Elementary Counting Research Articles

The spectrum of resolvable Bose triple systems

A classical construction of Bose produces a Steiner triple system of order 3n from a symmetric, idempotent latin square of order n, which exists whenever n is odd. In an application to access-balancing in storage systems, certain Bose triple systems play a central role. A natural question arises: For which orders v does there exist a resolvable Bose triple system? Elementary counting establishes the necessary condition that v≡9(mod18). For specific Bose triple systems that optimize an access metric, we show that v≡9(mod18) is also sufficient.

On the Evolution of Consanguinities

“Consanguinity” is a gender-neutral term for “fraternity” or “sorority.” Initially a consanguinity includes M male members and F female members. Each week a member, chosen at random, selects a new member, always of the same gender as the member making the selection. This model for evolution is isomorphic to the classic Pólya’s urn. The male and female members play the same roles as the red and black balls in the urn, and the procedure for selecting a new member is equivalent to drawing a ball from the urn, then replacing it and adding a new ball of the same color. It is well known that for Pólya’s urn, the proportion of red balls in the urn is a martingale. It follows that for a consanguinity, the proportion of the membership that is male is a martingale. Furthermore, being bounded, this martingale converges to a limit. For a martingale that is the sum of independent random variables, such as a symmetric random walk, there is also a well-known second-degree martingale from which the variance of the limiting distribution can be deduced. What the author discovered, in the process of solving his own examination problem, is that a similar martingale exists also for Pólya’s urn, even though in this case the number of red balls is the sum of random variables that are not independent. This new martingale can be used to calculate the variance of the limiting distribution. Traditionally, the probability that r red balls will be drawn from Pólya’s urn in n trials is derived by a rather tricky argument involving conditional probability. This article uses an obvious but overlooked simpler approach. Pólya’s formula for the probability that m male members will be chosen in n weeks is derived, without any mention of conditional probability, by an elementary counting argument, and its limit is shown to be a beta distribution.

Products of k atoms in Krull monoids

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor. For $k\in \mathbb N$, let $\mathcal U_k(H)$ denote the set of all $m\in \mathbb N$ with the following property: There exist atoms $u_1, \ldots, u_k, v_1, \ldots , v_m\in H$ such that $u_1\cdot\ldots\cdot u_k=v_1\cdot\ldots\cdot v_m$. It is well-known that the sets $\mathcal U_k (H)$ are finite intervals whose maxima $\rho_k(H)=\max \mathcal U_k(H) $ depend only on $G$. If $|G|\le 2$, then $\rho_k (H) = k$ for every $k \in \mathbb N$. Suppose that $|G| \ge 3$. An elementary counting argument shows that $\rho_{2k}(H)=k\mathsf D(G)$ and $k\mathsf D(G)+1\le \rho_{2k+1}(H)\le k\mathsf D(G)+\lfloor \frac{\mathsf D(G)}{2}\rfloor$ where $\mathsf D(G)$ is the Davenport constant. In \cite{Ga-Ge09b} it was proved that for cyclic groups we have $k\mathsf D(G)+1 = \rho_{2k+1}(H)$ for every $k \in \mathbb N$. In the present paper we show that (under a mild condition on the Davenport constant) for every noncyclic group there exists a $k^*\in \mathbb N$ such that $\rho_{2k+1}(H)= k\mathsf D(G)+\lfloor \frac{\mathsf D(G)}{2}\rfloor$ for every $k\ge k^*$. This confirms a conjecture of A. Geroldinger, D. Grynkiewicz, and P. Yuan in \cite{Ge-Gr-Yu15}.

Counting-based impossibility proofs for set agreement and renaming

Set agreement and renaming are two tasks that allow processes to coordinate, even when agreement is impossible. In k-set agreement, n processes must decide on at most k of their input values. While n-set agreement is trivially wait-free solvable by each process deciding on its input, (n−1)-set agreement is not wait-free solvable. In M-renaming, processes must decide on distinct names in a range of size M. For any number n of processes, (2n−1)-renaming is wait-free solvable, but surprisingly, (2n−2)-renaming is wait-free solvable if and only if n is not a prime power; the only previous lower bound on the number of names necessary for renaming, when n is not a prime power, is n+1. In adaptive renaming, M decreases when the number p of participants in the execution decreases. It is known that (2p−1)-adaptive renaming is wait-free solvable, while (2p−⌈pn−1⌉)-adaptive renaming is not.This paper presents counting-based proofs for the above mentioned impossibility results: n processes can wait-free solve neither (n−1)-set agreement nor (2p−⌈pn−1⌉)-adaptive renaming; if n is a prime power, n processes cannot wait-free solve (2n−2)-renaming. For an arbitrary number of processes, we give a lower bound for renaming, by reduction from renaming for a different number of processes, and relying on the distribution of prime numbers.Our proofs combine simple operational properties of a restricted set of executions with elementary counting arguments to show the existence of an execution violating the task’s conditions. This makes the proofs easier to understand, verify, and, we hope, extend.

Note on asymptotically good packings

AbstractGiven 𝓁<k<n, a partial Steiner (n, k, 𝓁)‐system 𝒫 is a family of k‐element subsets of an n‐element set such that every 𝓁‐element subset is contained in at most one member of 𝒫. The second author, followed by several others, verified a conjecture of P. Erdős and H. Hanani stating that, for every n, there exists a partial Steiner (n, k, 𝓁)‐system of size \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$${{\left( {1 - o\left( 1 \right)} \right)\left( {_{\ell }^{n} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - o\left( 1 \right)} \right)\left( {_{\ell }^{n} } \right)} {\left( {_{\ell }^{k} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {_{\ell }^{k} } \right)}}$$\end{document}. While all known proofs involve the “semi‐random method”, in this short manuscript we use an algebraic construction (European J Combin 16(1) (1995), 35–40; Problems Control Inform Theory/Problemy Upravlen Teor Inform 12(1) (1983), 3–10) and complement it by an elementary counting argument to obtain a simple proof of the above fact. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 448–455, 2009

HEISENBERG ACTION IN SKEIN THEORY AND EXTERNAL EDGE CONDITION

In this article we give an explicit description of the representation matrix of a Heisenberg type action constructed by Blanchet, Habegger, Masbaum and Vogel. We give the matrix in terms of a ribbon graph and its admissible colorings. We show that components of the representation matrix satisfies the {\it external edge condition}, which is a natural combinatorial/geometric condition for maps from the first homology of the graph. We give the explicit formula of the trace of the action in the case of surfaces with colored structure using the external edge condition, the Verlinde formula and elementary counting arguments. Our formula is a generalization of the results for a surface without colored structure, which are already known.

Aggregation rules that satisfy anonymity and neutrality

In the case of two alternatives, elementary counting techniques are used to determine the number of aggregation rules that satisfy anonymity and neutrality.

Projective ovoids and generalized quadrangles

Abstract We consider a simple condition defining a tetradic set of ovoids in a projective three-space over a finite field. By elementary counting and geometrical methods we establish the properties of a tetradic set and are able to give a purely synthetic construction of the class of generalized quadrangles of order (s, s 2) satisfying Property (G) at a flag. This includes the class of dual flock generalized quadrangles due to Kantor and Payne in the 1980's. We also show that the dual flock generalized quadrangles are characterised by Property (G) at a line.

More Combinatorial Proofs via Flagpole Arrangements

In their recent book [1], Arthur Benjamin and Jennifer Quinn derive a wide array of algebraic and number theoretic identities with clever applications of elementary counting principles. As they say, While [counting is] not necessarily the simplest approach, it offers another method to gain understanding of mathematical truths. To a combinatorialist, this kind of proof is the only right one. A sampling of the combinatorial approach was presented in [2], which also issued a challenge to the reader to find a combinatorial proof of the identity

Irrational proofs for three theorems of Stanley

We give new proofs of three theorems of Stanley on generating functions for the integer points in rational cones. The first relates the rational generating function σ v + K ( x ) ≔ ∑ m ∈ ( v + K ) ∩ Z d x m , where K is a rational cone and v ∈ R d , with σ − v + K ∘ ( 1 / x ) . The second theorem asserts that the generating function 1 + ∑ n ≥ 1 L P ( n ) t n of the Ehrhart quasi-polynomial L P ( n ) ≔ # ( n P ∩ Z d ) of a rational polytope P can be written as a rational function ν P ( t ) ( 1 − t ) dim P + 1 with nonnegative numerator ν P . The third theorem asserts that if P ⊆ Q , then ν P ≤ ν Q . Our proofs are based on elementary counting afforded by irrational decompositions of rational polyhedra.

Integrability of induction cocycles for Kac-Moody groups

We prove that whenever a Kac-Moody group over a finite field is a lattice of its buildings, it has a fundamental domain with respect to which the induction cocycle is Lp for any p ∈ [1;+∞). The proof uses elementary counting arguments for root group actions on buildings. The applications are the possibility to apply some lattice superrigidity, and the normal subgroup property for Kac-Moody lattices.

The Number of Nonisomorphic Two-dimensional Algebras over a Finite Field

In this paper, we derive explicit formulas for the number of nonisomorphic two-dimensional nonassociative algebras, possibly without a unit, over a finite field. The proof combines the first author’s general classification theory of two-dimensional nonassociative algebras over arbitrary base fields with elementary counting arguments which are primarily addressed to the problem of determining the number of orbits of a finite set acted upon by the group of integers mod 2. The number of nonisomorphic two-dimensional division algebras will also be determined.

A class of expandable polyhedral structures

A class of expandable polyhedral structures, the expandohedra, consisting of prismatic faces linked along edges by hinged plates, provides a model for the swelling of viruses. The finite breathing mode of the expandohedra, apparent in the model, and from geometric arguments, is not, however, detected by elementary counting techniques. Symmetry extended mobility criteria show that the breathing motion is part of a set of face mechanisms for many expandohedra. Numerical analysis using the singular value decomposition of the compatibility matrix confirms the completeness of the deductions from general symmetry theorems, derived for expandohedra.

The Alexander polynomial and finite type 3-manifold invariants

Using elementary counting methods, we calculate a universal perturbative invariant (also known as the LMO invariant) of a 3-manifold M, satisfying $H_1(M,\mathbb{Z})=\mathbb{Z}$ , in terms of the Alexander polynomial of M. We show that +1 surgery on a knot in the 3-sphere induces an injective map from finite type invariants of integral homology 3-spheres to finite type invariants of knots. We also show that weight systems of degree 2m on knots, obtained by applying finite type 3m invariants of integral homology 3-spheres, lie in the algebra of Alexander-Conway weight systems, thus answering the questions raised in [Ga].

Salvaging the library of life

Salvaging the Library of Life The next century may well be dominated by the biological sciences. Both biotechnology and advancing medicine promise much. But consider: We have only begun the elementary counting and description of the world biota. Each species contains genetic variations of potential use, and at least of scientific interest. Although about 1.4 million species have been given scientific names, estimates of the total number of species range up to roughly 30 million, with some people guessing 100 million. This means we may very well not know the species diversity of the world flora and fauna to the nearest order of magnitude. Yet time is running out in which we can even catalog our living wealth. Very conservative estimates of the current extinction rate give roughly 5000 species lost annually, at least several species per day. Some suspect the true level may be 30,000 per year. And so we accelerate toward a calamity unparalleled in planetary history. The best known cause of present day species extinction is the cutting of tropical forests. They have lost about 55% of their original cover of, say, two centuries ago-and are shrinking at the rate of 1.8% per year. We now co-opt about 40% of plant growth worldwide, favoring monocultural crops, which must greatly affect genetic diversity. Given the blunt economic and cultural forces at work, even slowing the rate of destruction seems doubtful in the immediate future. Worse, the rate seems doomed to increase, since its ultimate cause is human activity, and human numbers and expectations grow apace. To improve the economic lot of a swelling human tropical population would require at least a five-fold increase in economic activity there, bringing a crushing load on the already strained biosphere. Other biological zones such as coral reefs and oceanic islands also dwindle at alarming rates. Because warmer regions host the greatest species diversity, losses are most severe in precisely the tropical continents where our own numbers swell so alarmingly. Everywhere there are calls for a halt to tropical deforestation. In October 1991, I gave a reasonably optimistic speech about our future at California Institute of Technology’s Centennial Celebration. It was a festive occasion, studded with Nobel Prize winners, amusing stories, wry wisdom-a testament to the rise of science and technology in our era. Gregory Emford, Department of Physics, University of Cdifomia, Irvine, CA 92717 Jcwmaf of Social and fiolmionar9 S9mn.s 16(4jz367-374 ISSN: 016L7361 Copyright 0 1993 by JAI Press, Inc. AU rights of ~cproduction in soy form fwmai.

Elementary counting of cardinal and ordinal numbers by persons with mental retardation.

Cardinal and ordinal counting skills were assessed in 37 mentally retarded adult workers and 42 school children with mental retardation. The major results were the very poor performance of the younger children (essentially at chance beyond the numbers 1 and 2) and the overall marked inferiority in ordinal compared with cardinal counting.

On a Converse to the Tscgebotarev density theorem

AbstractUsing an elementary counting procedure on biquadratic polynomials over Zp it is shown that the probability distribution of odd, unramified rational primes according to decomposition type in a fixed dihedral numberfield is identical to the probility of separable quartic polynomials (mod p) whose roots generate numberfields with normal closure having Galois group isomorphic to D4, as p → ∞. This verifies a conjecture about a converse to the Tschebotarev density theorem. Further evidence in support of this conjecture is provided in quadratic and coubic numberfields.

Partitioning of the Plane by Lines

The purpose of this article is to determine the maximum number of regions into which n lines partition the plane—a problem that could be appropriate for some secondary school students. The solution, as presented here, draws on some basic geometric notions, some elementary counting techniques, and Euder’s formula for plane figures—that is, V — E + R = 1, where V is the number of vertices, E is the number of edges, and R is the number of regions.

Recent advances in low level counting techniques.

Problems involving low level counting techniques are encountered in many diverse fields including biological and chemical tracer studies, archaeo­ logical and geological dating (I), investigations of natural and induced radio­ activities, and the study of fundamental particles. Such a host of methods and variations on methods have been reported that an exhaustive catalog would constitute little more than an annotated bibliography. We have, there­ fore, limited our attention to a few selected examples which seem to be of special interest because of their versatility or their extreme sensitivity. We prefer to organize our discussion around specific problems rather than around types of detectors since this permits a comparison of alternative methods and provides practical examples of various techniques in actual operation. Alpha counting is arbitrarily excluded. For a discussion of photographic methods (radioautographs) the reader is referred to Norris & Woodruff (2) and refer­ ences cited therein. We will discuss the methods developed for the measure­ ment of natural radiocarbon and tritium, cosmic ray produced Be7, and the K(O content of the human body. DOUble-beta decay experiments are consid­ ered since they seem to constitute one of the most difficult of the outstanding problems and because of their importance from the viewpoint of theoretical nuclear physics. Finally, an account is given of recent attempts to detect the neutrino, probably the most difficult low level counting problem of all and one which seems finally on the verge of successful solution. Principles.-The fundamental principles of low level counting are simple: maximize the number of detected events of the desired kind (signal) and minimize all others (background). The number of detected events is, of course, the product of sample size, over-all counting efficiency, and counting time. However, the striking importance of the net sample counting rate is not always fully appreciated. It can be shown (3) from elementary counting statistics, that if the sample rate is small compared with background (which is usually the case in a truly low level problem) and if counting statistics is the limiting source of error (also usually true), then the proper parameter for the comparison of counting methods is S2/B, the ratio of the square of the net sample rate to the background rate. (Any sample strength can be used in making the comparison so long as the same strength is used in testing all methods.) The method giving the maximum value of S2/B will yield the