相关论文: On the reconstruction problem for Pascal lines
Cyclotomic polynomials are basic objects in Number Theory. Their properties depend on the number of distinct primes that intervene in the factorization of their order, and the binary case is thus the first nontrivial case. This paper sees…
We prove that certain polynomials previously introduced by the author can be identified with tau functions of Painlev\'e VI, obtained from one of Picard's algebraic solutions by acting with a four-dimensional lattice of B\"acklund…
In this paper, we introduce a new generalization of Pascal's triangle. The new object is called the hyperbolic Pascal triangle since the mathematical background goes back to regular mosaics on the hyperbolic plane. We describe precisely the…
For $0\leq k \leq n$, the number $C(n,k)$ represents the number of all lattice paths in the plane from the point $(0,0)$ to the point $(n,k)$, using steps $(1,0)$ and $(0,1)$, that never rise above the main diagonal $y=x$. The Fuss-Catalan…
This article demonstrates, using numerous examples of varying complexity, how one can visually prove summation formulas involving binomial coefficients by exclusively using the recurrence relation for binomial coefficients and its…
The main goal of the paper is to introduce methods which compute B\'ezier curves faster than Casteljau's method does. These methods are based on the spectral factorization of a $n\times n$ Bernstein matrix, $B^e_n(s)= P_nG_n(s)P_n^{-1}$,…
We show that 4-connected plane triangulations can be redrawn such that edges are represented by straight segments and the vertices are covered by a set of at most $\sqrt{2n}$ lines each of them horizontal or vertical. The same holds for all…
The $(n-\ell)$-deck of an $n$-vertex graph is the multiset of (unlabeled) subgraphs obtained from it by deleting $\ell$ vertices. An $n$-vertex graph is $\ell$-reconstructible if it is determined by its $(n-\ell)$-deck, meaning that no…
Three-term recurrences have infused stupendous amount of research in a broad spectrum of the sciences, such as orthogonal polynomials (in special functions) and lattice paths (in enumerative combinatorics). Among these are the Lucas…
We discover new P-time computable six-vertex models on planar graphs beyond Kasteleyn's algorithm for counting planar perfect matchings. We further prove that there are no more: Together, they exhaust all P-time computable six-vertex models…
In the present paper, we will show that three apparently disjoint objects: Galois representations arising from twenty-seven lines on a cubic surface (number theory and arithmetic algebraic geometry), Picard modular forms (automorphic…
We prove that a large family of graphs which are decomposable with respect to the modular decomposition can be reconstructed from their collection of vertex-deleted subgraphs.
A \emph{trace} of a sequence is generated by deleting each bit of the sequence independently with a fixed probability. The well-studied \emph{trace reconstruction} problem asks how many traces are required to reconstruct an unknown binary…
A binary $[n,k]$-linear code $\mathcal{C}$ is a $k$-dimensional subspace of $\mathbb{F}_2^n$. For $\boldsymbol{x}\in \mathbb{F}_2^n$, the set $\boldsymbol{x}+\mathcal{C}$ is a coset of $\mathcal{C}$. In this work we study a partial ordering…
The purpose of this note is to report, in narrative rather than rigorous style, about the nice geometry of $6$-division points on the Fermat cubic $F$ and various conics naturally attached to them. Most facts presented here were derived by…
Grothendieck's theory of dessins provides a bridge between algebraic numbers and combinatorics. This paper adds a new concept, called 'bias', to the bridge. This produces: (i) from a biased plane tree the construction of a sequence of…
Let $W^c(A_n)$ be the set of fully commutative elements in the $A_n$-type Coxeter group. Using only the settings of their canonical form, we recount $W^c(A_n)$ by the recurrence that is taken as a definition of the Catalan number $C_{n+1}$…
The trinomial transform of a sequence is a generalization of the well-known binomial transform, replacing binomial coefficients with trinomial coefficients. We examine Pascal-like triangles under trinomial transform, focusing on the ternary…
In this paper we provide some factorization theorems of the Poincar\'e series $P_C$ of a plane curve singularity $C$ depending on some key values of the semigroup of values of \(C\). These results yield an iterative computation of $P_C$ in…
The direct application of the definition of sorting in lattices is impractical because it leads to an algorithm with exponential complexity. In this paper we present for distributive lattices a recursive formulation to compute the sort of a…