PARITY RESULTS FOR PARTITIONS WHEREIN EACH PART APPEARS AN ODD NUMBER OF TIMES

2018 ◽  
Vol 99 (1) ◽  
pp. 51-55
Author(s):  
MICHAEL D. HIRSCHHORN ◽  
JAMES A. SELLERS
Keyword(s):  
Generating Function ◽  
Proof Techniques ◽  
One To One

We consider the function $f(n)$ that enumerates partitions of weight $n$ wherein each part appears an odd number of times. Chern [‘Unlimited parity alternating partitions’, Quaest. Math. (to appear)] noted that such partitions can be placed in one-to-one correspondence with the partitions of $n$ which he calls unlimited parity alternating partitions with smallest part odd. Our goal is to study the parity of $f(n)$ in detail. In particular, we prove a characterisation of $f(2n)$ modulo 2 which implies that there are infinitely many Ramanujan-like congruences modulo 2 satisfied by the function $f.$ The proof techniques are elementary and involve classical generating function dissection tools.

On a result of Rubin and Vere-Jones concerning subcritical branching processes

1976 ◽  
Vol 13 (04) ◽  
pp. 804-808
Author(s):  
Fred M. Hoppe
Keyword(s):  
Mass Distribution ◽  
Generating Function ◽  
Branching Processes ◽  
Probability Generating Function ◽  
Single Parameter ◽  
Limit Distributions ◽  
Arbitrary Mass ◽  
One To One ◽  
Watson Process ◽  
Conditional Limit

If a subcritical Galton-Watson process is initiated with an arbitrary mass distribution, then it is known that under certain conditions proper conditional limit distributions exist, depending on a single parameter. It is shown here that there is a one-to-one correspondence between these distributions and those arising from the process with a linear offspring probability generating function.

On a result of Rubin and Vere-Jones concerning subcritical branching processes

1976 ◽  
Vol 13 (4) ◽  
pp. 804-808 ◽  
Author(s):  
Fred M. Hoppe
Keyword(s):  
Mass Distribution ◽  
Generating Function ◽  
Branching Processes ◽  
Probability Generating Function ◽  
Single Parameter ◽  
Limit Distributions ◽  
Arbitrary Mass ◽  
One To One ◽  
Watson Process ◽  
Conditional Limit

If a subcritical Galton-Watson process is initiated with an arbitrary mass distribution, then it is known that under certain conditions proper conditional limit distributions exist, depending on a single parameter. It is shown here that there is a one-to-one correspondence between these distributions and those arising from the process with a linear offspring probability generating function.

scholarly journals Counting Segmented Permutations Using Bicoloured Dyck Paths

10.37236/1936 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Anders Claesson
Keyword(s):  
Generating Function ◽  
Chebyshev Polynomials ◽  
Dyck Path ◽  
Dyck Paths ◽  
One To One ◽  
Bivariate Generating Function

A bicoloured Dyck path is a Dyck path in which each up-step is assigned one of two colours, say, red and green. We say that a permutation $\pi$ is $\sigma$-segmented if every occurrence $o$ of $\sigma$ in $\pi$ is a segment-occurrence (i.e., $o$ is a contiguous subword in $\pi$). We show combinatorially the following two results: The $132$-segmented permutations of length $n$ with $k$ occurrences of $132$ are in one-to-one correspondence with bicoloured Dyck paths of length $2n-4k$ with $k$ red up-steps. Similarly, the $123$-segmented permutations of length $n$ with $k$ occurrences of $123$ are in one-to-one correspondence with bicoloured Dyck paths of length $2n-4k$ with $k$ red up-steps, each of height less than $2$. We enumerate the permutations above by enumerating the corresponding bicoloured Dyck paths. More generally, we present a bivariate generating function for the number of bicoloured Dyck paths of length $2n$ with $k$ red up-steps, each of height less than $h$. This generating function is expressed in terms of Chebyshev polynomials of the second kind.

scholarly journals The application of substitutional analysis to invariants

1935 ◽  
Vol 234 (733) ◽  
pp. 79-114 ◽  
Keyword(s):  
Characteristic Function ◽  
Finite Order ◽  
Generating Function ◽  
Generating Functions ◽  
Polynomial Function ◽  
General Theorem ◽  
Simple Extension ◽  
New Form ◽  
One To One ◽  
New System

In a previous paper it was proved that the generating function for any class of ternary concomitants might be obtained from the corresponding generating function for gradients (coefficient products) by multiplication by (1 — x ) (1 — y ) x — y ). A generating function for ternary gradients was given in Theorem III of that paper, but it is of such a character that it is useless for purposes of calculation.In this paper a new system of generating functions is obtained applicable to perpetuants or to forms of finite order, and also to binary, ternary, or any forms. In Section I a class of polynomial function f a ( z ) is discussed which appeared in the paper just quoted in connection with the generating function for binary gradients of particular substitutional form in the perpetuant case. In Section II a one to one correspondence between binary perpetuants of particular substitutional form and the terms of the corresponding generating function is obtained by means of the tableau notation ; and this is used to give a very simple extension of GRACE's Theorem on irreducible perpetuants to the case of perpetuants of particular substitutional form. Section III deals with the properties of the functions for forms of finite order which correspond to the functions f a ( z ) for perpetuants. The generating functions for the different substitutional classes must by addition give the generating function for types. This in some cases has been obtained independently. Thus there arise certain algebraic identities. In Section IV a general theorem is established covering all these identities. It is obtained by means of the Characteristic Function of SCHUR. The same method is then used to express the binary generating functions in a new form.

A Support Program

1994 ◽  
Vol 25 (2) ◽  
pp. 112-114 ◽  
Author(s):  
Henna Grunblatt ◽  
Lisa Daar
Keyword(s):  
Hearing Aids ◽  
School Counselor ◽  
Support Program ◽  
Educational Audiology ◽  
Part Program ◽  
One To One

A program for providing information to children who are deaf about their deafness and addressing common concerns about deafness is detailed. Developed by a school audiologist and the school counselor, this two-part program is geared for children from 3 years to 15 years of age. The first part is an educational audiology program consisting of varied informational classes conducted by the audiologist. Five topics are addressed in this part of the program, including basic audiology, hearing aids, FM systems, audiograms, and student concerns. The second part of the program consists of individualized counseling. This involves both one-to-one counseling sessions between a student and the school counselor, as well as conjoint sessions conducted—with the student’s permission—by both the audiologist and the school counselor.

Review of One to One: The Experience of Psychotherapy.

1989 ◽  
Vol 34 (10) ◽  
pp. 958-958
Author(s):  
No authorship indicated
Keyword(s):  
One To One

Influence of Rim Run-Out on the Nonuniformity of Tire-Wheel Assemblies

1994 ◽  
Vol 22 (2) ◽  
pp. 99-120 ◽  
Author(s):  
T. B. Rhyne ◽  
R. Gall ◽  
L. Y. Chang
Keyword(s):  
Finite Element ◽  
Radial Force ◽  
Membrane Model ◽  
Model Experiments ◽  
Force Variation ◽  
One To One ◽  
The One

Abstract An analytical membrane model is used to study how wheel imperfections are converted into radial force variation of the tire-wheel assembly. This model indicates that the radial run-out of the rim generates run-out of the tire-wheel assembly at slightly less than the one to one ratio that was expected. Lateral run-out of the rim is found to generate radial run-out of the tire-wheel assembly at a ratio that is dependent on the tire design and the wheel width. Finite element studies of a production tire validate and quantify the results of the membrane model. Experiments using a specially constructed precision wheel demonstrate the behavior predicted by the models. Finally, a population of production tires and wheels show that the lateral run-out of the rims contribute a significant portion to the assembly radial force variation. These findings might be used to improve match-mounting results by taking lateral rim run-out into account.

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