SYLVESTER S WAVE THEORY OF PARTITIONS
|
|
|
- Valerie Fields
- 5 years ago
- Views:
Transcription
1 SYLVESTER S WAVE THEORY OF PARTITIONS Georgia Southern University DIMACS July 11, 2011
2 A partition of an integer n is a representation of n as a sum of positive integers where order of summands (parts) does not matter.
3 A partition of an integer n is a representation of n as a sum of positive integers where order of summands (parts) does not matter ,
4 A partition of an integer n is a representation of n as a sum of positive integers where order of summands (parts) does not matter , so, we have five partions of 4.
5 Let p(n) denote the number of partitions of n.
6 Let p(n) denote the number of partitions of n. p(0) = 1
7 Let p(n) denote the number of partitions of n. p(0) = 1 p(1) = 1
8 Let p(n) denote the number of partitions of n. p(0) = 1 p(1) = 1 p(2) = 2
9 Let p(n) denote the number of partitions of n. p(0) = 1 p(1) = 1 p(2) = 2 p(3) = 3
10 Let p(n) denote the number of partitions of n. p(0) = 1 p(1) = 1 p(2) = 2 p(3) = 3 p(4) = 5
11 Let p(n) denote the number of partitions of n. p(0) = 1 p(1) = 1 p(2) = 2 p(3) = 3 p(4) = 5 p(5) = 7 p(6) = 11 p(7) = 15 p(8) = 22 p(9) = 30 p(10) = 42
12 Let p(n) denote the number of partitions of n. p(0) = 1 p(1) = 1 p(2) = 2 p(3) = 3 p(4) = 5 p(5) = 7 p(6) = 11 p(7) = 15 p(8) = 22 p(9) = 30 p(10) = 42 p(100) = 190, 569, 192 p(200) = 3, 972, 999, 029, 388
13 G. H. Hardy ( ) and S. Ramanujan ( )
14 G. H. Hardy and S. Ramanujan (1918) p(n) exp (π 2n/3) 4n, as n. 3
15 Hardy and Ramanujan (1918) p(n) = 1 2π 2 α n k k=1 0 h<k gcd(h,k)=1 ω h,k e 2πihn/k d dn ( exp π k ) 3 (n 1 24 ) 2 n O(n 1/4 ), with α an arbitrary constant and ω h,k a certain complex 24kth root of unity.
16 Hans Rademacher ( )
17 p(n) = 1 π 2 H. Rademacher (1937) A k (n) k d dn k=1 ( sinh π k 2 3 n 1 24 ( n 1 24 ) ),
18 where A k (n) = is a Kloosterman sum, 0 h<k gcd(h,k)=1 e πi(s(h,k) 2nh/k),
19 where A k (n) = is a Kloosterman sum, and s(h, k) = is a Dedekind sum. 0 h<k gcd(h,k)=1 k 1 r=1 ( r hr k k e πi(s(h,k) 2nh/k), hr 1 ) k 2
20 Ken Ono and Jan Bruinier
21 In 2011, Ken Ono and Jan Bruinier announced a new formula that expresses p(n) as a finite sum of certain algebraic numbers.
22 Let p(n, m) denote the number of partitions of n into parts m.
23 Let p(n, m) denote the number of partitions of n into parts m. 3+3 = = = = = =
24 Let p(n, m) denote the number of partitions of n into parts m. 3+3 = = = = = = So, p(6, 3) = 7
25 L. Euler ( )
26 ( 1 ) ( 1 ) ( 1 ) 1 x 1 x 2 1 x 3 = (1 + x + x 2 + x 3 + x 4 + ) (1 + x 2 + x 4 + x 6 + x 8 + ) (1 + x 3 + x 6 + x 9 + x 12 + )
27 ( 1 ) ( 1 ) ( 1 ) 1 x 1 1 x 2 1 x 3 = (1 + x 1 + x x x ) (1 + x 2 + x x x ) (1 + x 3 + x x x )
28 ( 1 ) ( 1 ) ( 1 ) 1 x 1 1 x 2 1 x 3 = (1 + x 1 + x x x ) (1 + x 2 + x x x ) (1 + x 3 + x x x ) =1 + x 1 + (x 2 + x 1+1 ) + (x 3 + x x ) +
29 ( 1 ) ( 1 ) ( 1 ) 1 x 1 1 x 2 1 x 3 = (1 + x 1 + x x x ) (1 + x 2 + x x x ) (1 + x 3 + x x x ) =1 + x 1 + (x 2 + x 1+1 ) + (x 3 + x x ) + = p(n, 3)x n. n=0
30 p(n)x n = n=0 k=1 1 1 x k.
31 Let x = e 2πiτ. x 1/24 η(τ) = p(n)x n, n=0
32 Let x = e 2πiτ. provided x 1/24 η(τ) = p(n)x n, n=0 Iτ > 0, i.e. x < 1.
33 Arthur Cayley
34 p(n, 3)x n 1 = (1 x)(1 x 2 )(1 x 3 ) n=0 = 17/72 1 x + 1/4 (1 x) 2 + 1/6 (1 x) 3 + 1/ x x x + x 2
35 p(n, 3)x n 1 = (1 x)(1 x 2 )(1 x 3 ) n=0 = 17/72 1 x + 1/4 (1 x) 2 + 1/6 (1 x) 3 + 1/ x x x + x 2 = x n n= (n+1)x n + 1 ( ) n + 2 x n n=0 cr 3 (1, 1, 0)x n n=0 cr 2 (1, 1)x n n=0 cr 3 (0, 1, 1)x n n=1
36 p(n, 3)x n 1 = (1 x)(1 x 2 )(1 x 3 ) n=0 = n=0 = 17/72 1 x + 1/4 (1 x) 2 + 1/6 (1 x) 3 + 1/ x x x + x 2 ( n ( n ) cr 2(1, 1) + cr 3 ( 2 9, 1 9, 1 9 ) ) x n
37 p(n, 3) = n ( n ) +cr 2 ( 1 8, 1 8 ) ( ) 2 +cr 3 9, 1 9, 1 9
38 p(n, 3) = n ( n = n ( n ) +cr 2 ( 1 8, 1 8 ) ( ) 2 +cr 3 9, 1 9, 1 9 ) ( 25 + cr 6 72, 17 72, 1 72, 7 72, 17 ) 72, 17 72
39 p(n, 3) = n ( n = n ( n ) +cr 2 ( 1 8, 1 8 ) ( ) 2 +cr 3 9, 1 9, 1 9 ) ( 25 + cr 6 72, 17 72, 1 72, 7 72, 17 ) 72, ( ) n + 2 = nearest integer to n
40 J. J. Sylvester ( )
41 Sylvester s theorem where W q,m = Res x=0 ρ prim qth root of 1 p(n, m) = m W q,m, q=1 ρ n e nx (1 ρ 1 e x )(1 ρ 2 e 2x ) (1 ρ m e mx )
42 e nx W 1,m =Res x=0 (1 e x )(1 e 2x ) (1 e mx ) e x(n+ 1 2 ( m)) =Res x=0 (e 1 2 x e 1 2 x )(e 1 2 2x e 1 2 2x ) (e 1 2 mx e 1 2 mx ) =Res x=0 e νx m(m+1) m j=1 2 sinh( 1 where ν = n + 2jx), 4.
43 W 1,1 = 1
44 W 1,1 = 1 W 1,2 = 1 2 ν
45 W 1,1 = 1 W 1,2 = 1 2 ν ( ) W 1,3 = 1 ν 2 3! 2! s 2 24 where s r = 1 r + 2 r + 3 r + + m r.
46 W 1,1 = 1 W 1,2 = 1 2 ν W 1,3 = 1 3! W 1,4 = 1 4! ( ν 2 2! s 2 24 ) ( ν 3 3! s 2 24 ν ) where s r = 1 r + 2 r + 3 r + + m r.
47 W 1,1 = 1 W 1,2 = 1 2 ν W 1,3 = 1 3! W 1,4 = 1 4! W 1,5 = 1 5! ( ν 2 2! s 2 24 ( ν 3 3! s 2 ) ) 24 ν ( ν 4 4! s 2 ν ! where s r = 1 r + 2 r + 3 r + + m r. ( s s 4 5 ))
48 W 1,m = 1 m! where ( ν m 1 (m 1)! J ν m 3 1 (m 3)! + J ν m 5 ) 2 (m 5)!, J 1 = s 2 24 ( J 2 = 1 s J 3 = ) 2 + s 4 5 ( s s 2s s )
49 J. W. L. Glaisher ( )
50 Glaisher calculated explicit formulas for W i,m where i = 1, 2, 3, 4, 5, 6.
51 Research project Program Shalosh to find explicit formulas for W i,m where i = 1, 2, 3, 4, 5, 6, 7, 8,....
52 Research project Program Shalosh to find explicit formulas for W i,m where i = 1, 2, 3, 4, 5, 6, 7, 8,.... Find an explicit formula for W i,m for arbitrary m.
53 Research project Program Shalosh to find explicit formulas for W i,m where i = 1, 2, 3, 4, 5, 6, 7, 8,.... Find an explicit formula for W i,m for arbitrary m. p(n) = p(n, n), so p(n) = m i=1 W i,n.
54 Numerical calculations show that for reasonable size n, p(n) = p(n, n) W 1,n.
55 Research project Find r = r(n) so that p(n) = p(n, n) = nearest integer to r W i,n. i=1
56 Theory of q-partial fractions." Augustine O. Munagi
57 p(n, 3)x n 1 = (1 x)(1 x 2 )(1 x 3 ) n=0 = 17/72 1 x + 1/4 (1 x) 2 + 1/6 (1 x) 3 + 1/ x x x + x 2
58 p(n, 3)x n 1 = (1 x)(1 x 2 )(1 x 3 ) n=0 = 17/72 1 x + 1/4 (1 x) 2 + 1/6 (1 x) 3 + 1/ x x x + x 2 = 0 1 x + 1/4 (1 x) 2 + 1/6 (1 x) 3 + 1/4 1 x 2 + 1/3 1 x 3
59 p(n, 3)x n 1 = (1 x)(1 x 2 )(1 x 3 ) n=0 = = 17/72 1 x + 1/4 (1 x) 2 + 1/6 (1 x) 3 + 1/ x x x + x 2 = 0 1 x + 1/4 (1 x) 2 + 1/6 (1 x) 3 + 1/4 1 x 2 + 1/3 1 x 3 ( n ( ) n cr 2(1, 0) + 1 ) 3 cr 3(1, 0, 0) n=0 x n
60 p(n, 3) = n ( n ) +cr 2 ( 1 8, 1 8 ) ( ) 2 +cr 3 9, 1 9, 1 9
61 p(n, 3) = n ( n = n ) +cr 2 ( 1 8, 1 8 ) ( ) 2 +cr 3 9, 1 9, 1 9 ( ) ( ) ( ) n cr 2, 0 + cr 3, 0, 0 2
62 Research project Find the q-partial fraction analog of Sylvester s wave theory.
63 Back to the beginning A composition of an integer n is a representation of n as a sum of positive integers where order of summands (parts) does matter.
64 Back to the beginning A composition of an integer n is a representation of n as a sum of positive integers where order of summands (parts) does matter and and and ,
65 Back to the beginning A composition of an integer n is a representation of n as a sum of positive integers where order of summands (parts) does matter and and and , so, we have eight compositions of 4.
66 Let c(n) denote the number of compositions of n.
67 Let c(n) denote the number of compositions of n. c(1) = 1
68 Let c(n) denote the number of compositions of n. c(1) = 1 c(2) = 2
69 Let c(n) denote the number of compositions of n. c(1) = 1 c(2) = 2 c(3) = 4
70 Let c(n) denote the number of compositions of n. c(1) = 1 c(2) = 2 c(3) = 4 c(4) = 8
71 Let c(n) denote the number of compositions of n. c(1) = 1 c(2) = 2 c(3) = 4 c(4) = 8 c(5) = 16 c(6) = 32 c(7) = 64
72 Let c(n) denote the number of compositions of n. c(1) = 1 c(2) = 2 c(3) = 4 c(4) = 8 c(5) = 16 c(6) = 32 c(7) = 64. c(n) = 2 n 1
73 P. A. MacMahon ( )
74 The graph of a composition The MacMahon graph of the composition of n = 12
75 The graph of a composition The MacMahon graph of the composition of n = 12 A composition of n with m parts has a MacMahon graph with n dashes and m 1 dots
76 The graph of a composition The MacMahon graph of the composition of n = 12 A composition of n with m parts has a MacMahon graph with n dashes and m 1 dots, and is in one-to-one correspondence with a bit sequence of length n 1 containing exactly m 1 ones.
77 The graph of a composition The MacMahon graph of the composition of n = 12 A composition of n with m parts has a MacMahon graph with n dashes and m 1 dots, and is in one-to-one correspondence with a bit sequence of length n 1 containing exactly m 1 ones ( )
78 The graph of a composition The MacMahon graph of the composition of n = 12 A composition of n with m parts has a MacMahon graph with n dashes and m 1 dots, and is in one-to-one correspondence with a bit sequence of length n 1 containing exactly m 1 ones ( ) Thus c(n) = 2 n 1.
79 Research project Use information about c(n) to derive a formula for p(n).
COMPOSITIONS, PARTITIONS, AND FIBONACCI NUMBERS
COMPOSITIONS PARTITIONS AND FIBONACCI NUMBERS ANDREW V. SILLS Abstract. A bijective proof is given for the following theorem: the number of compositions of n into odd parts equals the number of compositions
Efficient implementation of the Hardy-Ramanujan-Rademacher formula
Efficient implementation of the Hardy-Ramanujan-Rademacher formula or: Partitions in the quintillions Fredrik Johansson RISC-Linz July 10, 2013 2013 SIAM Annual Meeting San Diego, CA Supported by Austrian
Partitions and Rademacher s Exact Formula
Partitions and Rademacher s Exact Formula Anurag Sahay KVPY Reg. No. SX - 11011010 Indian Institute of Technology, Kanpur Rijul Saini KVPY Reg. No. SA - 11010155 Chennai Mathematical Institute June - July,
Partitions in the quintillions or Billions of congruences
Partitions in the quintillions or Billions of congruences Fredrik Johansson November 2011 The partition function p(n) counts the number of ways n can be written as the sum of positive integers without
Partitioning of Positive Integers
Partitioning of Positive Integers By Hossam Shoman Supervised by Dr. Ayman Badawi A thesis presented to the American University of Sharjah in the partial fulfillment of the requirements for the degree
The Hardy-Ramanujan-Rademacher Expansion of p(n)
The Hardy-Ramanujan-Rademacher Expansion of p(n) Definition 0. A partition of a positive integer n is a nonincreasing sequence of positive integers whose sum is n. We denote the number of partitions of
A Generalization of the Euler-Glaisher Bijection
A Generalization of the Euler-Glaisher Bijection p.1/48 A Generalization of the Euler-Glaisher Bijection Andrew Sills Georgia Southern University A Generalization of the Euler-Glaisher Bijection p.2/48
Modular forms, combinatorially and otherwise
Modular forms, combinatorially and otherwise p. 1/103 Modular forms, combinatorially and otherwise David Penniston Sums of squares Modular forms, combinatorially and otherwise p. 2/103 Modular forms, combinatorially
Exact Formulas for Invariants of Hilbert Schemes
Exact Formulas for Invariants of Hilbert Schemes N. Gillman, X. Gonzalez, M. Schoenbauer, advised by John Duncan, Larry Rolen, and Ken Ono September 2, 2018 1 / 36 Paper See the related paper at https://arxiv.org/pdf/1808.04431.pdf
COMPOSITIONS WITH A FIXED NUMBER OF INVERSIONS
COMPOSITIONS WITH A FIXED NUMBER OF INVERSIONS A. KNOPFMACHER, M. E. MAYS, AND S. WAGNER Abstract. A composition of the positive integer n is a representation of n as an ordered sum of positive integers
Applications of modular forms to partitions and multipartitions
Applications of modular forms to partitions and multipartitions Holly Swisher Oregon State University October 22, 2009 Goal The goal of this talk is to highlight some applications of the theory of modular
Combinatorics, Modular Forms, and Discrete Geometry
Combinatorics, Modular Forms, and Discrete Geometry / 1 Geometric and Enumerative Combinatorics, IMA University of Minnesota, Nov 10 14, 2014 Combinatorics, Modular Forms, and Discrete Geometry Peter Paule
Mock and quantum modular forms
Mock and quantum modular forms Amanda Folsom (Amherst College) 1 Ramanujan s mock theta functions 2 Ramanujan s mock theta functions 1887-1920 3 Ramanujan s mock theta functions 1887-1920 4 History S.
Generating Functions
Semester 1, 2004 Generating functions Another means of organising enumeration. Two examples we have seen already. Example 1. Binomial coefficients. Let X = {1, 2,..., n} c k = # k-element subsets of X
Course : Algebraic Combinatorics
Course 18.31: Algebraic Combinatorics Lecture Notes # 10 Addendum by Gregg Musiker (Based on Lauren Williams Notes for Math 19 at Harvard) February 5th - 7th, 009 1 Introduction to Partitions A partition
Thesis submitted in partial fulfillment of the requirement for The award of the degree of. Masters of Science in Mathematics and Computing
SOME n-color COMPOSITION Thesis submitted in partial fulfillment of the requirement for The award of the degree of Masters of Science in Mathematics and Computing Submitted by Shelja Ratta Roll no- 301203014
MULTIPLICATIVE PROPERTIES OF THE NUMBER OF k-regular PARTITIONS
MULTIPLICATIVE PROPERTIES OF THE NUMBER OF k-regular PARTITIONS OLIVIA BECKWITH AND CHRISTINE BESSENRODT Abstract. In a previous paper of the second author with K. Ono, surprising multiplicative properties
COMBINATORIAL APPLICATIONS OF MÖBIUS INVERSION
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 COMBINATORIAL APPLICATIONS OF MÖBIUS INVERSION MARIE JAMESON AND ROBERT P. SCHNEIDER (Communicated
Partition 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
Partitions with Parts in a Finite Set and with PartsOutsideaFiniteSet
Partitions with Parts in a Finite Set with PartsOutsideaFiniteSet Gert lmkvist CONTENTS Introduction Partitions into Parts in a Finite Set 3 Partitions with Parts Outside a Finite Set References Exact
CONGRUENCES FOR BROKEN k-diamond PARTITIONS
CONGRUENCES FOR BROKEN k-diamond PARTITIONS MARIE JAMESON Abstract. We prove two conjectures of Paule and Radu from their recent paper on broken k-diamond partitions. 1. Introduction and Statement of Results
CONGRUENCES MODULO SQUARES OF PRIMES FOR FU S k DOTS BRACELET PARTITIONS
CONGRUENCES MODULO SQUARES OF PRIMES FOR FU S k DOTS BRACELET PARTITIONS CRISTIAN-SILVIU RADU AND JAMES A SELLERS Dedicated to George Andrews on the occasion of his 75th birthday Abstract In 2007, Andrews
MODULAR FORMS ARISING FROM Q(n) AND DYSON S RANK
MODULAR FORMS ARISING FROM Q(n) AND DYSON S RANK MARIA MONKS AND KEN ONO Abstract Let R(w; q) be Dyson s generating function for partition ranks For roots of unity ζ it is known that R(ζ; q) and R(ζ; /q)
The Kth M-ary Partition Function
Indiana University of Pennsylvania Knowledge Repository @ IUP Theses and Dissertations (All) Fall 12-2016 The Kth M-ary Partition Function Laura E. Rucci Follow this and additional works at: http://knowledge.library.iup.edu/etd
Simultaneous Diophantine Approximation with Excluded Primes. László Babai Daniel Štefankovič
Simultaneous Diophantine Approximation with Excluded Primes László Babai Daniel Štefankovič Dirichlet (1842) Simultaneous Diophantine Approximation Given reals integers α,,...,, 1 α 2 α n Q r1,..., r n
THE CIRCLE METHOD, THE j FUNCTION, AND PARTITIONS
THE CIRCLE METHOD, THE j FUCTIO, AD PARTITIOS JEREMY BOOHER. Introduction The circle method was first used by Hardy and Ramanujan [] in 98 to investigate the asymptotic growth of the partition function,
RAMANUJAN S MOST BEAUTIFUL IDENTITY
RAMANUJAN S MOST BEAUTIFUL IDENTITY MICHAEL D. HIRSCHHORN Abstract We give a simple proof of the identity which for Hardy represented the best of Ramanujan. On the way, we give a new proof of an important
INFINITELY MANY CONGRUENCES FOR BROKEN 2 DIAMOND PARTITIONS MODULO 3
INFINITELY MANY CONGRUENCES FOR BROKEN 2 DIAMOND PARTITIONS MODULO 3 SILVIU RADU AND JAMES A. SELLERS Abstract. In 2007, Andrews and Paule introduced the family of functions k n) which enumerate the number
Rademacher-Type Formulas for Partitions and Overpartitions
Georgia Southern University Digital Commons@Georgia Southern Mathematical Sciences Faculty Publications Mathematical Sciences, Department of 2010 Rademacher-Type Formulas for Partitions and Overpartitions
congruences Shanghai, July 2013 Simple proofs of Ramanujan s partition congruences Michael D. Hirschhorn
s s m.hirschhorn@unsw.edu.au Let p(n) be the number of s of n. For example, p(4) = 5, since we can write 4 = 4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1 and there are 5 such representations. It was known to Euler
Some families of identities for the integer partition function
MATHEMATICAL COMMUNICATIONS 193 Math. Commun. 0(015), 193 00 Some families of identities for the integer partition function Ivica Martinja 1, and Dragutin Svrtan 1 Department of Physics, University of
SOME RECURRENCES FOR ARITHMETICAL FUNCTIONS. Ken Ono, Neville Robbins, Brad Wilson. Journal of the Indian Mathematical Society, 62, 1996, pages
SOME RECURRENCES FOR ARITHMETICAL FUNCTIONS Ken Ono, Neville Robbins, Brad Wilson Journal of the Indian Mathematical Society, 6, 1996, pages 9-50. Abstract. Euler proved the following recurrence for p(n),
More about partitions
Partitions 2.4, 3.4, 4.4 02 More about partitions 3 + +, + 3 +, and + + 3 are all the same partition, so we will write the numbers in non-increasing order. We use greek letters to denote partitions, often
Scott Ahlgren and Ken Ono. At first glance the stuff of partitions seems like child s play: 4 = = = =
ADDITION AND COUNTING: THE ARITHMETIC OF PARTITIONS Scott Ahlgren and Ken Ono At first glance the stuff of partitions seems like child s play: 4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1. Therefore,
Eulerian series in q-series and modular forms. Youn Seo Choi
Eulerian series in q-series and modular forms Youn Seo Choi abstrat Eulerian series is very interesting power series even though we do not know any single method to handle the general Eulerian series However,
DEDEKIND S ETA-FUNCTION AND ITS TRUNCATED PRODUCTS. George E. Andrews and Ken Ono. February 17, Introduction and Statement of Results
DEDEKIND S ETA-FUNCTION AND ITS TRUNCATED PRODUCTS George E. Andrews and Ken Ono February 7, 2000.. Introduction and Statement of Results Dedekind s eta function ηz, defined by the infinite product ηz
Addition and Counting: The Arithmetic of Partitions
Addition and Counting: The Arithmetic of Partitions Scott Ahlgren and Ken Ono At first glance the stuff of partitions seems like child s play: 4=3+1=2+2=2+1+1=1+1+1+1. Therefore, there are 5 partitions
ON DYSON S CRANK CONJECTURE AND THE UNIFORM ASYMPTOTIC BEHAVIOR OF CERTAIN INVERSE THETA FUNCTIONS
ON DYSON S CRANK CONJECTURE AND THE UNIFORM ASYMPTOTIC BEHAVIOR OF CERTAIN INVERSE THETA FUNCTIONS KATHRIN BRINGMANN AND JEHANNE DOUSSE Abstract In this paper we prove a longstanding conjecture by Freeman
A MODULAR IDENTITY FOR THE RAMANUJAN IDENTITY MODULO 35
A MODULAR IDENTITY FOR THE RAMANUJAN IDENTITY MODULO 35 Adrian Dan Stanger Department of Mathematics, Brigham Young University, Provo, Utah 84602, USA adrian@math.byu.edu Received: 3/20/02, Revised: 6/27/02,
Upper Bounds for Partitions into k-th Powers Elementary Methods
Int. J. Contemp. Math. Sciences, Vol. 4, 2009, no. 9, 433-438 Upper Bounds for Partitions into -th Powers Elementary Methods Rafael Jaimczu División Matemática, Universidad Nacional de Luján Buenos Aires,
q(m, n) -- q(m + 1, n) = p(m, n), n >~ 1. (2)
JOURNAL OF COMBINATORIAL THEORY 7, 56-61 (1969) A New Symmetry of Partitions FREEMAN J. DYSON* Belfer Graduate School of Science, Yeshiva University, New York, New York 10033 Communicated by D. Younger
Exact Formulas for the Generalized Sum-of-Divisors Functions
Exact Formulas for the Generalized Sum-of-Divisors Functions arxiv:705.03488v4 [math.nt] Sep 08 Maxie D. Schmidt School of Mathematics Georgia Institute of Technology Atlanta, GA 3033 USA maxieds@gmail.com
Is Analysis Necessary?
Is Analysis Necessary? Ira M. Gessel Brandeis University Waltham, MA gessel@brandeis.edu Special Session on Algebraic and Analytic Combinatorics AMS Fall Eastern Meeting University of Connecticut, Storrs
INFINITE FAMILIES OF STRANGE PARTITION CONGRUENCES FOR BROKEN 2-DIAMONDS
December 1, 2009 INFINITE FAMILIES OF STRANGE PARTITION CONGRUENCES FOR BROKEN 2-DIAMONDS PETER PAULE AND SILVIU RADU Dedicated to our friend George E. Andrews on the occasion of his 70th birthday Abstract.
Finding Divisors, Number of Divisors and Summation of Divisors. Jane Alam Jan
Finding Divisors, Number of Divisors and Summation of Divisors by Jane Alam Jan Finding Divisors If we observe the property of the numbers, we will find some interesting facts. Suppose we have a number
COMPUTATION OF q-partial FRACTIONS
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7 2007, #A25 COMPUTATION OF q-partial FRACTIONS Augustine O Munagi The John Knopfmacher Centre for Applicable Analysis and Number Theory, University
Constructing Class invariants
Constructing Class invariants Aristides Kontogeorgis Department of Mathematics University of Athens. Workshop Thales 1-3 July 2015 :Algebraic modeling of topological and computational structures and applications,
1. Introduction and statement of results This paper concerns the deep properties of the modular forms and mock modular forms.
MOONSHINE FOR M 4 AND DONALDSON INVARIANTS OF CP ANDREAS MALMENDIER AND KEN ONO Abstract. Eguchi, Ooguri, and Tachikawa recently conjectured 9] a new moonshine phenomenon. They conjecture that the coefficients
On Partition Functions and Divisor Sums
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.4 On Partition Functions and Divisor Sums Neville Robbins Mathematics Department San Francisco State University San Francisco,
PARITY RESULTS FOR BROKEN k DIAMOND PARTITIONS AND (2k + 1) CORES
PARITY RESULTS FOR BROKEN k DIAMOND PARTITIONS AND 2k + CORES SILVIU RADU AND JAMES A. SELLERS Abstract. In this paper we prove several new parity results for broken k-diamond partitions introduced in
Addition sequences and numerical evaluation of modular forms
Addition sequences and numerical evaluation of modular forms Fredrik Johansson (INRIA Bordeaux) Joint work with Andreas Enge (INRIA Bordeaux) William Hart (TU Kaiserslautern) DK Statusseminar in Strobl,
Honors Advanced Algebra Unit 3: Polynomial Functions November 9, 2016 Task 11: Characteristics of Polynomial Functions
Honors Advanced Algebra Name Unit 3: Polynomial Functions November 9, 2016 Task 11: Characteristics of Polynomial Functions MGSE9 12.F.IF.7 Graph functions expressed symbolically and show key features
Generalised Rogers Ramanujan identities and arithmetics. Ole Warnaar School of Mathematics and Physics
Generalised Rogers Ramanujan identities and arithmetics Ole Warnaar School of Mathematics and Physics Based on joint work with Nick Bartlett Michael Griffin Ken Ono Eric Rains History and motivation The
MATH 566, Final Project Alexandra Tcheng,
MATH 566, Final Project Alexanra Tcheng, 60665 The unrestricte partition function pn counts the number of ways a positive integer n can be resse as a sum of positive integers n. For example: p 5, since
Analytic and combinatorial explorations of partitions associated with the Rogers-Ramanujan identities and partitions with initial repetitions
Analytic and combinatorial explorations of partitions associated with the Rogers-Ramanujan identities and partitions with initial repetitions by Darlison Nyirenda A thesis submitted to the Faculty of Science,
Uniform convergence of N-dimensional Walsh Fourier series
STUDIA MATHEMATICA 68 2005 Uniform convergence of N-dimensional Walsh Fourier series by U. Goginava Tbilisi Abstract. We establish conditions on the partial moduli of continuity which guarantee uniform
Partitions, permutations and posets Péter Csikvári
Partitions, permutations and posets Péter Csivári In this note I only collect those things which are not discussed in R Stanley s Algebraic Combinatorics boo Partitions For the definition of (number) partition,
ON THE EXPLICIT CONSTRUCTION OF HIGHER DEFORMATIONS OF PARTITION STATISTICS
ON THE EXPLICIT CONSTUCTION OF HIGHE DEFOMATIONS OF PATITION STATISTICS KATHIN BINGMANN Abstract The modularity of the partition generating function has many important consequences, for example asymptotics
RECENT WORK ON THE PARTITION FUNCTION
RECENT WORK ON THE PARTITION FUNCTION JAN HENDRIK BRUINIER, AMANDA FOLSOM, ZACHARY A. KENT, AND KEN ONO Abstract. This expository article describes recent work by the authors on the partition function
On the Sylvester Denumerants for General Restricted Partitions
On the Sylvester Denumerants for General Restricted Partitions Geir Agnarsson Abstract Let n be a nonnegative integer and let ã = (a 1... a k be a k-tuple of positive integers. The term denumerant introduced
Arithmetic Relations for Overpartitions
Arithmetic Relations for Overpartitions Michael D. Hirschhorn School of Mathematics, UNSW, Sydney 2052, Australia m.hirschhorn@unsw.edu.au James A. Sellers Department of Mathematics The Pennsylvania State
#A8 INTEGERS 12 (2012) PARTITION OF AN INTEGER INTO DISTINCT BOUNDED PARTS, IDENTITIES AND BOUNDS
#A8 INTEGERS 1 (01) PARTITION OF AN INTEGER INTO DISTINCT BOUNDED PARTS, IDENTITIES AND BOUNDS Mohammadreza Bidar 1 Deartment of Mathematics, Sharif University of Technology, Tehran, Iran mrebidar@gmailcom
A Hardy-Ramanujan-Rademacher-type formula for (r, s)-regular partitions
The Ramanujan Journal manuscript No. will be inserted by the editor A Hardy-Ramanujan-Rademacher-type formula for r, s-regular partitions James Mc Laughlin Scott Parsell Received: date / Accepted: date
ASYMPTOTICS FOR RANK AND CRANK MOMENTS
ASYMPTOTICS FOR RANK AND CRANK MOMENTS KATHRIN BRINGMANN, KARL MAHLBURG, AND ROBERT C. RHOADES Abstract. Moments of the partition rank and crank statistics have been studied for their connections to combinatorial
FERMAT S WORLD A TOUR OF. Outline. Ching-Li Chai. Philadelphia, March, Samples of numbers. 2 More samples in arithemetic. 3 Congruent numbers
Department of Mathematics University of Pennsylvania Philadelphia, March, 2016 Outline 1 2 3 4 5 6 7 8 9 Some familiar whole numbers 1. Examples of numbers 2, the only even prime number. 30, the largest
Primary 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
AN ELEMENTARY APPROACH TO THE MACDONALD IDENTITIES* DENNIS STANTON
AN ELEMENTARY APPROACH TO THE MACDONALD IDENTITIES* DENNIS STANTON Abstract. Elementary proofs are given for the infinite families of Macdonald identities. The reflections of the Weyl group provide sign-reversing
Mathematical Induction Assignments
1 Mathematical Induction Assignments Prove the Following using Principle of Mathematical induction 1) Prove that for any positive integer number n, n 3 + 2 n is divisible by 3 2) Prove that 1 3 + 2 3 +
What is the crank of a partition? Daniel Glasscock, July 2014
What is the crank of a partition? Daniel Glasscock, July 2014 These notes complement a talk given for the What is...? seminar at the Ohio State University. Introduction The crank of an integer partition
Integer Partitions: An Overview
Integer Partitions: An Overview D Singh, PhD, A M Ibrahim, PhD, J N Singh, Ph D, S M Ladan,PhD, Abstract This paper describes fundamentals of integer partitions and basic tools for their construction and
HYPERBOLICITY OF THE PARTITION JENSEN POLYNOMIALS
HYPERBOLICITY OF THE PARTITION JENSEN POLYNOMIALS HANNAH LARSON AND IAN WAGNER Abstract Given an arithmetic function a : N R, one can associate a naturally defined, doubly infinite family of Jensen polynomials
ON THE POSITIVITY OF THE NUMBER OF t CORE PARTITIONS. Ken Ono. 1. Introduction
ON THE POSITIVITY OF THE NUMBER OF t CORE PARTITIONS Ken Ono Abstract. A partition of a positive integer n is a nonincreasing sequence of positive integers whose sum is n. A Ferrers graph represents a
Algebra Review. Instructor: Laszlo Babai Notes by Vincent Lucarelli and the instructor. June 15, 2001
Algebra Review Instructor: Laszlo Babai Notes by Vincent Lucarelli and the instructor June 15, 2001 1 Groups Definition 1.1 A semigroup (G, ) is a set G with a binary operation such that: Axiom 1 ( a,
The Ihara zeta function of the infinite grid
The Ihara zeta function of the infinite grid Bryan Clair Department of Mathematics and Computer Science Saint Louis University bryan@slu.edu November 3, 2013 An Integral I(k) = 1 2π 2π (2π) 2 log [ 1 k
ETA-QUOTIENTS AND ELLIPTIC CURVES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 11, November 1997, Pages 3169 3176 S 0002-9939(97)03928-2 ETA-QUOTIENTS AND ELLIPTIC CURVES YVES MARTIN AND KEN ONO (Communicated by
AN ARITHMETIC FORMULA FOR THE PARTITION FUNCTION
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 135, Number 11, November 007, Pages 3507 3514 S 000-9939(07)08883-1 Article electronically published on July 7, 007 AN ARITHMETIC FORMULA FOR THE
NEW IDENTITIES INVOLVING SUMS OF THE TAILS RELATED TO REAL QUADRATIC FIELDS KATHRIN BRINGMANN AND BEN KANE
NEW IDENTITIES INVOLVING SUMS OF THE TAILS RELATED TO REAL QUADRATIC FIELDS KATHRIN BRINGMANN AND BEN KANE To George Andrews, who has been a great inspiration, on the occasion of his 70th birthday Abstract.
arxiv:math/ v1 [math.nt] 28 Jan 2005
![arxiv:math/ v1 [math.nt] 28 Jan 2005](/thumbs/90/101994425.jpg "arxiv:math/ v1 [math.nt] 28 Jan 2005")
arxiv:math/0501528v1 [math.nt] 28 Jan 2005 TRANSFORMATIONS OF RAMANUJAN S SUMMATION FORMULA AND ITS APPLICATIONS Chandrashekar Adiga 1 and N.Anitha 2 Department of Studies in Mathematics University of
THE ARITHMETIC OF BORCHERDS EXPONENTS. Jan H. Bruinier and Ken Ono
THE ARITHMETIC OF BORCHERDS EXPONENTS Jan H. Bruinier and Ken Ono. Introduction and Statement of Results. Recently, Borcherds [B] provided a striking description for the exponents in the naive infinite
Electronic Theses and Dissertations
Eat Tenneee State Univerity Digital Common @ Eat Tenneee State Univerity Electronic Thee and Diertation Student Work 5-208 Vector Partition Jennifer French Eat Tenneee State Univerity Follow thi and additional
Series of Error Terms for Rational Approximations of Irrational Numbers
2 3 47 6 23 Journal of Integer Sequences, Vol. 4 20, Article..4 Series of Error Terms for Rational Approximations of Irrational Numbers Carsten Elsner Fachhochschule für die Wirtschaft Hannover Freundallee
Permutations 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
arxiv: v1 [math.co] 24 Sep 2017
![arxiv: v1 [math.co] 24 Sep 2017](/thumbs/92/110850760.jpg "arxiv: v1 [math.co] 24 Sep 2017")
k-planar Crossing Number of Random Graphs and Random Regular Graphs arxiv:1709.08136v1 [math.co] 24 Sep 2017 John Asplund 1 Thao Do 2 Arran Hamm 3 László Székely 4 Libby Taylor 5 Zhiyu Wang 4 Abstract
Integer Partitions. Student ID: Tutor: Roman Kotecky\Neil O Connell April 22, 2015
Integer Partitions Student ID: 35929 Tutor: Roman Kotecky\Neil O Connell April 22, 205 My study of Integer partitions came from a particular algebra assignment in which we were asked to find all the isomorphism
ALTERNATING STRANGE FUNCTIONS
ALTERNATING STRANGE FUNCTIONS ROBERT SCHNEIDER Abstract In this note we consider infinite series similar to the strange function F (q) of Kontsevich studied by Zagier, Bryson-Ono-Pitman-Rhoades, Bringmann-Folsom-
ALTERNATING STRANGE FUNCTIONS
ALTERNATING STRANGE FUNCTIONS ROBERT SCHNEIDER Abstract. In this note we consider infinite series similar to the strange function F (q) of Kontsevich studied by Zagier Bryson-Ono-Pitman-Rhoades Bringmann-Folsom-
Taylor Expansions with Finite Radius of Convergence Work of Robert Jentzsch, 1917
Polynomial Families with Interesting Zeros Robert Boyer Drexel University April 30, 2010 Taylor Expansions with Finite Radius of Convergence Work of Robert Jentzsch, 1917 Partial Sums of Geometric Series
Jacobi-Angelesco multiple orthogonal polynomials on an r-star
M. Leurs Jacobi-Angelesco m.o.p. 1/19 Jacobi-Angelesco multiple orthogonal polynomials on an r-star Marjolein Leurs, (joint work with Walter Van Assche) Conference on Orthogonal Polynomials and Holomorphic
For COURSE PACK and other PERMISSIONS, refer to entry on previous page. For more information, send to
COPYRIGHT NOTICE: Elias M. Stein and Rami Shakarchi: Complex Analysis is published by Princeton University Press and copyrighted, 2003, by Princeton University Press. All rights reserved. No part of this
Repeated Values of Euler s Function. a talk by Paul Kinlaw on joint work with Jonathan Bayless
Repeated Values of Euler s Function a talk by Paul Kinlaw on joint work with Jonathan Bayless Oct 4 205 Problem of Erdős: Consider solutions of the equations ϕ(n) = ϕ(n + k), σ(n) = σ(n + k) for fixed
AN ARITHMETIC FORMULA FOR THE PARTITION FUNCTION
AN ARITHMETIC FORMULA FOR THE PARTITION FUNCTION KATHRIN BRINGMANN AND KEN ONO 1 Introduction and Statement of Results A partition of a non-negative integer n is a non-increasing sequence of positive integers
e Integer Partition Function Kevin Durant
e Integer Partition Function Kevin Durant October 00 Abstract e function pn counts the number of integer partitions of a positive integer n. e purpose of this text is to deduce a remarable formula for
On the Ordinary and Signed Göllnitz-Gordon Partitions
On the Ordinary and Signed Göllnitz-Gordon Partitions Andrew V. Sills Department of Mathematical Sciences Georgia Southern University Statesboro, Georgia, USA asills@georgiasouthern.edu Version of October
Congruence Properties of Partition Function
CHAPTER H Congruence Properties of Partition Function Congruence properties of p(n), the number of partitions of n, were first discovered by Ramanujan on examining the table of the first 200 values of
Congruences for Fishburn numbers modulo prime powers
Congruences for Fishburn numbers modulo prime powers Partitions, q-series, and modular forms AMS Joint Mathematics Meetings, San Antonio January, 205 University of Illinois at Urbana Champaign ξ(3) = 5
Integer Partitions and Why Counting Them is Hard
Portland State University PDXScholar University Honors Theses University Honors College 3-3-207 Integer Partitions and Why Counting Them is Hard Jose A. Ortiz Portland State University Let us know how
Some congruences for Andrews Paule s broken 2-diamond partitions
Discrete Mathematics 308 (2008) 5735 5741 www.elsevier.com/locate/disc Some congruences for Andrews Paule s broken 2-diamond partitions Song Heng Chan Division of Mathematical Sciences, School of Physical
Math Assignment 11
Math 2280 - Assignment 11 Dylan Zwick Fall 2013 Section 8.1-2, 8, 13, 21, 25 Section 8.2-1, 7, 14, 17, 32 Section 8.3-1, 8, 15, 18, 24 1 Section 8.1 - Introduction and Review of Power Series 8.1.2 - Find
New asymptotic expansion for the Γ (z) function.
New asymptotic expansion for the Γ z function. Gergő Nemes Institute of Mathematics, Eötvös Loránd University 7 Budapest, Hungary September 4, 007 Published in Stan s Library, Volume II, 3 Dec 007. Link:
Solutions 2016 AB Exam
Solutions 206 AB Exam Texas A&M High School Math Contest October 22, 206. If (x, y) is a point on the circle x 2 + y 2 = and the distance from (x, y) to (0, ) is 6 5, what is the value of y? ANSWER: 7