Rigged configurations are combinatorial objects originating from the Bethe Ansatz, that label highest weight crystal elements. In this paper a new unrestricted set of rigged configurations is introduced for types ADE by constructing a crystal struc...
Circle packings with specified patterns of tangencies form a discrete counterpart of analytic functions. In this paper we study univalent packings (with a combinatorial closed disk as tangent graph) which are embedded in (or fill) a bounded, simply c...
We characterize the fundamental group of a locally finite graph G with ends combinatorially, as a group of infinite words. Our characterization gives rise to a canonical embedding of this group in the inverse limit of the (free) fundamental groups...
We provide new results on combinatorial characterizations of covering properties in end spaces and ray spaces. In particular, we characterize the Lindelöf degree, the extent, the Rothberger property, $σ$-compactness and the Menger property for ray,...
We consider normal affine T-varieties X endowed with an action of finite abelian group G commuting with the action of T. For such varieties we establish the existence of G-equivariant geometrico-combinatorial presentations in the sense of Altmann and...
We define a far-reaching generalization of Schnyder woods which encompasses many classical combinatorial structures on planar graphs. Schnyder woods are defined for planar triangulations as certain triples of spanning trees covering the triangulati...
In many combinatorial games, one can prove that the first player wins under best play using a simple but non-constructive argument called strategy-stealing. This work is about the complexity behind these proofs: how hard is it to actually find a winn...
We consider two families of Drinfeld twists generated from a simple Jordanian twist further twisted with 1-cochains. Using combinatorial identities, they are presented as a series expansion in the dilatation and momentum generators. These twists inte...
In the projective plane, we consider congruences of straight lines with the combinatorics of the square grid and with all elementary quadrilaterals possessing touching inscribed conics. The inscribed conics of two combinatorially neighbouring quadril...
Let $Γ=Γ(A)$ denote a simple strongly connected digraph with vertex set $X$, diameter $D$, and let $\{A_0,A:=A_1,A_2,\ldots,A_D\}$ denote the set of distance-$i$ matrices of $Γ$. Let $\{R_i\}_{i=0}^D$ denote a partition of $X\times X$, where $R_i=...
All-or-nothing transforms (AONT) were proposed by Rivest as a message preprocessing technique for encrypting data to protect against brute-force attacks, and have numerous applications in cryptography and information security. Later the unconditional...
We formalize a combinatorial principle, called the 3XOR principle, due to Feige, Kim and Ofek (2006), as a family of unsatisfiable propositional formulas for which refutations of small size in any propositional proof system that possesses the feasibl...
Several combinatorial identities are presented, involving Stirling functions of the second kind with a complex variable. The identities involve also Stirling numbers of the first kind, binomial coefficients and harmonic numbers....
Code for paper "Mapping the combinatorial coding between olfactory receptors and perception with deep learning" (⭐ 10)
Recently, Guo and Zeng discovered two families of polynomials featuring in a q-analogue of Faulhaber's formula for the sums of powers and a q-analogue of Gessel-Viennot's formula involving Salie's coefficients for the alternating sums of powers. In...
The theorem on squaring a rectangle from a tiling of a quadrilateral (Schramm and Cannon-Floyd-Parry) gives a combinatorial version of the Riemann mapping theorem. We elucidate by example (the dumbbell) some of the limitations of rectangle-squaring...
We present an LLT-type formula for a general power of the nabla operator applied to the Cauchy product for the modified Macdonald polynomials, and use it to deduce a new proof of the generalized shuffle theorem describing $\nabla^k e_n$, and the Elia...
Roitman's combinatorial principle $Δ$ is equivalent to monotone normality of the nabla product, $\nabla (ω+1)^ω$. If $\{ X_n : n\in ω\}$ is a family of metrizable spaces and $\nabla_n X_n$ is monotonically normal, then $\nabla_n X_n$ is hereditar...
The reversible pebble game is a combinatorial game played on rooted DAGs. This game was introduced by Bennett (1989) motivated by applications in designing space efficient reversible algorithms. Recently, Chan (2013) showed that the reversible pebble...
The notion of a reduced expression for a double coset in a Coxeter group was introduced by Williamson, and recent work of Elias and Ko has made this theory more accessible and combinatorial. One result of Elias-Ko is that any coset admits a reduced e...