Generalized Pascal triangle for binomial coefficients of words

A universal sequence of integers generating balanced Steinhaus figures modulo an odd number

Picking binomial coefficients which cannot be divided by a given prime from Pascal's triangle, we find that they form a set with self-similarity. Essouabri studied on a class of meromorphic functions associated to the above set. These functions are related to fractal geometry and it is a problem whether such a function has a non-real pole on its axis of absolute convergence. Essouabri gave a proof of existence of such a non-real pole in the simplest case. The keys of his proof are Stein's and Wilson's estimates on how fast the points multiply in Pascal's triangle modulo a prime. This article will give an extension of Essouabri's result to some cases with certain ways to count the points in Pascal's triangle modulo a prime which are different from the traditional one.

Counting: Basic Subsets of a Set Pascal's Triangle Binomial Coefficient Identities Counting: Intermediate Finding a Polynomial The Upward-Extended Pascal's Triangle Recurrence Relations and Fibonacci Numbers Counting: Advanced Generating Functions and Making Change Integer Triangles Rook Paths and Queen Paths Discrete Probability Probability Spaces and Distributions Markov Chains Random Tournaments Number Theory Divisibility of Factorials and Binomial Coefficients Covering Systems Partitions of an Integer Information Theory What Is Surprise? A Coin-Tossing Game Shannon's Theorems Games A Little Graph Theory Background The Ramsey Game Tic-Tac-Toe and Animal Games Algorithms Counters Listing Permutations and Combinations Sudoku Solving and Polycube Packing Appendix A: Hints and Solutions to Exercises Appendix B: Notation Bibliography Index

I read with interest Thomas Byrne's article: "An Extension of Pascal's Triangle and Work concerning Binomial Expansion Coefficients" in the February 1990 issue. On page 28 Mr. Byrne states: "I was seeking a mechanism to generate multinomial expansion coefficients but this doesn't do it . . . Maybe a three dimensional extension will prove useful."

Mass spectrometry (MS) is widely used for isotopic studies of metabolism in which detailed information about biochemical processes is obtained from the analysis of isotope incorporation into metabolites. The biological value of such experiments is dependent on the accuracy of the isotopic measurements. Using MS, isotopologue distributions are measured from the quantitative analysis of isotopic clusters. These measurements are prone to various biases, which can occur during the experimental workflow and/or MS analysis. The lack of relevant standards limits investigations of the quality of the measured isotopologue distributions. To meet that need, we developed a complete theoretical and experimental framework for the biological production of metabolites with fully controlled and predictable labeling patterns. This strategy is valid for different isotopes and different types of metabolisms and organisms, and was applied to two model microorganisms, Pichia augusta and Escherichia coli, cultivated on (13)C-labeled methanol and acetate as sole carbon source, respectively. The isotopic composition of the substrates was designed to obtain samples in which the isotopologue distribution of all the metabolites should give the binomial coefficients found in Pascal's triangle. The strategy was validated on a liquid chromatography-tandem mass spectrometry (LC-MS/MS) platform by quantifying the complete isotopologue distributions of different intracellular metabolites, which were in close agreement with predictions. This strategy can be used to evaluate entire experimental workflows (from sampling to data processing) or different analytical platforms in the context of isotope labeling experiments.

The Pascal triangle is a geometric representation of binomial coefficients in triangular form. We utilize this formalism to deterministically arrange silver nanocylinders of different sizes (30, 60, and 90 nm) on a triangle and numerically study their near-field optical properties. We show that near-field intensities at specific points on this triangle depend on the wavelength and angle of incidence. From the wavelength-dependent studies at various junctions of nanocylinders, we obtain maximum near-field intensity at 350 and 380 nm. By varying the angle of incidence of the TM-polarized plane wave, we find systematic variation in the near-field intensity at different junctions of the geometry. Our study will lead to insights in designing controllable electromagnetic hot spots for chip-based plasmonic devices.

(1999). Relating Geometry and Algebra in the Pascal Triangle, Hexagon, Tetrahedron, and Cuboctahedron Part I: Binomial Coefficients, Extended Binomial Coefficients and Preparation for Further Work. The College Mathematics Journal: Vol. 30, No. 3, pp. 170-186.

Inspired by a recent beautiful construction of Armin Straub and Wadim Zudilin, that ‘tweaked’ the sum of the sth powers of the n-th row of Pascal's triangle, getting instead of sequences of numbers, sequences of rational functions, we do the same for general binomial coefficients sums, getting a practically unlimited supply of Apéry limits. While getting what we call ‘major Apéry miracles’, proving irrationality of the associated constants (i.e. the so-called Apéry limits) is very rare, we do get, every time, at least a ‘minor Apéry miracle’ where an explicit constant, defined as an (extremely slowly converging) limit of some explicit sequence, is expressed as an Apéry limit of some recurrence, with some initial conditions, thus enabling a very fast computation of that constant, with exponentially decaying error.

Linear cellular automata, finite automata and Pascal's triangle

A property of superposability is defined here for Treillis Automata as an equivalent of the linearity property of Cellular Automata. We take advantage of the binary local transition function of TAs, which enables us to use an infix notation for it, to study this property in an algebraic framework. We show that a TA is superposable iff its local transition function operates on its set of states as a commutative monoid law and that the general case of commutative monoids is reducible to the case of Abelian groups in which the space-time diagrams obtained are isomorphic to a superposition of figures representing the triangle of binomial coefficients (Pascal's triangle) modulo integers.KeywordsCellular AutomatonInitial ConfigurationNeutral ElementBinomial CoefficientCommutative MonoidThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Unit Determinants in Generalized Pascal Triangles

A finite word w ∈ {0, 1}* is balanced if for every equal-length factors u and v of every cyclic shift of w we have ||u|1 - |v|1| ≤ 1. This new class of finite words was defined in [O. Jenkinson and L. Q. Zamboni, Characterisations of balanced words via orderings, Theoret. Comput. Sci.310(1–3) (2004) 247–271]. In [O. Jenkinson, Balanced words and majorization, Discrete Math. Algorithms Appl.1(4) (2009) 463–484], there was proved several results considering finite balanced words and majorization. One of the main results was that the base-2 orbit of the balanced word is the least element in the set of orbits with respect to partial sum. It was also proved that the product of the elements in the base-2 orbit of a word is maximized precisely when the word is balanced. It turns out that the words 0q-p1p have similar extremal properties, opposite to the balanced words, which makes it meaningful to call these words the most unbalanced words. This paper contains the counterparts of the results mentioned above. We will prove that the orbit of the word u = 0q-p1p is the greatest element in the set of orbits with respect to partial sum and that it has the smallest product. We will also prove that u is the greatest element in the set of orbits with respect to partial product.

G. Fici proved that a finite word has a minimal suffix automaton if and only if all its left special factors occur as prefixes. He called LSP all finite and infinite words having this latter property. We characterize here infinite LSP words in terms of S-adicity. More precisely we provide a finite set of morphisms S and an automaton \(\mathcal{A}\) such that an infinite word is LSP if and only if it is S-adic and all its directive words are recognizable by \(\mathcal{A}\).

On generating binary words palindromically

Many topological spaces are simply sets of points (atoms) endowed with a topology. Some spaces, however, have elements that are functions, matrices, or other non-atomic items. Another special type of space has elements that are themselves subsets of another space; these spaces are called hyperspaces. Hyperspaces are metric spaces, and the metric defined on them is called the Hausdorff metric. Hyperspaces wllose points are the closed subsets and hyperspaces whose points are the closed connected subsets (of metric spaces) have been extensively studied. Also, hyperspaces of closed and convex subsets of a bounded convex set in Euclidean space are of great interest in geometry. See Lay [3]. In recent years, geometers and topological dynamicists have explored spaces of closed and bounded subsets of the plane in connection with the study of fractals. One of the major results in the theory is that the hyperspace of closed subsets of a closed interval of real numbers is homeomorphic with the Hilbert cube, and with the space IX, the countable product of unit intervals. For more general information about the Hausdorff metric and spaces of subsets, see Devaney [2] and Sieradski [6]. The main result of this paper is a topological characterization of the space of closed subsets of a convergent sequence of points. The proof given here provides a homneornorphic embedding of the space in the plane, E2. The result was first proved by Pelczynski [5]. He studied arbitrary compact zero-dimensional metric spaces, so his proofs are much more widely applicable, but they are also somewhat technical. Our proof depends only on some well-known results in the theory of metric spaces, and is therefore accessible to advanced undergraduate mathematics majors. We also prove that for no convergent sequence of real numbers is there an isometric embedding of the hyperspace in Euclidean space, E,1, for any n. For other results and discussion, see Nadler [4]. Let (X, d) denote a compact metric space. The hyperspace (2X, D) of X is the metric space whose points are the closed nonempty subsets of X and whose metric is the Hausdorff metric D given by