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The attempt is to give a formal concpet of system, and with this provide a definition of category, that will also satisfy the definition of a system. An axiomatic base is given, for constructing the group of integers. In the process, we…
A. Bondal's symplectic groupoid of triangular bilinear forms induces a Poisson structure on the space $\mathcal{A}_n$ of $n \times n$ unipotent upper-triangular matrices. It is governed by the classical $\mathfrak{so}(n)$ reflection…
We obtain a combinatorial formula related to the shear transformation for semi-invariants of binary forms, which implies the classical characterization of semi-invariants in terms of a differential operator. Then, we present a combinatorial…
We present the first representation of the general term of the Rayleigh-Schr\"odinger series for quasidegenerate systems. Each term of the series is represented by a tree and there is a straightforward relation between the tree and the…
We employ the notions of `sequential function' and `interrogation' (dialogue) in order to define new partial combinatory algebra structures on sets of functions. These structures are analyzed using J. Longley's preorder-enriched category of…
For $A$ a gentle algebra, and $X$ and $Y$ string modules, we construct a combinatorial basis for Hom($X,\tau Y$). We use this to describe support $\tau$-tilting modules for $A$. We give a combinatorial realization of maps in both directions…
This is the second part of a work initiated in \cite{GaHa}, where we constructed a model category, $\Qt$, for set theory. In the present paper we use this model category to introduce homotopy-theoretic intuitions to set theory. Our main…
We consider conjugation action of symmetric group on the semigroup of all partial functions and develop a machinery to investigate character formulas and multiplicities. In particular, we determine nilpotent matrices whose orbit under…
We introduce a formalism based on a combinatorial notion of cell complex subject to an inclusion-reversing duality operation. Our main goal is to open the way for a functorial definition of field theories in a context where no manifold or…
Polynomial functors are useful in the theory of data types, where they are often called containers. They are also useful in algebra, combinatorics, topology, and higher category theory, and in this broader perspective the polynomial aspect…
Let $\mathfrak{g}$ be an untwisted affine Lie algebra with associated Weyl group $W_a$. To any level 0 weight $\gamma$ we associate a weighted graph $\Gamma_\gamma$ that encodes the orbit of $\gamma$ under the action $W_a$. We show that the…
We introduce a new definition for the species of type B, or H-species, analog to the classical species (of type A), but on which we consider the action of the groups Bn of signed permutations. We are interested in algebraic structure on…
Partially commutative monoids provide a powerful tool to study graphs, viewingwalks as words whose letters, the edges of the graph, obey a specific commutation rule. A particularclass of traces emerges from this framework, the hikes, whose…
In this work we introduce and study various generalizations of the notion of increasingly labelled trees, where the label of a child node is always larger than the label of its parent node, to multilabelled tree families, where the nodes in…
We introduce a family of sequence transformations, defined via partial Bell polynomials, that may be used for a systematic study of a wide variety of problems in enumerative combinatorics. This family includes some of the transformations…
We give a combinatorial description (including explicit differential-form bases) for the cohomology groups of the space of n distinct nonzero complex numbers, with coefficients in rank-one local systems which are of finite monodromy around…
Orbit harmonics is a tool in combinatorial representation theory which promotes the (ungraded) action of a linear group $G$ on a finite set $X$ to a graded action of $G$ on a polynomial ring quotient by viewing $X$ as a $G$-stable point…
Aguiar and Mahajan introduced a cohomology theory for the twisted coalgebras of Joyal, with particular interest in the computation of their second cohomology group, which gives rise to their deformations. We use the Koszul duality theory…
The birational $R$-matrix is a transformation that appears in the theory of geometric crystals, the study of total positivity in loop groups, and discrete dynamical systems. This $R$-matrix gives rise to an action of the symmetric group…
Two of the pillars of combinatorics are the notion of choosing an arbitrary subset of a set with $n$ elements (which can be done in $2^n$ ways), and the notion of choosing a $k$-element subset of a set with $n$ elements (which can be done…