DOCSLIB.ORG

(12) Patent Application Publication (10) Pub. No.: US 2016/0328253 A1 MAJUMDAR (43) Pub

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
(12) Patent Application Publication (10) Pub. No.: US 2016/0328253 A1 MAJUMDAR (43) Pub US 2016.0328253A1 (19) United States (12) Patent Application Publication (10) Pub. No.: US 2016/0328253 A1 MAJUMDAR (43) Pub. Date: Nov. 10, 2016 (54) QUANTON REPRESENTATION FOR (52) U.S. Cl. EMULATING QUANTUM-LIKE CPC ............... G06F 9/455 (2013.01); G06F 17/18 COMPUTATION ON CLASSICAL (2013.01); G06N 99/002 (2013.01) PROCESSORS (57) ABSTRACT (71) Applicant: KYNDI, INC., Redwood City, CA (US) The Quanton virtual machine approximates Solutions to (72) Inventor: Arun MAJUMDAR, Alexandria, VA NP-Hard problems in factorial spaces in polynomial time. (US) The data representation and methods emulate quantum com puting on classical hardware but also implement quantum (73) Assignee: KYNDI, INC., Redwood City, CA (US) computing if run on quantum hardware. The Quanton uses permutations indexed by Lehmer codes and permutation (21) Appl. No.: 15/147,751 operators to represent quantum gates and operations. A (22) Filed: May 5, 2016 generating function embeds the indexes into a geometric object for efficient compressed representation. A nonlinear Related U.S. Application Data directional probability distribution is embedded to the mani fold and at the tangent space to each index point is also a (60) Provisional application No. 62/156,955, filed on May linear probability distribution. Simple vector operations on 5, 2015. the distributions correspond to quantum gate operations. The Publication Classification Quanton provides features of quantum computing: Superpo sitioning, quantization and entanglement Surrogates. Popu (51) Int. Cl. lations of Quantons are evolved as local evolving gate G06F 9/455 (2006.01) operations solving problems or as solution candidates in an G06N 99/00 (2006.01) Estimation of Distribution algorithm. The Quanton repre G06F 7/8 (2006.01) sentation and methods are fully parallel on any hardware. Patent Application Publication US 2016/0328253 A1 Patent Application Publication Nov. 10, 2016 Sheet 2 of 47 US 2016/0328253 A1 Patent Application Publication Nov. 10, 2016 Sheet 3 of 47 US 2016/0328253 A1 Patent Application Publication Nov. 10, 2016 Sheet 4 of 47 US 2016/0328253 A1 i Patent Application Publication Nov. 10, 2016 Sheet 5 of 47 US 2016/0328253 A1 Patent Application Publication Nov. 10, 2016 Sheet 6 of 47 US 2016/0328253 A1 i Patent Application Publication Nov. 10, 2016 Sheet 7 of 47 US 2016/0328253 A1 ºes\^k-·// olnyno, -------------------------------------------------------------------------------------------------------(wolºno)©5 Patent Application Publication Nov. 10, 2016 Sheet 8 of 47 US 2016/0328253 A1 Patent Application Publication Nov. 10, 2016 Sheet 9 of 47 US 2016/0328253 A1 @ @ | ||||||20 9 |× ||9 cu: | | 10 | Patent Application Publication Nov. 10, 2016 Sheet 10 of 47 US 2016/0328253 A1 ? |3 vivo)@anlongus Patent Application Publication Nov. 10, 2016 Sheet 11 of 47 US 2016/0328253 A1 Patent Application Publication Nov. 10, 2016 Sheet 12 of 47 US 2016/0328253 A1 :~~~~-.. --------------------------------------------------------------------|3{,}3&3&&3&&-?????????! |Nouvzravnieºwwe Patent Application Publication Nov. 10, 2016 Sheet 13 of 47 US 2016/0328253 A1 @ ze?í?ezcùyzízèlevizészlº /><N) @ Patent Application Publication Nov. 10, 2016 Sheet 14 of 47 US 2016/0328253 A1 X; . g 3 x • ?ex s X e s 48 .* 3. : wis a w ea s: s X s: : : 1. ea. w a. &:: Patent Application Publication Nov. 10, 2016 Sheet 15 of 47 US 2016/0328253 A1 Patent Application Publication Nov. 10, 2016 Sheet 16 of 47 US 2016/0328253 A1 Patent Application Publication Nov. 10, 2016 Sheet 17 of 47 US 2016/0328253 A1 Patent Application Publication Nov. 10, 2016 Sheet 18 of 47 US 2016/0328253 A1 Patent Application Publication Nov. 10, 2016 Sheet 19 of 47 US 2016/0328253 A1 HIV/50SE83|d 31\/5)\TOHHOL (q€)el @ |alvºSawad Patent Application Publication Nov. 10, 2016 Sheet 20 of 47 US 2016/0328253 A1 S. S r N g SC N w re-w S. s N da o asses N SP se y LS ses SO St. s k s Patent Application Publication Nov. 10, 2016 Sheet 21 of 47 US 2016/0328253 A1 LÍÑOHIO(AHHVO-LOHT3S)H3OJOVTT^+{3T8ISH3N3H)WO?NVITO Patent Application Publication Nov. 10, 2016 Sheet 22 of 47 US 2016/0328253 A1 eJoySuo?eugdoJosS33æpaudpuejossaponsaugumaqaq Patent Application Publication Nov. 10, 2016 Sheet 23 of 47 US 2016/0328253 A1 Patent Application Publication Nov. 10, 2016 Sheet 24 of 47 US 2016/0328253 A1 DOZaun3}+ Patent Application Publication Nov. 10, 2016 Sheet 25 of 47 US 2016/0328253 A1 GOzaun3}+ Patent Application Publication Nov. 10, 2016 Sheet 26 of 47 US 2016/0328253 A1 : ^. s wir S is 83 k y x ax x r as Patent Application Publication Nov. 10, 2016 Sheet 27 of 47 US 2016/0328253 A1 Patent Application Publication Nov. 10, 2016 Sheet 28 of 47 US 2016/0328253 A1 Patent Application Publication Nov. 10, 2016 Sheet 29 of 47 US 2016/0328253 A1 Patent Application Publication Nov. 10, 2016 Sheet 30 of 47 US 2016/0328253 A1 Patent Application Publication Nov. 10, 2016 Sheet 31 of 47 US 2016/0328253 A1 @ @ Patent Application Publication Nov. 10, 2016 Sheet 32 of 47 US 2016/0328253 A1 i Patent Application Publication Nov. 10, 2016 Sheet 33 of 47 US 2016/0328253 A1 Patent Application Publication Nov. 10, 2016 Sheet 34 of 47 US 2016/0328253 A1 Patent Application Publication Nov. 10, 2016 Sheet 35 of 47 US 2016/0328253 A1 Patent Application Publication Nov. 10, 2016 Sheet 36 of 47 US 2016/0328253 A1 i Patent Application Publication Nov. 10, 2016 Sheet 37 of 47 US 2016/0328253 A1 S. : & g Patent Application Publication Nov. 10, 2016 Sheet 38 of 47 US 2016/0328253 A1 Patent Application Publication Nov. 10, 2016 Sheet 39 of 47 US 2016/0328253 A1 & & 8 & & & & 8 x^x as a sat a: (S) Patent Application Publication Nov. 10, 2016 Sheet 40 of 47 US 2016/0328253 A1 Patent Application Publication Nov. 10, 2016 Sheet 41 of 47 US 2016/0328253 A1 Patent Application Publication Nov. 10, 2016 Sheet 42 of 47 US 2016/0328253 A1 Patent Application Publication Nov. 10, 2016 Sheet 43 of 47 US 2016/0328253 A1 Patent Application Publication Nov. 10, 2016 Sheet 44 of 47 US 2016/0328253 A1 ---*******--. **?***************************************?***************************************g** --- Oxo (3)(i)(a)(3)vem (3) (3) (E) Patent Application Publication Nov. 10, 2016 Sheet 45 of 47 US 2016/0328253 A1 @ @(suappe)Jes?A3aèpopdaud?æT apop 0(sue?jnq)des?aeuepopdeuue1 EN|5)NESON[C]OONENYHE||||\fd Patent Application Publication Nov. 10, 2016 Sheet 46 of 47 US 2016/0328253 A1 Wswae-?ae!!!!!!!!! ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~). (3) (3) (3)(E) Patent Application Publication Nov. 10, 2016 Sheet 47 of 47 US 2016/0328253 A1 i US 2016/0328253 A1 Nov. 10, 2016 QUANTON REPRESENTATION FOR data volume or size of the description of the data, its EMULATING QUANTUM-LIKE high-dimensionality, dynamics or intertwined relationships COMPUTATION ON CLASSICAL with other data. Altogether, these issues make it very com PROCESSORS plicated to separate data into distinct classes and analyze. 0008 Training data samples are assumed to exist when CROSS-REFERENCE TO RELATED many times there are no training data for synthesizing new APPLICATIONS Solutions to creative problems (such as synthesis of new 0001. This application is based upon and claims the drugs or materials design). The same underlying distribution benefit of priority to provisional U.S. Application No. in classical approaches draws training samples from the same sets as the problem solution data: however, due to the 62/156.955, filed May 5, 2015, the entire contents of which sparsity and noise in data collection and sometimes-large are incorporated herein by reference. variations of the inputs, the assumptions about the congru BACKGROUND ence between training and solution data cannot be made or when a new model that did not exist before has to be created. 0002 1. Field of the Disclosure 0009. In settings where decisions, based on stratified 0003. The present disclosure relates to a probabilistic data, need to be made (Such as data in ontologies, and polynomial Turing Machine computing model that emulates taxonomies or, other strata) we can expect an effect on the quantum-like computing and performs several practical data response due to the stratum to which data belongs. In a processing functions, and, more particularly, to a system of model framework, a shift due to stratum membership can representation and computing method in both classical significantly influence the estimate of distribution of some probabilistic and quantum computing or quantum emulation outcome given a particular set of covariates and data. on classical computers by reformulating the computation of Membership within a particular stratum can impact the value functions by permutations and embeds these into a prob of the distribution of interest. However, in many cases one ability space, and incorporating methods of topological does not wish to explicitly estimate these stratum-level quantum computing. effects: rather, one seeks to estimate other parameters of 0004 2. Description of the Related Art interest—such as linear coefficients associated with other 0005. The universal model of computing can be referred features that are observed across strata and account for any to as a virtual machine that is traditionally called the Turing non-linearity in the estimates. For a concrete example, Machine, and in accordance with the Church-Turing thesis, consider the case of a classical conditional likelihood model the virtual machine is built on the foundation of computation approach, wherein one conditions on the histogram of the as function evaluation whose archetype is the lambda observations in the stratum. This conditional likelihood is calculus. There are many other ways in which to design a invariant to any stratum-level linear effects, thus removing Turing Machine. Such as, for example, using pi-calculus as them as a contributing factor (in the likelihood) in the model a foundation, or using probability theoretic models to build described herein.
Load more

Recommended publications
  • The Twelvefold Way, the Nonintersecting Circles Problem, and Partitions of Multisets
    Turkish Journal of Mathematics Turk J Math (2019) 43: 765 – 782 http://journals.tubitak.gov.tr/math/ © TÜBİTAK Research Article doi:10.3906/mat-1805-72 The twelvefold way, the nonintersecting circles problem, and partitions of multisets Toufik MANSOUR1,, Madjid MIRZVAZIRI2, Daniel YAQUBI2;∗ 1Department of Mathematics, University of Haifa, Haifa, Israel 2Department of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran Received: 14.05.2018 • Accepted/Published Online: 30.01.2019 • Final Version: 27.03.2019 Abstract: Let n be a nonnegative integer and A = fa1; : : : ; akg be a multiset with k positive integers such that a1 6 ··· 6 ak . In this paper, we give a recursive formula for partitions and distinct partitions of positive integer n with respect to a multiset A. We also consider the extension of the twelvefold way. By using this notion, we solve the nonintersecting circles problem, which asks to evaluate the number of ways to draw n nonintersecting circles in the plane regardless of their sizes. The latter also enumerates the number of unlabeled rooted trees with n + 1 vertices. Key words: Multiset, partitions and distinct partitions, twelvefold way, nonintersecting circles problem, rooted trees, Wilf partitions 1. Introduction A partition of n is a sequence λ1 > λ2 > ··· > λk of positive integers such that λ1 + λ2 + ··· + λk = n (see [2]). We write λ ` n to denote that λ is a partition of n. The nonzero integers λ in λ are called parts of λ. P k j j The number of parts of λ is the length of λ, denoted by `(λ), and λ = k>1 λk is the weight of λ.
    [Show full text]
  • BASIC DISCRETE MATHEMATICS Contents 1. Introduction 2 2. Graphs
    BASIC DISCRETE MATHEMATICS DAVID GALVIN, DEPARTMENT OF MATHEMATICS, UNIVERSITY OF NOTRE DAME Abstract. This document includes lecture notes, homework and exams from the Spring 2017 incarnation of Math 60610 | Basic Discrete Mathematics, a graduate course offered by the Department of Mathematics at the University of Notre Dame. The notes have been written in a single pass, and as such may well contain typographical (and sometimes more substantial) errors. Comments and corrections will be happily received at [email protected]. Contents 1. Introduction 2 2. Graphs and trees | basic definitions and questions 3 3. The extremal question for trees, and some basic properties 5 4. The enumerative question for trees | Cayley's formula 6 5. Proof of Cayley's formula 7 6. Pr¨ufer's proof of Cayley 12 7. Otter's formula 15 8. Some problems 15 9. Some basic counting problems 18 10. Subsets of a set 19 11. Binomial coefficient identities 20 12. Some problems 25 13. Multisets, weak compositions, compositions 30 14. Set partitions 32 15. Some problems 37 16. Inclusion-exclusion 39 17. Some problems 45 18. Partitions of an integer 47 19. Some problems 49 20. The Twelvefold way 49 21. Generating functions 51 22. Some problems 59 23. Operations on power series 61 24. The Catalan numbers 62 25. Some problems 69 26. Some examples of two-variable generating functions 70 27. Binomial coefficients 71 28. Delannoy numbers 71 29. Some problems 74 30. Stirling numbers of the second kind 76 Date: Spring 2017; version of December 13, 2017. 1 2 DAVID GALVIN, DEPARTMENT OF MATHEMATICS, UNIVERSITY OF NOTRE DAME 31.
    [Show full text]
  • Combinatorics—
    The Ohio State University October 10, 2016 Combinatorics | 1 Functions and sets Enumerative combinatorics (or counting), at its heart, is all about finding functions between different sets in such a way that reveals their size. A little combinatorics goes a long way: you'll probably run into it in every discipline that involves numbers. Beyond numbers and sets, there is not much additional formal theory needed to get started. For sets A and B; a function f : A ! B is any assignment of elements of B defined for every element of A: All f needs to do to be a function from A to B is that there is a rule defined for obtaining f(a) 2 B for every element of a 2 A: In some situations, it can be non-obvious that a rule in unambiguous (we will see examples), in which case it may be necessary to prove that f is well-defined. We will be interested in counting, which motivates the following three definitions. Definition 1: A function f : A ! B is injective if and only if for all x; y 2 A; f(x) = f(y) ) x = y: These are sometimes called one-to-one functions. Think of injective functions as rules for placing A inside of B in such a way that we can undo our steps. Lemma 1: A function f : A ! B is injective if and only if there is a function g : B ! A so that 8x 2 A g(f(x)) = x: This function g is called a left-inverse for f Proof.
    [Show full text]
  • Homomesies Lurking in the Twelvefold Way
    Homomesies Lurking in the Twelvefold Way Tom Roby (UConn) Describing joint research with Michael Joseph & Michael LaCroix Special Session on Enumerative Combinatorics AMS Central Sectional Meeting University of St. Thomas Minneapolis, MN USA 28 October 2016 (Friday) Slides for this talk are available online (or will be soon) at http://www.math.uconn.edu/~troby/research.html Homomesies Lurking in the Twelvefold Way Tom Roby (UConn) Describing joint research with Michael Joseph & Michael LaCroix Special Session on Enumerative Combinatorics AMS Central Sectional Meeting University of St. Thomas Minneapolis, MN USA 28 October 2016 (Friday) Slides for this talk are available online (or will be soon) at http://www.math.uconn.edu/~troby/research.html Abstract Abstract: Given a group acting on a finite set of combinatorial objects, one can often find natural statistics on these objects which are homomesic, i.e., over each orbit of the action, the average value of the statistic is the same. Since the notion was codified a few years ago, homomesic statistics have been uncovered in a wide variety of situations within dynamical algebraic combinatorics. We discuss several examples lurking in Rota's Twelvefold Way related to actions on injections, surjections (joint work with Michael Joseph), and bijections/permutations (joint work with Michael LaCroix) of finite sets. Acknowledgments This seminar talk discusses joint work with Michael Joseph and Michael La Croix. Thanks to James Propp for suggesting the study of whirling and of Foatic actions, as well as earlier collaborations on the homomesy phenomenon. Please feel free to interrupt with questions or comments. Outline Actions, orbits, and homomesy; The Twelvefold Way; Foatic actions on Sn.
    [Show full text]
  • A Species Approach to the Twelvefold
    A species approach to Rota’s twelvefold way Anders Claesson 7 September 2019 Abstract An introduction to Joyal’s theory of combinatorial species is given and through it an alternative view of Rota’s twelvefold way emerges. 1. Introduction In how many ways can n balls be distributed into k urns? If there are no restrictions given, then each of the n balls can be freely placed into any of the k urns and so the answer is clearly kn. But what if we think of the balls, or the urns, as being identical rather than distinct? What if we have to place at least one ball, or can place at most one ball, into each urn? The twelve cases resulting from exhaustively considering these options are collectively referred to as the twelvefold way, an account of which can be found in Section 1.4 of Richard Stanley’s [8] excellent Enumerative Combinatorics, Volume 1. He attributes the idea of the twelvefold way to Gian-Carlo Rota and its name to Joel Spencer. Stanley presents the twelvefold way in terms of counting functions, f : U V, between two finite sets. If we think of U as a set of balls and V as a set of urns, then requiring that each urn contains at least one ball is the same as requiring that f!is surjective, and requiring that each urn contain at most one ball is the same as requiring that f is injective. To say that the balls are identical, or that the urns are identical, is to consider the equality of such functions up to a permutation of the elements of U, or up to a permutation of the elements of V.
    [Show full text]
  • Richard Stanley's Twelvefold Way
    Richard Stanley's Twelvefold Way August 31, 2009 Many combinatorial problems can be framed as counting the number of ways to allocate balls to urns, subject to various conditions. Richard Stanley invented the \twelvefold way" to organize these results into a table with twelve entries. See his book Enumerative Combinatorics, Volume 1. Let b represent the number of balls available and u the number of urns. The following table gives the number of ways to partition the balls among the urns according to the various states of labeled or unlabeled and subject to certain restrictions. The column headed \≤ 1" corresponds to requiring that there be no more than one ball in each urn. Similarly, the column headed \≥ 1" corresponds to requiring at least one ball in each urn. Balls Urns unrestricted ≤ 1 ≥ 1 b labeled labeled u (u)b u!S(b; u) unlabeled labeled u u u b b b-u labeled unlabeled u i=1 S(b; i) [b ≤ u] S(b; u) unlabeled unlabeled u P i=1 pi(b) [b ≤ u] pu(b) P For convenient cross referencing, we will refer to each of the cases by three symbols. The rst character is an l or a u depending on whether the balls 1 are labeled or unlabeled. The second character similarly indicates whether the urns are labeled or unlabeled. The nal character is one the regular expression symbols *, ?, or + indicating no restrictions, at most one ball per urn, and at least one ball per urn respectively. 1 Labeled balls, labeled urns, unrestricted (ll*) This is the number of b-tuples of u things.
    [Show full text]
  • The Twelvefold Way
    The Twelvefold Way Lukas Bulwahn February 23, 2021 Abstract This entry provides all cardinality theorems of the Twelvefold Way. The Twelvefold Way [1, 5, 6] systematically classifies twelve related combinatorial problems concerning two finite sets, which include count- ing permutations, combinations, multisets, set partitions and number partitions. This development builds upon the existing formal develop- ments [2, 3, 4] with cardinality theorems for those structures. It pro- vides twelve bijections from the various structures to different equiva- lence classes on finite functions, and hence, proves cardinality formulae for these equivalence classes on finite functions. Contents 1 Preliminaries 4 1.1 Additions to Finite Set Theory ................. 4 1.2 Additions to Equiv Relation Theory .............. 4 1.2.1 Counting Sets by Splitting into Equivalence Classes . 5 1.3 Additions to FuncSet Theory .................. 5 1.4 Additions to Permutations Theory ............... 6 1.5 Additions to List Theory ..................... 6 1.6 Additions to Disjoint Set Theory ................ 7 1.7 Additions to Multiset Theory .................. 7 1.8 Additions to Number Partitions Theory ............ 8 1.9 Cardinality Theorems with Iverson Function .......... 8 2 Main Observations on Operations and Permutations 9 2.1 Range Multiset .......................... 9 2.1.1 Existence of a Suitable Finite Function ........ 9 2.1.2 Existence of Permutation ................ 9 2.2 Domain Partition ......................... 9 2.2.1 Existence of a Suitable Finite Function ........ 9 2.2.2 Equality under Permutation Application ........ 10 2.2.3 Existence of Permutation ................ 10 2.3 Number Partition of Range ................... 11 1 2.3.1 Existence of a Suitable Finite Function ........ 11 2.3.2 Equality under Permutation Application .......
    [Show full text]
  • Discrete Structures I – Combinatorics
    Discrete Structures I – Combinatorics Lectures by Prof. Dr. Stefan Felsner Notes taken by Elisa Haubenreißer Summer term 2011 2 Contents 1 Introductory Examples 5 1.1 Derangements..................................... .. 5 1.2 TheOfficersProblem.................................. 9 2 Basic Counting 13 n 2.1 Models for k ...................................... 14 2.2 Extending binomial coefficients . ... 15 3 Fibonacci Numbers 17 3.1 Models for F -numbers .................................. 18 3.2 FormulasforFibonacciNumbers . .... 19 3.3 The -numberSystem.................................. 21 F 3.4 ContinuedFractionsandContinuants . ..... 22 4 The Twelvefold Way 25 5 Formal Power Series 33 5.1 Bernoullinumbersandsummation . ... 34 5.2 Compositionofseries................................. .. 35 5.3 RootsofFPS ....................................... 36 5.4 CatalanNumbers.................................... 37 6 Solvinglinearrecurrencesviageneratingfunctions 39 6.1 How to compute spanning trees of Gn ......................... 40 6.2 An example with exponential generating function . ....... 42 7 q-Enumeration 43 7.1 Another model for the q-binomial............................ 47 8 Finite Sets And Posets 49 8.1 Posets–Partiallyorderedsets. ..... 50 8.2 Thediagramofaposet ................................ 51 8.3 k-intersecting families . 52 8.4 Shadows – Another prooffor Sperner’s theorem . ......... 53 8.5 Erd¨os-Ko-RadofromKruskal-Katona. ........ 55 8.6 Symmetric chain decompositions . ... 57 8.7 An application: Dedekinds problem . ... 60 9 Duality
    [Show full text]
  • Lecture Notes Combinatorics
    Lecture Notes Combinatorics Lecture by Torsten Ueckerdt (KIT) Problem Classes by Jonathan Rollin (KIT) Lecture Notes by Stefan Walzer (TU Ilmenau) Last updated: July 19, 2016 1 Contents 0 What is Combinatorics?4 1 Permutations and Combinations 10 1.1 Basic Counting Principles...................... 10 1.1.1 Addition Principle...................... 10 1.1.2 Multiplication Principle................... 10 1.1.3 Subtraction Principle.................... 11 1.1.4 Bijection Principle...................... 11 1.1.5 Pigeonhole Principle..................... 11 1.1.6 Double counting....................... 12 1.2 Ordered Arrangements { Strings, Maps and Products...... 12 1.2.1 Permutations......................... 13 1.3 Unordered Arrangements { Combinations, Subsets and Multisets................ 14 1.4 Multinomial Coefficients....................... 16 1.5 The Twelvefold Way { Balls in Boxes............... 22 1.5.1 U L: n Unlabeled Balls in k Labeled Boxes...... 22 1.5.2 L ! U: n Labeled Balls in k Unlabeled Boxes...... 24 1.5.3 L ! L: n Labeled Balls in k Labeled Boxes....... 26 1.5.4 U ! U: n Unlabeled Balls in k Unlabeled Boxes.... 28 1.5.5 Summary:! The Twelvefold Way............... 29 1.6 Binomial Coefficients { Examples and Identities.......... 30 1.7 Permutations of Sets......................... 34 1.7.1 Cycle Decompositions.................... 35 1.7.2 Transpositions........................ 38 1.7.3 Derangements......................... 40 2 Inclusion-Exclusion-Principle and M¨obiusInversion 44 2.1 The Inclusion-Exclusion Principle.................. 44 2.1.1 Applications......................... 46 2.1.2 Stronger Version of PIE................... 51 2.2 M¨obiusInversion Formula...................... 52 3 Generating Functions 57 3.1 Newton's Binomial Theorem..................... 61 3.2 Exponential Generating Functions................. 62 3.3 Recurrence Relations........................
    [Show full text]
  • Let's Expand Rota's Twelvefold Way for Counting Partitions!
    (Submitted to the American Mathematical Monthly.) Let's Expand Rota's Twelvefold Way For Counting Partitions! Robert A. Proctor Department of Mathematics University of North Carolina Chapel Hill, North Carolina 27599 rap -at- email.unc.edu January 5, 2007 Dedicated to the memory of Gian-Carlo Rota. 1. Introduction Let's suppose we have a diamond, an emerald, and a ruby. How many ways are there to store these gems in small velvet bags? The bags are identical, and we can use any number of them. Unused bags are ignored. Since the gems can move around within the bags, we may as well list the gems within each bag in alphabetical order. Since the bags are identical, we may as well record the bags in alphabetical order, according to their first elements. Using inner {}'s for the bags and outer {}'s to collect the bags, here are the possibilities: { {D}, {E}, {R} }, { {D,E}, {R} }, { {D,R}, {E} }, { {D}, {E,R} }, { {D,E,R} }. So there are 5 storage configurations. These are the "partitions" of the set {D, E, R}. For m ≥ 0, let B(m) denote the number of partitions of a set of m distinguishable items. So B(3) = 5. The counts B(0), B(1), B(2), B(3), … are called the "Bell numbers". Surprisingly, these numbers also arise in a Maclaurin series expansion: ex–1 xm e = ∑m≥0B(m)——— . m! There are dozens of variations of this notion of partition. If autographed baseballs were stored in socks, the order in which the baseballs were shoved into each sock would matter.
    [Show full text]
  • Lecture #17: Distributions (Tucker Section 5.4) David Mix Barrington 21 October 2016 Distributions
    CMPSCI 575/MATH 513 Combinatorics and Graph Theory Lecture #17: Distributions (Tucker Section 5.4) David Mix Barrington 21 October 2016 Distributions • Objects in Boxes • Unlimited Repetition: All Functions • No Repetition: One-to-One Functions • Restricted Repetition • Counting Onto Functions • The Twelvefold Way Objects in Boxes • Today we’re going to look at a new framework to describe our basic counting problems, that of distributions. • The general picture is that we have r objects that we are going to place into n boxes. We are thus choosing a function f: O → B, where O has size r and B has size n. • But there are multiple versions of the problem of counting these functions, depending on our definitions. Objects in Boxes • Suppose the objects in O are labeled with the numbers 1,…,r. (They are distinct.) Then if we put object 1 in box f(1), 2 in f(2), and so forth, our function is defined by the sequence of numbers f(1)f(2)…f(r). • But if the objects are identical, then each of our functions is characterized by just the number of objects we put in each box. Examples of Objects in Boxes • Suppose we have 100 diplomats who must each be assigned to one of five continents. If the diplomats are distinct, there are 5100 ways to do this. • If we further insist that there be 20 diplomats assigned to each continent, we can consider that the 100 diplomats are being mapped into 100 positions by a bijection, in one of 100! ways, and then correct for the over counting to get 100!/ (20!)5, which Tucker also writes as P(100; 20, 20, 20, 20, 20).
    [Show full text]
  • Combinatorics: the Art of Counting
    Prepublication copy provided to Dr Bruce Sagan. Please give confirmation to AMS by September 21, 2020. Combinatorics: The Art of Counting Not for print or electronic distribution. This file may not be posted electronically. Prepublication copy provided to Dr Bruce Sagan. Please give confirmation to AMS by September 21, 2020. Not for print or electronic distribution. This file may not be posted electronically. Prepublication copy provided to Dr Bruce Sagan. Please give confirmation to AMS by September 21, 2020. GRADUATE STUDIES IN MATHEMATICS 210 Combinatorics: The Art of Counting Bruce E. Sagan Not for print or electronic distribution. This file may not be posted electronically. Prepublication copy provided to Dr Bruce Sagan. Please give confirmation to AMS by September 21, 2020. Marco Gualtieri Bjorn Poonen Gigliola Staffilani (Chair) Jeff A. Viaclovsky 2020 Mathematics Subject Classification. Primary 05-01; Secondary 06-01. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-210 Library of Congress Cataloging-in-Publication Data Names: Sagan, Bruce Eli, 1954- author. Title: Combinatorics : the art of counting / Bruce E. Sagan. Description: Providence, Rhode Island : American Mathematical Society, [2020] | Series: Gradu- ate studies in mathematics, 1065-7339 ; 210 | Includes bibliographical references and index. | Identifiers: LCCN 2020025345 | ISBN 9781470460327 (paperback) | ISBN 9781470462802 (ebook) Subjects: LCSH: Combinatorial analysis–Textbooks. | AMS: Combinatorics – Instructional ex- position (textbooks, tutorial papers, etc.). | Order, lattices, ordered algebraic structures – Instructional exposition (textbooks, tutorial papers, etc.). Classification: LCC QA164 .S24 2020 | DDC 511/.6–dc23 LC record available at https://lccn.loc.gov/2020025345 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research.
    [Show full text]