Generating functions and congruences for 9-regular and 27-regular partitions in 3 colours

Let $b_{\ell;3}(n)$ denote the number of $\ell$-regular partitions of $n$ in 3 colours. In this paper, we find some general generating functions and new infinite families of congruences modulo arbitrary powers of $3$ when $\ell\in\{9,27\}$. For instance, for positive integers $n$ and $k$, we have\begin{align*}b_{9;3}\left(3^k\cdot n+3^k-1\right)&\equiv0~\left(\textup{mod}~3^{2k}\right),\\b_{27;3}\left(3^{2k+3}\cdot n+\dfrac{3^{2k+4}-13}{4}\right)&\equiv0~\left(\textup{mod}~3^{2k+5}\right).\end{align*}

Short Generating Functions for some Semigroup Algebras

The Electronic Journal of Combinatorics ◽

10.37236/1729 ◽

2003 ◽

Vol 10 (1) ◽

Cited By ~ 23

Author(s):

Graham Denham

Keyword(s):

Rational Function ◽

Generating Functions ◽

Hilbert Series ◽

Simple Expression ◽

Arbitrary Field ◽

Graded Algebra ◽

Semigroup Algebras ◽

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

Asymptotic counting theorems for primitive juggling patterns

International Journal of Number Theory ◽

10.1142/s1793042119500568 ◽

2019 ◽

Vol 15 (05) ◽

pp. 1037-1050

Author(s):

Erik R. Tou

Keyword(s):

Prime Number ◽

Generating Functions ◽

Arbitrary Number ◽

Mathematical Theory ◽

Prime Number Theorem ◽

Prime Numbers ◽

Theoretical Framework ◽

Positive Integers ◽

Infinite Set ◽

Fixed Period

The mathematics of juggling emerged after the development of siteswap notation in the 1980s. Consequently, much work was done to establish a mathematical theory that describes and enumerates the patterns that a juggler can (or would want to) execute. More recently, mathematicians have provided a broader picture of juggling sequences as an infinite set possessing properties similar to the set of positive integers. This theoretical framework moves beyond the physical possibilities of juggling and instead seeks more general mathematical results, such as an enumeration of juggling patterns with a fixed period and arbitrary number of balls. One problem unresolved until now is the enumeration of primitive juggling sequences, those fundamental juggling patterns that are analogous to the set of prime numbers. By applying analytic techniques to previously-known generating functions, we give asymptotic counting theorems for primitive juggling sequences, much as the prime number theorem gives asymptotic counts for the prime positive integers.

Generating Functions for Permutations which Contain a Given Descent Set

The Electronic Journal of Combinatorics ◽

10.37236/299 ◽

2010 ◽

Vol 17 (1) ◽

Author(s):

Jeffrey Remmel ◽

Manda Riehl

Keyword(s):

Symmetric Group ◽

Generating Functions ◽

Symmetric Function ◽

Schur Functions ◽

Permutation Statistics ◽

Finite Set ◽

Set Of Permutations ◽

Descent Set ◽

A large number of generating functions for permutation statistics can be obtained by applying homomorphisms to simple symmetric function identities. In particular, a large number of generating functions involving the number of descents of a permutation $\sigma$, $des(\sigma)$, arise in this way. For any given finite set $S$ of positive integers, we develop a method to produce similar generating functions for the set of permutations of the symmetric group $S_n$ whose descent set contains $S$. Our method will be to apply certain homomorphisms to symmetric function identities involving ribbon Schur functions.

Formulas involving sums of powers, special numbers and polynomials arising from p-adic integrals, trigonometric and generating functions

Publications de l Institut Mathematique ◽

10.2298/pim2022103k ◽

2020 ◽

Vol 108 (122) ◽

pp. 103-120

Author(s):

Neslihan Kilar ◽

Yilmaz Simsek

Keyword(s):

Generating Functions ◽

Bernoulli Polynomials ◽

Stirling Numbers ◽

Sums Of Powers ◽

Alternating Sums ◽

Special Numbers And Polynomials ◽

Bernoulli Polynomials And Numbers ◽

Euler Polynomials And Numbers ◽

Historical Remarks ◽

The formula for the sums of powers of positive integers, given by Faulhaber in 1631, is proven by using trigonometric identities and some properties of the Bernoulli polynomials. Using trigonometric functions identities and generating functions for some well-known special numbers and polynomials, many novel formulas and relations including alternating sums of powers of positive integers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the Fubini numbers, the Stirling numbers, the tangent numbers are also given. Moreover, by applying the Riemann integral and p-adic integrals involving the fermionic p-adic integral and the Volkenborn integral, some new identities and combinatorial sums related to the aforementioned numbers and polynomials are derived. Furthermore, we serve up some revealing and historical remarks and observations on the results of this paper.

PARTITIONS WITH AN ARBITRARY NUMBER OF SPECIFIED DISTANCES

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972719000650 ◽

2019 ◽

Vol 101 (1) ◽

pp. 35-39 ◽

Author(s):

BERNARD L. S. LIN

Keyword(s):

Explicit Formula ◽

Generating Function ◽

Generating Functions ◽

Arbitrary Number ◽

For positive integers $t_{1},\ldots ,t_{k}$, let $\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$ (respectively $p(n,t_{1},t_{2},\ldots ,t_{k})$) be the number of partitions of $n$ such that, if $m$ is the smallest part, then each of $m+t_{1},m+t_{1}+t_{2},\ldots ,m+t_{1}+t_{2}+\cdots +t_{k-1}$ appears as a part and the largest part is at most (respectively equal to) $m+t_{1}+t_{2}+\cdots +t_{k}$. Andrews et al. [‘Partitions with fixed differences between largest and smallest parts’, Proc. Amer. Math. Soc.143 (2015), 4283–4289] found an explicit formula for the generating function of $p(n,t_{1},t_{2},\ldots ,t_{k})$. We establish a $q$-series identity from which the formulae for the generating functions of $\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$ and $p(n,t_{1},t_{2},\ldots ,t_{k})$ can be obtained.

Descents after maxima in compositions

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.1251 ◽

2014 ◽

Vol Vol. 16 no. 1 (Combinatorics) ◽

Author(s):

Aubrey Blecher ◽

Charlotte Brennan ◽

Arnold Knopfmacher

Keyword(s):

Generating Functions ◽

Mellin Transforms ◽

Asymptotic Expressions ◽

International Audience ◽

Average Size ◽

Combinatorics International audience We consider compositions of n, i.e., sequences of positive integers (or parts) (σi)i=1k where σ1+σ2+...+σk=n. We define a maximum to be any part which is not less than any other part. The variable of interest is the size of the descent immediately following the first and the last maximum. Using generating functions and Mellin transforms, we obtain asymptotic expressions for the average size of these descents. Finally, we show with the use of a simple bijection between the compositions of n for n>1, that on average the descent after the last maximum is greater than the descent after the first.

On Some two way Classifications of Integers

Canadian Mathematical Bulletin ◽

10.4153/cmb-1959-013-x ◽

1959 ◽

Vol 2 (2) ◽

pp. 85-89 ◽

Cited By ~ 10

Author(s):

J. Lambek ◽

L. Moser

Keyword(s):

Generating Functions ◽

Similar Theorem ◽

In this note we use the method of generating functions to show that there is a unique way of splitting the non-negative integers into two classes in such a way that the sums of pairs of distinct integers will be the same (with same multiplicities) for both classes. We prove a similar theorem for products of positive integers and consider some related problems.

Evaluation of the convolution sums ∑ak+bl+cm=nσ(k)σ(l)σ(m) with lcm(a,b,c) = 7,8 or 9

International Journal of Number Theory ◽

10.1142/s1793042118500999 ◽

2018 ◽

Vol 14 (06) ◽

pp. 1637-1650 ◽

Author(s):

Yoon Kyung Park

Keyword(s):

Modular Form ◽

Generating Functions ◽

Convolution Sums ◽

Divisor Functions ◽

It is known that the generating functions of divisor functions are quasimodular forms of weight [Formula: see text]. Hence their product is a quasimodular form of higher weight. In this paper, we evaluate the convolution sums [Formula: see text] for all positive integers [Formula: see text] with [Formula: see text] or [Formula: see text] using theory of modular form.

Greatest Descents after any Maxima in Compositions

The Electronic Journal of Combinatorics ◽

10.37236/5303 ◽

2015 ◽

Vol 22 (4) ◽

Author(s):

Margaret Archibald ◽

Aubrey Blecher ◽

Charlotte Brennan ◽

Arnold Knopfmacher

Keyword(s):

Generating Functions ◽

Mellin Transforms ◽

Asymptotic Expressions ◽

In this paper, compositions of $n$ are studied. These are sequences of positive integers $(\sigma_i)_{i=1}^k$ whose sum is $n$. We define a maximum to be a part which is greater than or equal to all other parts. We investigate the size of the descents immediately following any maximum and we focus particularly on the largest and average of these, obtaining the generating functions in each case. Using Mellin transforms, we obtain asymptotic expressions for these quantities.

Record statistics in integer compositions

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2691 ◽

2009 ◽

Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽

Author(s):

Arnold Knopfmacher ◽

Toufik Mansour

Keyword(s):

Generating Functions ◽

Natural Numbers ◽

Mean Values ◽

Fixed Subset ◽

Record Statistics ◽

Weak Records ◽

International Audience ◽

International audience A $\textit{composition}$ $\sigma =a_1 a_2 \ldots a_m$ of $n$ is an ordered collection of positive integers whose sum is $n$. An element $a_i$ in $\sigma$ is a strong (weak) $\textit{record}$ if $a_i> a_j (a_i \geq a_j)$ for all $j=1,2,\ldots,i-1$. Furthermore, the position of this record is $i$. We derive generating functions for the total number of strong (weak) records in all compositions of $n$, as well as for the sum of the positions of the records in all compositions of $n$, where the parts $a_i$ belong to a fixed subset $A$ of the natural numbers. In particular when $A=\mathbb{N}$, we find the asymptotic mean values for the number, and for the sum of positions, of records in compositions of $n$.