On the Density of Languages Representing Finite Set Partitions 1
|
|
- Justina Natalie Johnston
- 6 years ago
- Views:
Transcription
1 Journal of Integer Sequences, Vol , Article On the Density of Languages Representing Finite Set Partitions Nelma Moreira and Rogério Reis DCC-FC & LIACC Universidade do Porto R. do Campo Alegre Porto Portugal nam@ncc.up.pt rvr@ncc.up.pt Abstract We present a family of regular languages representing partitions of a set of n elements in less or equal c parts. The density of those languages is given by partial sums of Stirling numbers of second kind for which we obtain explicit formulas. We also determine the limit frequency of those languages. This work was motivated by computational representations of the configurations of some numerical games. The languages L c Consider a game where natural numbers are to be placed, by increasing order, in a fixed number of columns, subject to some specific constraints. In these games column order is irrelevant. Numbering the columns, game configurations can be seen as sequences of column numbers where the successive integers are placed. For instance, the string 23 stands for a configuration where, 2, 4 were placed in the first column, 3 was placed in the second and 5 was placed in the third. Because column order is irrelevant, and to have a unique representation for each configuration, it is not allowed to place an integer in the Work partially funded by Fundação para a Ciência e Tecnologia FCT and Program POSI.
2 kth column if the k th is still empty, for any k >. Blanchard and al. [BHR04] and Reis and al. [RMP04] used this kind of representation to study the possible configurations of sum-free games. Given c columns, let N c = {,..., c}. We are interested in studying the set of game configurations as strings in N c, i.e., in the set of finite sequences of elements of N c. Game configurations can be characterised by the following language L c N c : L c = {a a 2 a k N c i N k, a i max{a,..., a i } + }. For c = 4, there are only 5 strings in L 4 of length 4, instead of the total possible 256 in N 4 4 : Given a finite set Σ, a regular expression r.e. α over Σ represents a regular language Lα Σ and is inductively defined by: is a r.e and L = ; ɛ empty string is a r.e and Lɛ = {ɛ}; a Σ is a r.e and La = {a}; if α and α 2 are r.e., α +α 2, α α 2 and α are r.e., respectively with Lα + α 2 = Lα Lα 2, Lα α 2 = Lα Lα 2 and Lα = Lα, where we assume the usual precedence of the operators see [HMU00]. A regular expression α is unambiguous if for each w Lα there is only one path through α that matches w. Theorem.. For all c, L c is a regular language. Proof. For c =, we have L = L. We define by induction on c, a family of regular expressions: It is trivial to see that For instance, α 4 is α =, c α c = α c + j + + j. 2 α c = i= j= j= i j + + j It is also obvious that Lα c Lα c, for c >. For any c, we prove that L c = Lα c. L c Lα c : If x Lα c it is obvious that x L c. L c Lα c : By induction on the length of x L c : If x = then x Lα Lα c. Suppose that for any string x of length n, x Lα c. Let x = n + and x = ya, where a N c and y Lα c. Let c = max{a i a i y}. If c = c, obviously x Lα c. If c < c, then y Lα c, and x Lα c + Lα c. 2
3 2 Counting the strings of L c The density of a language L over a finite set Σ, ρ L n, is the number of strings of length n that are in L, i.e., In particular, the density of L c is ρ L n = L Σ n. ρ Lc n = L c N n c. Using generating functions we can determine a closed form for ρ Lc n. Recall that, a ordinary generating function for a sequence {a n } is a formal series see [GKP94] Gz = a n z n. i=0 If Az and Bz are generating functions for the density functions of the languages represented by unambiguous regular expressions A and B, and A + B, AB and A are also unambiguous r.e., we have that Az + Bz, AzBz and, are the generating Az functions for the density functions of the corresponding languages see [SF96], page 378. As α c are unambiguous regular expressions, from 3, we obtain the following generating function for {ρ Lc n}: Notice that T c z = i i= j= z jz = S i z = i j= z i i= i j= jz z i jz. are the generating functions for the Stirling numbers of second kind Sn, i = i i j i j n, i! j j=0 which are, for each n, the number of ways of partitioning a set of n elements into i nonempty sets see [GKP94] and A Then, a closed form for the density of L c, ρ Lc n, is given by ρ Lc n = Sn, i, 4 i= i.e., a partial sum of Stirling numbers of second kind. In Table we present the values of ρ Lc n, for c =..8 and n =..3. For some sequences, we also indicate the corresponding number in Sloane s On-Line Encyclopedia 3
4 of Integer Sequences [Slo03]. The closed forms were calculated using the Maple computer algebra system [Hec03]. From expression 4, it is also easy to see that Theorem 2.. lim ρ L c n = B n, c where B n are the Bell numbers, i.e., for each n, the number of ways a set of n elements can be partitioned into nonempty subsets. Proof. Bell numbers, B n, can be defined by the sum n B n = Sn, i. i= And, as Sn, i = 0 for i > n, we have n lim ρ L c n = Sn, i + lim c i= c i=n+ Sn, i = B n. In Table, for each c, the subsequence for n c coincides with the first c elements of B n A0000. Moreover, we can express ρ Lc n, and then the partial sums of Stirling numbers of second kind, as a generic linear combination of nth powers of k, for k N c. Let S j n, i denote the jth term in the summation of a Stirling number Sn, i, i.e., S j n, i = i i! j i j n. j Lemma 2.. For all n and 0 i n, Proof. S n, i + = S 0 n, i = S n, i +. 5 i +! i + i n = i! in = S 0 n, i. Applying 5 in the summation 4 of ρ Lc n, each term Sn, i simplifies the subterm S n, i with the subterm S 0 n, i, for i 2. We obtain ρ L n = S 0 n,, ρ L2 n = S 0 n, 2, ρ Lc n = S 0 n, c + i S j n, i, for c > 2; i=3 = cn i c! + i=3 S j n, i, for c > 2. 4
5 c ρ Lc n OEIS,,,,,,,,,,,,,, n, 2, 4, 8, 6, 32, 64, 28, 256, 52, 024, 2048, 4096, 892, n + 2, 2, 5, 4, 4, 22, 365, 094, 328, 9842, 29525, 88574, 26572,... A n + 4 2n + 3, 2, 5, 5, 5, 87, 75, 2795, 05, 43947, 75275, , ,... A n + 2 3n + 6 2n + 3 8, 2, 5, 5, 52, 202, 855, 3845, 8002, 86472, , , ,... A n n + 8 3n n + 30, 2, 5, 5, 52, 203, 876, 4, 20648, 09299, 60492, , ,... A n n n + 6 3n n , 2, 5, 5, 52, 203, 877, 439, 20, 579, , , ,... A n n n n n n , 2, 5, 5, 52, 203, 877, 440, 246, 5929, , , ,... A Table : Density functions of L c, for c =..8. 5
6 If the sums are rearranged such that i = k + j, we have ρ Lc n = cn c 2 c! + c k S j n, k + j, 6 k= where S j n, k + j = k n k + j k + j j 7 = kn k! j. 8 Replacing 8 into equation 6 we get ρ Lc n = cn c 2 c! + k= k n c k k! j. 9 In equation 9, the coefficients of k n, k c, can be calculated using the following recurrence relation: And, we have γ = ; γ c = γ c + c, for c > ; c! γk c = γc k, for c > and 2 k c. k Theorem 2.2. For all c, ρ Lc n = γkk c n. 0 k= From the expression 0, the closed forms in Table are easily derived. Finally, we can obtain the limit frequency of L c in N c. Since ρ Nc n = cn, and lim n k c n = 0, for k c 2, we have Theorem 2.3. For all c, lim n ρ Lc n ρ Nc n = c!. 6
7 3 A bijection between strings of L c and partitions of finite sets The connection between the density of L c and Stirling numbers of second kind is not accidental. Each string of L c with length n corresponds to a partition of N n with no more than c parts. This correspondence can be made explicit as follows. Let a a 2 a n be a string of L c. This string corresponds to the partition {A j } j Nc of N n with c = max{a,..., a n }, such that for each i N n, i A ai. For example, the string 23 corresponds to the partition {{, 2}, {3}, {4}} of N 4 into 3 parts. This defines a bijection. That each string corresponds to a unique partition is obvious. Given a partition {A j } j Nc of N n with c c, we can construct the string b b n, such that for i N n, b i = j if i A j. For the partition {{, 2}, {3}, {4}}, we obtain Counting the strings of L c of length equal or less than a certain value Although strings of L c of arbitrary length represent game configurations, for computational reasons 2 we consider all game configurations with the same length, padding with zeros the positions of integers not yet in one of the c columns. In this way, we obtain the languages L 0 c = L c {0 }. So, determining the number of strings of length equal or less than n that are in L c is tantamount to determining the density of L 0 c, i.e., n = L 0 c {0} N c n. As seen in Section 2, and because L c {0 } = Lα c 0, the generating function T cz of n can be obtained as the product of a generating function for ρ Lc n, T c z, by a generating function for ρ {0} n, e.g., z. Thus, the generating function for {ρ L 0 n} is c T cz = T c z z = z i z i j= jz and a closed form for n is as expected n = i= n m= i= where m starts at because Sm, i = 0, for i > m. 2 The data structures used in the programs are arrays of fixed length. Sm, i, 7
8 Using expression 9 in we have ρ L 0 n = n; ρ L 0 2 n = 2 n ; n n = m= c m c! + c 2 k= k m k! = cn+ c c c c! + n = c k j c n c c c! + n If we use the equation 0, we have Theorem 4.. For all c, Proof. j, for c > 2; c 2 k n+ k c k + k k! j n = nγ c + + c 2 j k n k k! γk ckn+ k. k c k j. n = n γkk c m = m= k= γk c k= m= n k m = nγ c + γk ckn+ k. k In the Table 2 we present the values of n, for c =..8 and n =..3. As before, the limiting sequence as c is the sequence of partial sums of Bell numbers. Finally, we determine the limit frequency of L 0 c in N c {0}. Notice that ρ Nc {0} n = cn+ c, as it is a sum of the first n terms of a geometric progression of ratio c. We have, Theorem 4.2. For all c, lim n n ρ Nc {0} n = c!. 8
9 c n OEIS n, 2, 3, 4, 5, 6, 7, 8, 9, 0,, 2, 3, n, 3, 7, 5, 3, 63, 27, 255, 5, 023, 2047, 4095, 89,... A n + 2 n 4, 3, 8, 22, 63, 85, 550, 644, 4925, 4767, 44292, 32866, ,... A n + 2 2n + 3 n 5 9, 3, 8, 23, 74, 26, 976, 377, 4822, 58769, , 9349, , n + 8 3n + 3 2n n 5 32, 3, 8, 23, 75, 277, 32, 4977, 22979, 0945, 53456, 26093, ,... A n n + 2 3n n + 30 n , 3, 8, 23, 75, 278, 54, 5265, 2593, 3522, , 43983, ,... A n n n n n n 3 64, 3, 8, 23, 75, 278, 55, 5294, 26404, 4583, , , , n n n n n n n , 3, 8, 23, 75, 278, 55, 5295, 2644, 42370, 89729, , ,... Table 2: Density functions of L 0 c, for c =..8. 9
10 Proof. lim n n ρ Nc {0} n = lim n cn+ cc nc c j c c!c n+ + c n+ c 2 + lim k n+ kc c k j n k k!c n+ = c lim c! n lim c n+ n c n+ c j nc + lim n c n+ = c!. c 2 + c c k j k k! lim n k c n c c n c n+ 5 Conclusion In this note we presented a family of regular languages representing finite set partitions and studied their densities. Although it is well-known that the number of partitions of a set of n elements into no more than c nonempty sets is given by partial sums of Stirling numbers of second kind, we determined explicit formulas for their closed forms, as linear combinations of k n, for k N c. We also determined the limit frequency of those languages, which gives an estimate of the space saved with those representations. Acknowledgements We thank Miguel Filgueiras and the referee for their comments in previous versions of the manuscript. We also thank J. Shallit for pointing us a connection between the density of L c and the Bell numbers. References [BHR04] Peter Blanchard, Frank Harary, and Rogério Reis, Partitions into sum-free sets, Integers: Electronic Journal of Combinatorial Number Theory, to appear, [GKP94] Roland Graham, Donald Knuth, and Oren Patashnik. Addison-Wesley, 994. Concrete Mathematics. [Hec03] Andri Heck. Introduction to Maple. Springer, 3rd edition,
11 [HMU00] John E. Hopcroft, Rajeev Motwani, and Jeffrey D. Ullman. Introduction to Automata Theory, Languages and Computation. Addison-Wesley, 2nd edition, [RMP04] Rogério Reis, Nelma Moreira, and João Pedro Pedroso. Educated brute-force to get h4. Technical Report DCC , DCC-FC& LIACC, Universidade do Porto, July [SF96] Robert Sedgewick and Philippe Flajolet. Analysis of Algorithms. Addison-Wesley, 996. [Slo03] N.J.A. Sloane. The On-line Encyclopedia of Integer Sequences, njas/sequences Mathematics Subject Classification: Primary 05A5; Secondary 05A8, B73, 68Q45. Keywords: Partions of sets, Stirling numbers, regular languages, enumeration Concerned with sequences A0000 A A00705 A00758 A A A A A A A A Received October ; revised version received April Published in Journal of Integer Sequences, May Return to Journal of Integer Sequences home page.
On the density of languages representing finite set partitions
On the density of languages representing finite set partitions Nelma Moreira Rogério Reis Tehnial Report Series: DCC-04-07 Departamento de Ciênia de Computadores Fauldade de Ciênias & Laboratório de Inteligênia
More informationDescriptional Complexity of Formal Systems (Draft) Deadline for submissions: March 20, 2007 Final versions: May 30, 2007
DCFS 2007 Descriptional Complexity of Formal Systems (Draft) Deadline for submissions: March 20, 2007 Final versions: May 30, 2007 Exact Generation of Minimal Acyclic Deterministic Finite Automata Marco
More informationExact Generation of Minimal Acyclic Deterministic Finite Automata
Exact Generation of Minimal Acyclic Deterministic Finite Automata Marco Almeida Nelma Moreira Rogério Reis LIACC, Faculdade de Ciências, Universidade do Porto Departamento de Ciência de Computadores Rua
More informationCombinatorial Interpretations of a Generalization of the Genocchi Numbers
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.6 Combinatorial Interpretations of a Generalization of the Genocchi Numbers Michael Domaratzki Jodrey School of Computer Science
More informationSequences that satisfy a(n a(n)) = 0
Sequences that satisfy a(n a(n)) = 0 Nate Kube Frank Ruskey October 13, 2005 Abstract We explore the properties of some sequences for which a(n a(n)) = 0. Under the natural restriction that a(n) < n the
More informationA REFINED ENUMERATION OF p-ary LABELED TREES
Korean J. Math. 21 (2013), No. 4, pp. 495 502 http://dx.doi.org/10.11568/kjm.2013.21.4.495 A REFINED ENUMERATION OF p-ary LABELED TREES Seunghyun Seo and Heesung Shin Abstract. Let T n (p) be the set of
More informationPermutations with Inversions
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 4 2001, Article 01.2.4 Permutations with Inversions Barbara H. Margolius Cleveland State University Cleveland, Ohio 44115 Email address: b.margolius@csuohio.edu
More informationOn the Partitions of a Number into Arithmetic Progressions
1 3 47 6 3 11 Journal of Integer Sequences, Vol 11 (008), Article 0854 On the Partitions of a Number into Arithmetic Progressions Augustine O Munagi and Temba Shonhiwa The John Knopfmacher Centre for Applicable
More informationAcyclic Automata with easy-to-find short regular expressions
Acyclic Automata with easy-to-find short regular expressions José João Morais Nelma Moreira Rogério Reis Technical Report Series: DCC-2005-03 Departamento de Ciência de Computadores Faculdade de Ciências
More informationAntimirov and Mosses s rewrite system revisited
Antimirov and Mosses s rewrite system revisited Marco Almeida Nelma Moreira Rogério Reis LIACC, Faculdade de Ciências, Universidade do Porto Departamento de Ciência de Computadores Rua do Campo Alegre,
More informationSome thoughts concerning power sums
Some thoughts concerning power sums Dedicated to the memory of Zoltán Szvetits (929 20 Gábor Nyul Institute of Mathematics, University of Debrecen H 00 Debrecen POBox 2, Hungary e mail: gnyul@scienceunidebhu
More informationTernary Modified Collatz Sequences And Jacobsthal Numbers
1 47 6 11 Journal of Integer Sequences, Vol. 19 (016), Article 16.7.5 Ternary Modified Collatz Sequences And Jacobsthal Numbers Ji Young Choi Department of Mathematics Shippensburg University of Pennsylvania
More informationEVALUATION OF A FAMILY OF BINOMIAL DETERMINANTS
EVALUATION OF A FAMILY OF BINOMIAL DETERMINANTS CHARLES HELOU AND JAMES A SELLERS Abstract Motivated by a recent work about finite sequences where the n-th term is bounded by n, we evaluate some classes
More informationGenerating All Permutations by Context-Free Grammars in Chomsky Normal Form
Generating All Permutations by Context-Free Grammars in Chomsky Normal Form Peter R.J. Asveld Department of Computer Science, Twente University of Technology P.O. Box 217, 7500 AE Enschede, the Netherlands
More informationCounting Palindromes According to r-runs of Ones Using Generating Functions
3 47 6 3 Journal of Integer Sequences, Vol. 7 (04), Article 4.6. Counting Palindromes According to r-runs of Ones Using Generating Functions Helmut Prodinger Department of Mathematics Stellenbosch University
More informationCounting k-marked Durfee Symbols
Counting k-marked Durfee Symbols Kağan Kurşungöz Department of Mathematics The Pennsylvania State University University Park PA 602 kursun@math.psu.edu Submitted: May 7 200; Accepted: Feb 5 20; Published:
More informationThe Average Transition Complexity of Glushkov and Partial Derivative Automata
The Average Transition Complexity of Glushkov and Partial Derivative Automata Sabine Broda, António Machiavelo, Nelma Moreira, Rogério Reis sbb@dcc.fc.up.pt,ajmachia@fc.up.pt,{nam,rvr}@dcc.fc.up.pt CMUP,
More informationA Note about the Pochhammer Symbol
Mathematica Moravica Vol. 12-1 (2008), 37 42 A Note about the Pochhammer Symbol Aleksandar Petoević Abstract. In this paper we give elementary proofs of the generating functions for the Pochhammer symbol
More informationMath 324 Summer 2012 Elementary Number Theory Notes on Mathematical Induction
Math 4 Summer 01 Elementary Number Theory Notes on Mathematical Induction Principle of Mathematical Induction Recall the following axiom for the set of integers. Well-Ordering Axiom for the Integers If
More informationGeneralized Akiyama-Tanigawa Algorithm for Hypersums of Powers of Integers
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol 16 (2013, Article 1332 Generalized Aiyama-Tanigawa Algorithm for Hypersums of Powers of Integers José Luis Cereceda Distrito Telefónica, Edificio Este
More informationGenerating All Permutations by Context-Free Grammars in Chomsky Normal Form
Generating All Permutations by Context-Free Grammars in Chomsky Normal Form Peter R.J. Asveld Department of Computer Science, Twente University of Technology P.O. Box 217, 7500 AE Enschede, the Netherlands
More informationCS 275 Automata and Formal Language Theory
CS 275 Automata and Formal Language Theory Course Notes Part II: The Recognition Problem (II) Chapter II.4.: Properties of Regular Languages (13) Anton Setzer (Based on a book draft by J. V. Tucker and
More informationClimbing an Infinite Ladder
Section 5.1 Section Summary Mathematical Induction Examples of Proof by Mathematical Induction Mistaken Proofs by Mathematical Induction Guidelines for Proofs by Mathematical Induction Climbing an Infinite
More informationApplication of Logic to Generating Functions. Holonomic (P-recursive) Sequences
Application of Logic to Generating Functions Holonomic (P-recursive) Sequences Johann A. Makowsky Faculty of Computer Science, Technion - Israel Institute of Technology, Haifa, Israel http://www.cs.technion.ac.il/
More informationSome Examples of Lexicographic Order Algorithms and some Open Combinatorial Problems
Some Examples of Lexicographic Order Algorithms and some Open Combinatorial Problems Dimitar Vandev A general reasoning based on the lexicographic order is studied. It helps to create algorithms for generation
More informationInvestigating Geometric and Exponential Polynomials with Euler-Seidel Matrices
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 14 (2011), Article 11.4.6 Investigating Geometric and Exponential Polynomials with Euler-Seidel Matrices Ayhan Dil and Veli Kurt Department of Mathematics
More informationAUGUSTINE O. MUNAGI. Received 9 September 2004
k-complementing SUBSETS OF NONNEGATIVE INTEGERS AUGUSTINE O. MUNAGI Received 9 September 2004 Acollection{S 1,S 2,...} of nonempty sets is called a complementing system of subsets for a set X of nonnegative
More informationLinks Between Sums Over Paths in Bernoulli s Triangles and the Fibonacci Numbers
Links Between Sums Over Paths in Bernoulli s Triangles and the Fibonacci Numbers arxiv:1611.09181v1 [math.co] 28 Nov 2016 Denis Neiter and Amsha Proag Ecole Polytechnique Route de Saclay 91128 Palaiseau
More informationMATH 324 Summer 2011 Elementary Number Theory. Notes on Mathematical Induction. Recall the following axiom for the set of integers.
MATH 4 Summer 011 Elementary Number Theory Notes on Mathematical Induction Principle of Mathematical Induction Recall the following axiom for the set of integers. Well-Ordering Axiom for the Integers If
More informationAN EXPLICIT FORMULA FOR BERNOULLI POLYNOMIALS IN TERMS OF r-stirling NUMBERS OF THE SECOND KIND
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 46, Number 6, 2016 AN EXPLICIT FORMULA FOR BERNOULLI POLYNOMIALS IN TERMS OF r-stirling NUMBERS OF THE SECOND KIND BAI-NI GUO, ISTVÁN MEZŐ AND FENG QI ABSTRACT.
More informationSloping Binary Numbers: A New Sequence Related to the Binary Numbers
Sloping Binary Numbers: A New Sequence Related to the Binary Numbers David Applegate, Internet and Network Systems Research Center, AT&T Shannon Labs, 180 Park Avenue, Florham Park, NJ 0793 0971, USA (Email:
More informationAn Application of Catalan Numbers on Cayley Tree of Order 2: Single Polygon Counting
BULLETIN of the Malaysian Mathematical Sciences Society http://math.usm.my/bulletin Bull. Malays. Math. Sci. Soc. (2) 31(2) (2008), 175 183 An Application of Catalan Numbers on Cayley Tree of Order 2:
More informationLengyel's Constant Page 1 of 5
Lengyel's Constant Page 1 of 5 Lengyel's Constant Set Partitions and Bell Numbers Let S be a set with n elements. The set of all subsets of S has elements. By a partition of S we mean a disjoint set of
More informationarxiv: v1 [math.co] 4 Mar 2010
NUMBER OF COMPOSITIONS AND CONVOLVED FIBONACCI NUMBERS arxiv:10030981v1 [mathco] 4 Mar 2010 MILAN JANJIĆ Abstract We consider two type of upper Hessenberg matrices which determinants are Fibonacci numbers
More informationOn the Number of Non-Equivalent Linear Transducers
On the Number of Non-Equivalent Linear Transducers Ivone Amorim António Machiavelo Rogério Reis email: {ivone.amorim,rvr}@dcc.fc.up.pt, ajmachia@fc.up.pt DCC-FC & CMUP, Universidade do Porto Rua do Campo
More informationMOMENTS OF INERTIA ASSOCIATED WITH THE LOZENGE TILINGS OF A HEXAGON
Séminaire Lotharingien de Combinatoire 45 001, Article B45f MOMENTS OF INERTIA ASSOCIATED WITH THE LOZENGE TILINGS OF A HEXAGON ILSE FISCHER Abstract. Consider the probability that an arbitrary chosen
More informationOn the indecomposability of polynomials
On the indecomposability of polynomials Andrej Dujella, Ivica Gusić and Robert F. Tichy Abstract Applying a combinatorial lemma a new sufficient condition for the indecomposability of integer polynomials
More informationThe Gift Exchange Problem
The Gift Exchange Problem David Applegate and N. J. A. Sloane (a), AT&T Shannon Labs, 180 Park Ave., Florham Park, NJ 07932-0971, USA. (a) Corresponding author. Email: david@research.att.com, njas@research.att.com.
More informationStirling Numbers of the 1st Kind
Daniel Reiss, Colebrook Jackson, Brad Dallas Western Washington University November 28, 2012 Introduction The set of all permutations of a set N is denoted S(N), while the set of all permutations of {1,
More informationCounting Palindromic Binary Strings Without r-runs of Ones
1 3 47 6 3 11 Journal of Integer Sequences, Vol. 16 (013), Article 13.8.7 Counting Palindromic Binary Strings Without r-runs of Ones M. A. Nyblom School of Mathematics and Geospatial Science RMIT University
More informationConstant Time Generation of Set Partitions
930 PAPER Special Section on Selected Papers from the 17th Workshop on Circuits and Systems in Karuizawa Constant Time Generation of Set Partitions Shin-ichiro KAWANO a) and Shin-ichi NAKANO b), Members
More informationPartition of Integers into Distinct Summands with Upper Bounds. Partition of Integers into Even Summands. An Example
Partition of Integers into Even Summands We ask for the number of partitions of m Z + into positive even integers The desired number is the coefficient of x m in + x + x 4 + ) + x 4 + x 8 + ) + x 6 + x
More informationCSCE 222 Discrete Structures for Computing. Dr. Hyunyoung Lee
CSCE 222 Discrete Structures for Computing Sequences and Summations Dr. Hyunyoung Lee Based on slides by Andreas Klappenecker 1 Sequences 2 Sequences A sequence is a function from a subset of the set of
More informationON DIVISIBILITY OF SOME POWER SUMS. Tamás Lengyel Department of Mathematics, Occidental College, 1600 Campus Road, Los Angeles, USA.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 (007, #A4 ON DIVISIBILITY OF SOME POWER SUMS Tamás Lengyel Department of Mathematics, Occidental College, 600 Campus Road, Los Angeles, USA
More informationThe Generating Functions for Pochhammer
The Generating Functions for Pochhammer Symbol { }, n N Aleksandar Petoević University of Novi Sad Teacher Training Faculty, Department of Mathematics Podgorička 4, 25000 Sombor SERBIA and MONTENEGRO Email
More informationGeneralized Near-Bell Numbers
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 12 2009, Article 09.5.7 Generalized Near-Bell Numbers Martin Griffiths Department of Mathematical Sciences University of Esse Wivenhoe Par Colchester
More informationarxiv: v1 [math.co] 22 May 2014
Using recurrence relations to count certain elements in symmetric groups arxiv:1405.5620v1 [math.co] 22 May 2014 S.P. GLASBY Abstract. We use the fact that certain cosets of the stabilizer of points are
More informationMergible States in Large NFA
Mergible States in Large NFA Cezar Câmpeanu a Nicolae Sântean b Sheng Yu b a Department of Computer Science and Information Technology University of Prince Edward Island, Charlottetown, PEI C1A 4P3, Canada
More informationCOMPLEMENTARY FAMILIES OF THE FIBONACCI-LUCAS RELATIONS. Ivica Martinjak Faculty of Science, University of Zagreb, Zagreb, Croatia
#A2 INTEGERS 9 (209) COMPLEMENTARY FAMILIES OF THE FIBONACCI-LUCAS RELATIONS Ivica Martinjak Faculty of Science, University of Zagreb, Zagreb, Croatia imartinjak@phy.hr Helmut Prodinger Department of Mathematics,
More informationPolyexponentials. Khristo N. Boyadzhiev Ohio Northern University Departnment of Mathematics Ada, OH
Polyexponentials Khristo N. Boyadzhiev Ohio Northern University Departnment of Mathematics Ada, OH 45810 k-boyadzhiev@onu.edu 1. Introduction. The polylogarithmic function [15] (1.1) and the more general
More informationThe Number of Inversions in Permutations: A Saddle Point Approach
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 6 (23), Article 3.2.8 The Number of Inversions in Permutations: A Saddle Point Approach Guy Louchard Département d Informatique CP 212, Boulevard du
More informationA LINEAR BINOMIAL RECURRENCE AND THE BELL NUMBERS AND POLYNOMIALS
Applicable Analysis and Discrete Mathematics, 1 (27, 371 385. Available electronically at http://pefmath.etf.bg.ac.yu A LINEAR BINOMIAL RECURRENCE AND THE BELL NUMBERS AND POLYNOMIALS H. W. Gould, Jocelyn
More informationEnumerating Binary Strings
International Mathematical Forum, Vol. 7, 2012, no. 38, 1865-1876 Enumerating Binary Strings without r-runs of Ones M. A. Nyblom School of Mathematics and Geospatial Science RMIT University Melbourne,
More informationTHE LOG-BEHAVIOR OF THE SEQUENCE FOR THE PARTIAL SUM OF A LOG-CONVEX SEQUENCE. 1. Introduction
SARAJEVO JOURNAL OF MATHEMATICS Vol.13 (26), No.2, (2017), 163 178 DOI: 10.5644/SJM.13.2.04 THE LOG-BEHAVIOR OF THE SEQUENCE FOR THE PARTIAL SUM OF A LOG-CONVEX SEQUENCE FENG-ZHEN ZHAO Abstract. In this
More informationOn Certain Sums of Stirling Numbers with Binomial Coefficients
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 18 (2015, Article 15.9.6 On Certain Sums of Stirling Numbers with Binomial Coefficients H. W. Gould Department of Mathematics West Virginia University
More informationHere, logx denotes the natural logarithm of x. Mrsky [7], and later Cheo and Yien [2], proved that iz;.(»)=^io 8 "0(i).
Curtis Cooper f Robert E* Kennedy, and Milo Renberg Dept. of Mathematics, Central Missouri State University, Warrensburg, MO 64093-5045 (Submitted January 1997-Final Revision June 1997) 0. INTRODUCTION
More informationTransformation formulas for the generalized hypergeometric function with integral parameter differences
Transformation formulas for the generalized hypergeometric function with integral parameter differences A. R. Miller Formerly Professor of Mathematics at George Washington University, 66 8th Street NW,
More informationPropositional Logic, Predicates, and Equivalence
Chapter 1 Propositional Logic, Predicates, and Equivalence A statement or a proposition is a sentence that is true (T) or false (F) but not both. The symbol denotes not, denotes and, and denotes or. If
More informationCounting Peaks and Valleys in a Partition of a Set
1 47 6 11 Journal of Integer Sequences Vol. 1 010 Article 10.6.8 Counting Peas and Valleys in a Partition of a Set Toufi Mansour Department of Mathematics University of Haifa 1905 Haifa Israel toufi@math.haifa.ac.il
More informationGenerating All Circular Shifts by Context-Free Grammars in Chomsky Normal Form
Generating All Circular Shifts by Context-Free Grammars in Chomsky Normal Form Peter R.J. Asveld Department of Computer Science, Twente University of Technology P.O. Box 217, 7500 AE Enschede, the Netherlands
More informationInternational Journal of Pure and Applied Mathematics Volume 21 No , THE VARIANCE OF SAMPLE VARIANCE FROM A FINITE POPULATION
International Journal of Pure and Applied Mathematics Volume 21 No. 3 2005, 387-394 THE VARIANCE OF SAMPLE VARIANCE FROM A FINITE POPULATION Eungchun Cho 1, Moon Jung Cho 2, John Eltinge 3 1 Department
More informationLEVEL GENERATING TREES AND PROPER RIORDAN ARRAYS
Applicable Analysis and Discrete Mathematics, (008), 69 9 Available electronically at http://pefmathetfbgacyu LEVEL GENERATING TREES AND PROPER RIORDAN ARRAYS D Baccherini, D Merlini, R Sprugnoli We introduce
More informationSums of Squares and Products of Jacobsthal Numbers
1 2 47 6 2 11 Journal of Integer Sequences, Vol. 10 2007, Article 07.2.5 Sums of Squares and Products of Jacobsthal Numbers Zvonko Čerin Department of Mathematics University of Zagreb Bijenička 0 Zagreb
More informationEnumerating split-pair arrangements
Journal of Combinatorial Theory, Series A 115 (28 293 33 www.elsevier.com/locate/jcta Enumerating split-pair arrangements Ron Graham 1, Nan Zang Department of Computer Science, University of California
More informationSection Summary. Sequences. Recurrence Relations. Summations. Examples: Geometric Progression, Arithmetic Progression. Example: Fibonacci Sequence
Section 2.4 Section Summary Sequences. Examples: Geometric Progression, Arithmetic Progression Recurrence Relations Example: Fibonacci Sequence Summations Introduction Sequences are ordered lists of elements.
More informationFractal structure of the block-complexity function
Fractal structure of the block-complexity function Konstantinos KARAMANOS and Ilias KOTSIREAS Institut des Hautes Études Scientifiques 35, route de Chartres 91440 Bures-sur-Yvette (France) Avril 2008 IHES/M/08/24
More informationFunctions. Given a function f: A B:
Functions Given a function f: A B: We say f maps A to B or f is a mapping from A to B. A is called the domain of f. B is called the codomain of f. If f(a) = b, then b is called the image of a under f.
More informationHanoi Graphs and Some Classical Numbers
Hanoi Graphs and Some Classical Numbers Sandi Klavžar Uroš Milutinović Ciril Petr Abstract The Hanoi graphs Hp n model the p-pegs n-discs Tower of Hanoi problem(s). It was previously known that Stirling
More informationEnumeration of Automata, Languages, and Regular Expressions
Enumeration of Automata, Languages, and Regular Expressions Jeffrey Shallit School of Computer Science University of Waterloo Waterloo, Ontario N2L 3G1 Canada shallit@cs.uwaterloo.ca http://www.cs.uwaterloo.ca/~shallit
More informationThe Fixed String of Elementary Cellular Automata
The Fixed String of Elementary Cellular Automata Jiang Zhisong Department of Mathematics East China University of Science and Technology Shanghai 200237, China zsjiang@ecust.edu.cn Qin Dakang School of
More information3515ICT: Theory of Computation. Regular languages
3515ICT: Theory of Computation Regular languages Notation and concepts concerning alphabets, strings and languages, and identification of languages with problems (H, 1.5). Regular expressions (H, 3.1,
More informationQUADRANT MARKED MESH PATTERNS IN 132-AVOIDING PERMUTATIONS III
#A39 INTEGERS 5 (05) QUADRANT MARKED MESH PATTERNS IN 3-AVOIDING PERMUTATIONS III Sergey Kitaev Department of Computer and Information Sciences, University of Strathclyde, Glasgow, United Kingdom sergey.kitaev@cis.strath.ac.uk
More informationAutomata and Formal Languages - CM0081 Non-Deterministic Finite Automata
Automata and Formal Languages - CM81 Non-Deterministic Finite Automata Andrés Sicard-Ramírez Universidad EAFIT Semester 217-2 Non-Deterministic Finite Automata (NFA) Introduction q i a a q j a q k The
More informationCHAPTER 8: EXPLORING R
CHAPTER 8: EXPLORING R LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN In the previous chapter we discussed the need for a complete ordered field. The field Q is not complete, so we constructed
More informationSection Summary. Definition of a Function.
Section 2.3 Section Summary Definition of a Function. Domain, Codomain Image, Preimage Injection, Surjection, Bijection Inverse Function Function Composition Graphing Functions Floor, Ceiling, Factorial
More informationA Combinatorial Interpretation of the Numbers 6 (2n)! /n! (n + 2)!
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.3 A Combinatorial Interpretation of the Numbers 6 (2n)! /n! (n + 2)! Ira M. Gessel 1 and Guoce Xin Department of Mathematics Brandeis
More informationTewodros Amdeberhan, Dante Manna and Victor H. Moll Department of Mathematics, Tulane University New Orleans, LA 70118
The -adic valuation of Stirling numbers Tewodros Amdeberhan, Dante Manna and Victor H. Moll Department of Mathematics, Tulane University New Orleans, LA 7011 Abstract We analyze properties of the -adic
More informationA shrinking lemma for random forbidding context languages
Theoretical Computer Science 237 (2000) 149 158 www.elsevier.com/locate/tcs A shrinking lemma for random forbidding context languages Andries van der Walt a, Sigrid Ewert b; a Department of Mathematics,
More informationPrimary classes of compositions of numbers
Annales Mathematicae et Informaticae 41 (2013) pp. 193 204 Proceedings of the 15 th International Conference on Fibonacci Numbers and Their Applications Institute of Mathematics and Informatics, Eszterházy
More informationPolya theory (With a radar application motivation)
Polya theory (With a radar application motivation) Daniel Zwillinger, PhD 5 April 200 Sudbury --623 telephone 4 660 Daniel I Zwillinger@raytheon.com http://www.az-tec.com/zwillinger/talks/2000405/ http://nesystemsengineering.rsc.ray.com/training/memo.polya.pdf
More informationMATH39001 Generating functions. 1 Ordinary power series generating functions
MATH3900 Generating functions The reference for this part of the course is generatingfunctionology by Herbert Wilf. The 2nd edition is downloadable free from http://www.math.upenn. edu/~wilf/downldgf.html,
More informationThe Generalization of Quicksort
The Generalization of Quicksort Shyue-Horng Shiau Chang-Biau Yang Ý Department of Computer Science and Engineering National Sun Yat-Sen University Kaohsiung, Taiwan 804, Republic of China shiaush@cse.nsysu.edu.tw
More informationAlgorithmic Approach to Counting of Certain Types m-ary Partitions
Algorithmic Approach to Counting of Certain Types m-ary Partitions Valentin P. Bakoev Abstract Partitions of integers of the type m n as a sum of powers of m (the so called m-ary partitions) and their
More informationOn a Balanced Property of Compositions
On a Balanced Property of Compositions Miklós Bóna Department of Mathematics University of Florida Gainesville FL 32611-8105 USA Submitted: October 2, 2006; Accepted: January 24, 2007; Published: March
More informationNotes by Zvi Rosen. Thanks to Alyssa Palfreyman for supplements.
Lecture: Hélène Barcelo Analytic Combinatorics ECCO 202, Bogotá Notes by Zvi Rosen. Thanks to Alyssa Palfreyman for supplements.. Tuesday, June 2, 202 Combinatorics is the study of finite structures that
More informationLinear Recurrence Relations for Sums of Products of Two Terms
Linear Recurrence Relations for Sums of Products of Two Terms Yan-Ping Mu College of Science, Tianjin University of Technology Tianjin 300384, P.R. China yanping.mu@gmail.com Submitted: Dec 27, 2010; Accepted:
More informationarxiv: v1 [math.co] 30 Mar 2013
ENUMERATING MAXIMAL TATAMI MAT COVERINGS OF SQUARE GRIDS WITH v VERTICAL DOMINOES arxiv:1304.0070v1 [math.co] 30 Mar 013 ALEJANDRO ERICKSON AND FRANK RUSKEY Abstract. We enumerate a certain class of monomino-domino
More informationTheory of Computation
Theory of Computation COMP363/COMP6363 Prerequisites: COMP4 and COMP 6 (Foundations of Computing) Textbook: Introduction to Automata Theory, Languages and Computation John E. Hopcroft, Rajeev Motwani,
More informationOn the Sequence A and Its Combinatorial Interpretations
1 2 47 6 2 11 Journal of Integer Sequences, Vol. 9 (2006), Article 06..1 On the Sequence A079500 and Its Combinatorial Interpretations A. Frosini and S. Rinaldi Università di Siena Dipartimento di Scienze
More informationRunning Modulus Recursions
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 13 (2010), Article 10.1.6 Running Modulus Recursions Bruce Dearden and Jerry Metzger University of North Dakota Department of Mathematics Witmer Hall
More informationarxiv: v1 [cs.fl] 4 Feb 2013
Incomplete Transition Complexity of Basic Operations on Finite Languages Eva Maia, Nelma Moreira, Rogério Reis arxiv:1302.0750v1 [cs.fl] 4 Fe 2013 CMUP & DCC, Faculdade de Ciências da Universidade do Porto
More informationZhi-Wei Sun Department of Mathematics, Nanjing University Nanjing , People s Republic of China
Ramanuan J. 40(2016, no. 3, 511-533. CONGRUENCES INVOLVING g n (x n ( n 2 ( 2 0 x Zhi-Wei Sun Deartment of Mathematics, Naning University Naning 210093, Peole s Reublic of China zwsun@nu.edu.cn htt://math.nu.edu.cn/
More informationConway s RATS Sequences in Base 3
3 47 6 3 Journal of Integer Sequences, Vol. 5 (0), Article.9. Conway s RATS Sequences in Base 3 Johann Thiel Department of Mathematical Sciences United States Military Academy West Point, NY 0996 USA johann.thiel@usma.edu
More informationF. T. HOWARD AND CURTIS COOPER
SOME IDENTITIES FOR r-fibonacci NUMBERS F. T. HOWARD AND CURTIS COOPER Abstract. Let r 1 be an integer. The r-generalized Fibonacci sequence {G n} is defined as 0, if 0 n < r 1; G n = 1, if n = r 1; G
More informationPattern avoidance in compositions and multiset permutations
Pattern avoidance in compositions and multiset permutations Carla D. Savage North Carolina State University Raleigh, NC 27695-8206 Herbert S. Wilf University of Pennsylvania Philadelphia, PA 19104-6395
More informationPoly-Bernoulli Numbers and Eulerian Numbers
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 21 (2018, Article 18.6.1 Poly-Bernoulli Nubers and Eulerian Nubers Beáta Bényi Faculty of Water Sciences National University of Public Service H-1441
More informationHAMMING DISTANCE FROM IRREDUCIBLE POLYNOMIALS OVER F Introduction and Motivation
HAMMING DISTANCE FROM IRREDUCIBLE POLYNOMIALS OVER F 2 GILBERT LEE, FRANK RUSKEY, AND AARON WILLIAMS Abstract. We study the Hamming distance from polynomials to classes of polynomials that share certain
More informationThe Number of Inversions and the Major Index of Permutations are Asymptotically Joint-Independently-Normal
The Number of Inversions and the Major Index of Permutations are Asymptotically Joint-Independently-Normal Andrew BAXTER 1 and Doron ZEILBERGER 1 Abstract: We use recurrences (alias difference equations)
More informationFIBONACCI NUMBERS AND DECIMATION OF BINARY SEQUENCES
FIBONACCI NUMBERS AND DECIMATION OF BINARY SEQUENCES Jovan Dj. Golić Security Innovation, Telecom Italia Via Reiss Romoli 274, 10148 Turin, Italy (Submitted August 2004-Final Revision April 200) ABSTRACT
More informationCombinatorial properties of the numbers of tableaux of bounded height
Combinatorial properties of the numbers of tableaux of bounded height Marilena Barnabei, Flavio Bonetti, and Matteo Sibani Abstract We introduce an infinite family of lower triangular matrices Γ (s), where
More information