scholarly journals The Distribution of Ascents of Size d or More in Partitions of n

2008 ◽  
Vol 17 (4) ◽  
pp. 495-509 ◽  
Author(s):  
CHARLOTTE BRENNAN ◽  
ARNOLD KNOPFMACHER ◽  
STEPHAN WAGNER
Keyword(s):  
Positive Integer ◽  
Finite Sequence ◽  
Limiting Distribution ◽  
Distinct Part ◽  
The Mean ◽  
Fixed Positive Integer ◽  
Positive Integers

A partition of a positive integer n is a finite sequence of positive integers a1, a2, . . ., ak such that a1+a2+ċ ċ ċ+ak=n and ai+1 ≥ ai for all i. Let d be a fixed positive integer. We say that we have an ascent of size d or more if ai+1 ≥ ai+d.We determine the mean, the variance and the limiting distribution of the number of ascents of size d or more (equivalently, the number of distinct part sizes of multiplicity d or more) in the partitions of n.

scholarly journals The distribution of ascents of size d or more in compositions

2009 ◽  
Vol Vol. 11 no. 1 (Combinatorics) ◽  
Author(s):  
Charlotte Brennan ◽  
Arnold Knopfmacher
Keyword(s):  
Positive Integer ◽  
Nonnegative Integer ◽  
Finite Sequence ◽  
Limiting Distribution ◽  
International Audience ◽  
The Mean ◽  
Average Size ◽  
Positive Integers ◽  
Mean Variance

Combinatorics International audience A composition of a positive integer n is a finite sequence of positive integers a(1), a(2), ..., a(k) such that a(1) + a(2) + ... + a(k) = n. Let d be a fixed nonnegative integer. We say that we have an ascent of size d or more if a(i+1) >= a(i) + d. We determine the mean, variance and limiting distribution of the number of ascents of size d or more in the set of compositions of n. We also study the average size of the greatest ascent over all compositions of n.

A Combinatorial Theorem

1966 ◽  
Vol 9 (4) ◽  
pp. 515-516
Author(s):  
Paul G. Bassett
Keyword(s):  
Positive Integer ◽  
Combinatorial Theorem ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Positive Integers

Let n be an arbitrary but fixed positive integer. Let Tn be the set of all monotone - increasing n-tuples of positive integers:1Define2In this note we prove that ϕ is a 1–1 mapping from Tn onto {1, 2, 3,…}.

Upper and Lower Bounds for a Function Related to Brown's Lemma

Integers ◽  
2010 ◽  
Vol 10 (6) ◽  
Author(s):  
Hayri Ardal
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Fixed Positive Integer ◽  
Positive Integers

AbstractThe well-known Brown's lemma says that for every finite coloring of the positive integers, there exist a fixed positive integer

scholarly journals On the generalized Ramanujan-Nagell equation $ x^2+(2k-1)^y = k^z $ with $ k\equiv 3 $ (mod 4)

2021 ◽  
Vol 6 (10) ◽  
pp. 10596-10601
Author(s):  
Yahui Yu ◽  
◽  
Jiayuan Hu ◽  
Keyword(s):  
Positive Integer ◽  
Unsolved Problem ◽  
Integer Solution ◽  
Positive Integer Solution ◽  
Elementary Methods ◽  
Fixed Positive Integer ◽  
Almost All ◽  
Terai’S Conjecture ◽  
Positive Integers

<abstract><p>Let $ k $ be a fixed positive integer with $ k &gt; 1 $. In 2014, N. Terai <sup>[<xref ref-type="bibr" rid="b6">6</xref>]</sup> conjectured that the equation $ x^2+(2k-1)^y = k^z $ has only the positive integer solution $ (x, y, z) = (k-1, 1, 2) $. This is still an unsolved problem as yet. For any positive integer $ n $, let $ Q(n) $ denote the squarefree part of $ n $. In this paper, using some elementary methods, we prove that if $ k\equiv 3 $ (mod 4) and $ Q(k-1)\ge 2.11 $ log $ k $, then the equation has only the positive integer solution $ (x, y, z) = (k-1, 1, 2) $. It can thus be seen that Terai's conjecture is true for almost all positive integers $ k $ with $ k\equiv 3 $(mod 4).</p></abstract>

scholarly journals GENERALIZED CONTINUED FRACTION EXPANSIONS WITH CONSTANT PARTIAL DENOMINATORS

2018 ◽  
Vol 107 (02) ◽  
pp. 272-288
Author(s):  
TOPI TÖRMÄ
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Continued Fraction ◽  
Positive Real Number ◽  
Rational Numbers ◽  
Positive Real ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Continued Fraction Expansions ◽  
Positive Integers

We study generalized continued fraction expansions of the form $$\begin{eqnarray}\frac{a_{1}}{N}\frac{}{+}\frac{a_{2}}{N}\frac{}{+}\frac{a_{3}}{N}\frac{}{+}\frac{}{\cdots },\end{eqnarray}$$ where $N$ is a fixed positive integer and the partial numerators $a_{i}$ are positive integers for all $i$ . We call these expansions $\operatorname{dn}_{N}$ expansions and show that every positive real number has infinitely many $\operatorname{dn}_{N}$ expansions for each $N$ . In particular, we study the $\operatorname{dn}_{N}$ expansions of rational numbers and quadratic irrationals. Finally, we show that every positive real number has, for each $N$ , a $\operatorname{dn}_{N}$ expansion with bounded partial numerators.

scholarly journals SHIFTED AND SHIFTLESS PARTITION IDENTITIES II

2007 ◽  
Vol 03 (01) ◽  
pp. 43-84 ◽  
Author(s):  
FRANK G. GARVAN ◽  
HAMZA YESILYURT
Keyword(s):  
Positive Integer ◽  
Theta Function ◽  
Computer Search ◽  
Modular Functions ◽  
Partition Identities ◽  
Partition Identity ◽  
Function Identity ◽  
Theta Function Identity ◽  
Fixed Positive Integer ◽  
Positive Integers

Let S and T be sets of positive integers and let a be a fixed positive integer. An a-shifted partition identity has the form [Formula: see text] Here p(S,n) is the number partitions of n whose parts are elements of S. For all known nontrivial shifted partition identities, the sets S and T are unions of arithmetic progressions modulo M for some M. In 1987, Andrews found two 1-shifted examples (M = 32, 40) and asked whether there were any more. In 1989, Kalvade responded with a further six. In 2000, the first author found 59 new 1-shifted identities using a computer search and showed how these could be proved using the theory of modular functions. Modular transformation of certain shifted identities leads to shiftless partition identities. Again let a be a fixed positive integer, and S, T be distinct sets of positive integers. A shiftless partition identity has the form [Formula: see text] In this paper, we show, except in one case, how all known 1-shifted and shiftless identities follow from a four-parameter theta-function identity due to Jacobi. New shifted and shiftless partition identities are proved.

Fixed points and iterations of mean-type mappings

2012 ◽  
Vol 10 (6) ◽  
Author(s):  
Janusz Matkowski
Keyword(s):  
Fixed Point ◽  
Positive Integer ◽  
Fixed Points ◽  
Metric Space ◽  
Real Interval ◽  
Type Mapping ◽  
Continuous Map ◽  
The Mean ◽  
Fixed Positive Integer ◽  
Pythagorean Harmony

AbstractLet (X, d) be a metric space and T: X → X a continuous map. If the sequence (T n)n∈ℕ of iterates of T is pointwise convergent in X, then for any x ∈ X, the limit $$\mu _T (x) = \mathop {\lim }\limits_{n \to \infty } T^n (x)$$ is a fixed point of T. The problem of determining the form of µT leads to the invariance equation µT ○ T = µT, which is difficult to solve in general if the set of fixed points of T is not a singleton. We consider this problem assuming that X = I p, where I is a real interval, p ≥ 2 a fixed positive integer and T is the mean-type mapping M =(M 1,...,M p) of I p. In this paper we give the explicit forms of µM for some classes of mean-type mappings. In particular, the classical Pythagorean harmony proportion can be interpreted as an important invariance equality. Some applications are presented. We show that, in general, the mean-type mappings are not non-expansive.

scholarly journals Two elementary commutativity theorems for generalized boolean rings

1997 ◽  
Vol 20 (2) ◽  
pp. 409-411
Author(s):  
Vishnu Gupta
Keyword(s):  
Positive Integer ◽  
Identity Element ◽  
Negative Integer ◽  
Free Ring ◽  
Boolean Rings ◽  
Commutativity Theorems ◽  
Fixed Positive Integer ◽  
Positive Integers ◽  
Torsion Free

In this paper we prove that ifRis a ring with1as an identity element in whichxm−xn∈Z(R)for allx∈Rand fixed relatively prime positive integersmandn, one of which is even, thenRis commutative. Also we prove that ifRis a2-torsion free ring with1in which(x2k)n+1−(x2k)n∈Z(R)for allx∈Rand fixed positive integernand non-negative integerk, thenRis commutative.

scholarly journals The first ascent of size $d$ or more in compositions

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Charlotte Brennan ◽  
Arnold Knopfmacher
Keyword(s):  
Positive Integer ◽  
Nonnegative Integer ◽  
Finite Sequence ◽  
Initial Height ◽  
International Audience ◽  
Positive Integers ◽  
Average Position

International audience A composition of a positive integer $n$ is a finite sequence of positive integers $a_1, a_2, \ldots, a_k$ such that $a_1+a_2+ \cdots +a_k=n$. Let $d$ be a fixed nonnegative integer. We say that we have an ascent of size $d$ or more at position $i$, if $a_{i+1}\geq a_i+d$. We study the average position, initial height and end height of the first ascent of size $d$ or more in compositions of $n$ as $n \to \infty$.

scholarly journals On Counting Permutations by Pairs of Congruence Classes of Major Index

10.37236/1638 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Hélène Barcelo ◽  
Robert Maule ◽  
Sheila Sundaram
Keyword(s):  
Positive Integer ◽  
Symmetric Group ◽  
Combinatorial Proof ◽  
Major Index ◽  
Sigma 1 ◽  
Fixed Positive Integer ◽  
Positive Integers ◽  
Combinatorial Methods ◽  
Congruence Classes

For a fixed positive integer $n,$ let $S_n$ denote the symmetric group of $n!$ permutations on $n$ symbols, and let maj${(\sigma)}$ denote the major index of a permutation $\sigma.$ Fix positive integers $k < \ell\leq n,$ and nonnegative integers $i,j.$ Let $m_n(i\backslash k; j\backslash \ell)$ denote the cardinality of the set $\{\sigma\in S_n:$ maj$(\sigma)\equiv i \pmod k,$ maj$ (\sigma^{-1})\equiv j \pmod \ell\}.$ In this paper we use combinatorial methods to investigate these numbers. Results of Gordon and Roselle imply that when $k,$ $\ell$ are relatively prime, $$ m_n(i\backslash k; j\backslash \ell)= { {n!}\over{k\cdot\ell}} .$$ We give a combinatorial proof of this in the case when $\ell$ divides $n-1$ and $k$ divides $n.$

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