, and a permutation $π\in S_n$ that specifies how to compose these functions to yield the SDS map $[G,\{f_{v_i}\}_{i=1}^n,π]\colon A^n\to A^n$. In this paper, we study symmetric invertible SDS defined over the cycle graph $C_n$ using the set of states $\mathbb F_2$. These are, in other words, asynchronous elementary cellular automata (ECA) defined using ECA rules 150 and 105. Each of these SDS defines a group action on the set …
We study the representation theory of three towers of algebras which are related to the symmetric groups and their Hecke algebras. The first one is constructed as the algebras generated simultaneously by the elementary transpositions and the elementary sorting operators acting on permutations. The two others are the monoid algebras of nondecreasing functions and nondecreasing parking functions. For these three towers, we describe the structure of simple and indecomposable projective modules, together with the Cartan map. The Grothendieck algebras and coalgebras given respectively by the induct…
We provide an explicit description of the quantum product of multi-symmetric functions using the elementary multi-symmetric functions introduced by Vaccarino.
