Related papers: On the Stern sequence and its twisted version
For a simple graph, we introduce a notion of the star sequence and prove that the star sequence and the frequently sequences of a graph are inverses of each other from a combinatorial point of view. As a consequence, we express the general…
Rota's basis conjecture states that in any square array of vectors whose rows are bases of a fixed vector space the vectors can be rearranged within their rows in such a way that afterwards not only the rows are bases, but also the columns.…
In this paper we study a natural generalization of the Stern-Brocot sequences which comes from the introduction of weighted mediants. We focus our attention on the case $k = 3$, in which $(2a + c)/(2b + d)$ and $(a + 2c)/(b + 2d)$ are the…
First, we shall formulate and prove Theorem of Lie-Kolchin type for a cone and derive some algebro-geometric consequences. Next, inspired by a recent result of Dinh and Sibony we pose a conjecture of Tits type for a group of automorphisms…
A simplicial complex is $r$-conic if every subcomplex of at most $r$ vertices is contained in the star of a vertex. A $4$-conic complex is simply connected. We prove that an $8$-conic complex is $2$-connected. In general a $(2n+1)$-conic…
In 1967 Herbert Scarf suggested a new proof of Brouwer's fixed point theorem based on a combinatorial analogue of Sperner's lemma. Scarf presented his arguments in very geometric language, even purely combinatorial ones. Recently H. Petri…
We present two related conjectures, arising in work on i-matchings in random r-regular bipartite graphs. The conjectures themselves are easily stated and involve only basic properties of convergent power series. One formulation involves…
For a simple graph we introduce notions of the double star sequence, the double star frequently sequence and prove that these sequences are inverses of each other. As a consequence, we express the general second Zagreb index in terms of the…
The first author introduced a sequence of polynomials (\cite{8}, sequence A174531) defined recursively. One of the main results of this study is proof of the integrality of its coefficients.
The aim of our paper is to make high-precision positional stellar catalogs by compiling large astrophysical data bulk and large astrometrical surveys. The data reliability and uniqueness is the primary request. The common precision of 1 arc…
English translation of "Solitony Ricciego" (Wiadomo\'sci Matematyczne 48, 2012, no. 1, pp. 1-32). Despite the general-sounding title, the text covers just a few narrow topics: Perelman's proof of the fact that compact Ricci solitons are of…
Deciding the positivity of a sequence defined by a linear recurrence with polynomial coefficients and initial condition is difficult in general. Even in the case of recurrences with constant coefficients, it is known to be decidable only…
A classical result of Euler states that the tangent numbers are an alternating sum of Eulerian numbers. A dual result of Roselle states that the secant numbers can be obtained by a signed enumeration of derangements. We show that both…
The distribution of certain Mahonian statistic (called $\mathrm{BAST}$) introduced by Babson and Steingr\'{i}msson over the set of permutations that avoid vincular pattern $1\underline{32}$, is shown bijectively to match the distribution of…
To investigate hyperbinary expansions of a nonnegative integer~$n$, an edge-labeled directed graph $A(n)$ has recently been introduced. After pointing out some new simple facts about its cyclomatic number, we give a relatively simple…
At a crossroads of calculus and combinatorics, the generating function of secant and tangent numbers (Euler numbers) provides enumeration of alternating permutations. In this article, we present a new refinement of Euler numbers to answer…
We prove several combinatorial properties of suffix arrays, including a characterization of suffix arrays through a bijection with a certain well-defined class of permutations. Our approach is based on the characterization of…
Stern's diatomic sequence with its intrinsic repetition and refinement structure between consecutive powers of $2$ gives rise to a rather natural probability measure on the unit interval. We construct this measure and show that it is purely…
The identification of causal relations is a cornerstone of the scientific method. Traditional approaches to this task are based on classical statistics. However, such classical approaches do not apply in the quantum domain, where a broader…
Proietti et al. (arXiv:1902.05080) reported on an experiment designed to settle, or at least to throw light upon, the paradox of Wigner's friend. Without questioning the rigor or ingenuity of the experimental protocol, I argue that its…