Related papers: P\'olya theory for species with an equivariant gro…
We give an Atiyah-Patodi-Singer index theory construction of the bundle of fermionic Fock spaces parametrized by vector potentials in odd space dimensions and prove that this leads in a simple manner to the known Schwinger terms…
We endow the category of bialgebras over a pair of operads in distribution with a cofibrantly generated model category structure. We work in the category of chain complexes over a field of characteristic zero. We split our construction in…
We give a new construction of the Joyal model structure on the category of simplicial sets, and we provide a simple characterization of the fibrations in it. We characterize the inner anodyne maps in terms of categorical equivalences and…
We develop an algebraic formalism for perturbative quantum field theory (pQFT) which is based on Joyal's combinatorial species. We show that certain basic structures of pQFT are correctly viewed as algebraic structures internal to species,…
We develop a calculus based on graph enumeration for $S_n$-equivariant motivic invariants of graphically stratified moduli spaces. We apply our theory to the Deligne--Mumford moduli space $\overline{\mathcal{M}}_{g, n}$ and to the space of…
The combinatorial interpretation of the persistence diagram as a M\"obius inversion was recently shown to be functorial. We employ this discovery to recast the Persistent Homology Transform of a geometric complex as a representation of a…
We continue our investigation of binary actions of simple groups. In this paper, we demonstrate a connection between the graph $\Gamma(\mathcal{C})$ based on the conjugacy class $\mathcal{C}$ of the group $G$, which was introduced in our…
In light of the grammar given by Ji for the $(\alpha,\beta)$-Eulerian polynomials introduced by Carlitz and Scoville, we provide a labeling scheme for increasing binary trees. In this setting, we obtain a combinatorial interpretation of the…
We define iterated monodromy groups of more general structures than partial self-covering. This generalization makes it possible to define a natural notion of a combinatorial model of an expanding dynamical system. We prove that a naturally…
This is a further investigation of our approach to group actions in homological algebra in the settings of homology of {\Gamma}-simplicial groups, particularly of {\Gamma}-equivariant homology and cohomology of {\Gamma}-groups. This…
Modeling binary and categorical data is one of the most commonly encountered tasks of applied statisticians and econometricians. While Bayesian methods in this context have been available for decades now, they often require a high level of…
Symmetric Hilbert spaces such as the bosonic and the fermionic Fock spaces over some `one particle space' $\K$ are formed by certain symmetrization procedures performed on the full Fock space. We investigate alternative ways of…
A combinatorial category Disks was introduced by Andr\'e Joyal to play a role in his definition of weak omega-category. He defined the category Theta to be dual to Disks. In the ensuing literature, a more concrete description of Theta was…
Representation theory of the symmetric group $\mathfrak{S}_n$ has a very distinctive combinatorial flavor. The conjugacy classes as well as the irreducible characters are indexed by integer partitions $\lambda \vdash n$. We introduce class…
We show that the theory of sorting by reversals fits into the well-established theory of circuit partitions of 4-regular multigraphs (which also involves the combinatorial structures of circle graphs and delta-matroids). In this way, we…
The discrete Green's functions are the pseudoinverse (or the inverse) of the Laplacian (or its variations) of a graph. In this paper, we will give combinatorial interpretations of Green's functions in terms of enumerating trees and forests…
We study several kinds of polynomial ensembles of derivative type which we propose to call P\'olya ensembles. These ensembles are defined on the spaces of complex square, complex rectangular, Hermitian, Hermitian anti-symmetric and…
This paper generalizes the basic notions of additive and multiplicative combinatorics to the setting of group actions: if $G$ is a group acting on a set $X$, and we have subsets $A\subseteq G$ and $Y\subseteq X$ such that the set of pairs…
Percolation theory can be used to describe the structural properties of complex networks using the generating function formulation. This mapping assumes that the network is locally tree-like and does not contain short-range loops between…
Combinatorial categories satisfy a stronger form of Yoneda Lemma, namely, the isomorphism type of an object can be recovered by counting the number of homomorphisms from all other objects into it. In this work, we show that this property…