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Many combinatorial problems can be formulated as a polynomial optimization problem that can be solved by state-of-the-art methods in real algebraic geometry. In this paper we explain many important methods from real algebraic geometry, we…
The \emph{$q,t$-Catalan numbers} $C_n(q,t)$ are polynomials in $q$ and $t$ that reduce to the ordinary Catalan numbers when $q=t=1$. These polynomials have important connections to representation theory, algebraic geometry, and symmetric…
We define Boolean algebras in the linear context and study its symmetric powers. We give explicit formulae for products in symmetric Boolean algebras of various dimensions. We formulate symmetric forms of the inclusion-exclusion principle.
We introduce a new family of symmetric functions, which are $q$-analogues of products of Schur functions defined in terms of ribbon tableaux. These functions can be interpreted in terms of the Fock space representation of the quantum affine…
Symmetric functions, which take as input an unordered, fixed-size set, are known to be universally representable by neural networks that enforce permutation invariance. These architectures only give guarantees for fixed input sizes, yet in…
We apply matrix methods to arithmetic functions by associating matrices to the functions in a manner drawn from the theory of symmetric functions. Then we study the characteristic polynomials of the associated matrices.
We give a review of modern approaches to constructing formal solutions to integrable hierarchies of mathematical physics, whose coefficients are answers to various enumerative problems. The relationship between these approaches and…
En esta serie de tres articulos, damos una exposicion de varios resultados y problemas abiertos en tres areas de la combinatoria algebraica y geometrica: las matrices totalmente no negativas, las representaciones del grupo simetrico, y los…
The characteristic map for the symmetric group is an isomorphism relating the representation theory of the symmetric group to symmetric functions. An analogous isomorphism is constructed for the symmetric space of symplectic forms over a…
Harmonic sums and their generalizations are extremely useful in the evaluation of higher-order perturbative corrections in quantum field theory. Of particular interest have been the so-called nested sums,where the harmonic sums and their…
This article serves as an introduction to several recent developments in the study of quasisymmetric functions. The focus of this survey is on connections between quasisymmetric functions and the combinatorial Hopf algebra of noncommutative…
For any homomorphism V on the space of symmetric functions, we introduce an operation which creates a q-analog of V. By giving several examples we demonstrate that this quantization occurs naturally within the theory of symmetric functions.…
Analogues of classical combinatorial identities for elementary and homogeneous symmetric functions with coefficients in Yanigian are discussed. As a corollary, similar relations are deduced for shifted Schur functions.
The elementary symmetric partition function is a map on the set of partitions. It sends a partition lambda to the partition whose parts are the summands in the evaluation of the elementary symmetric function on the parts of lambda. These…
Multiple harmonic-like numbers are studied using the generating function approach. A closed form is stated for binomial sums involving these numbers and two additional parameters. Several corollaries and examples are presented which are…
This article is devoted to the study of several algebras which are related to symmetric functions, and which admit linear bases labelled by various combinatorial objects: permutations (free quasi-symmetric functions), standard Young…
In this paper, we review the theory of time space-harmonic polynomials developed by using a symbolic device known in the literature as the classical umbral calculus. The advantage of this symbolic tool is twofold. First a moment…
We derive an explicit sum formula for symmetric Macdonald polynomials. Our expression contains multiple sums over the symmetric group and uses the action of Hecke generators on the ring of polynomials. In the special cases $t=1$ and $q=0$,…
This is a chapter for the upcoming book "Handbook of Combinatorial Algebraic Geometry". Comments are welcome! We cover symmetric function identities tied to delta operators, as well as representation-theoretic and geometric avatars of these…
On the vertex operator algebra associated with rank one lattice we derive a general formula for products of vertex operators in terms of generalized homogeneous symmetric functions. As an application we realize Jack symmetric functions of…