Generating Functions

Mafia (also called Werewolf) is a party game. The participants are divided into two competing groups: citizens and a mafia. The objective is to eliminate the opponent group. The game consists of two consecutive phases (day and night) and a certain set of actions (e.g. lynching during day). The mafia members have additional powers (knowing each other, killing during night) whereas the citizens are more numerous.

We propose a simple mathematical model of the game, which is essentially a pure death process with discrete time. We find the closed-form solutions for the mafia winning-chance w(n,m) as well as for the evolution of the game. Moreover, we investigate the discrete properties of results, as well as their continuous-time approximations.

It turns out that a relatively small number of the mafia members, i.e. proportional to the square root of the total number of players, gives equal winning-chance for both groups. Furthermore, the game strongly depends on the parity of the total number of players.

We consider compositions of n, i.e., sequences of positive integers (or parts) (σi)ki=1 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.

The theory of partitions has interested some of the best minds since the 18
th
century. In 1742, Leonhard Euler established the generating f unction of P(n).
Godf rey Harold Hardy said that Sriniv asa Ramanujan was the f irst, and up to
now the only , mathematician to discov er any such properties of P(n). In 1981, S.
Barnard and J.M. Child stated that the diff erent ty pes of partitions of n in
symbolic f orm. In this paper, diff erent ty pes of partitions of n are also explained
with symbolic f orm. In 1952, E. Grosswald quoted that the line ar Diophantine
equation has distinct solutions; the set of solution is the number of partitions of
n. This paper prov es theorem 1 with the help of certain restrictions. In 1965,
Godf rey Harold Hardy and E. M. Wright stated that the ‘Conv ergence The orem’
conv erges inside the unit circle. Theorem 2 has been prov ed here with easier
mathematical calculations. In 1853, British mathematician Norman Macleod
Ferrers explained a partition graphically by an array of dots or nodes. In this
paper, graphic representation of partitions, conjugate partitions and self -
conjugate partitions are described with the help of examples.

Special functions are expressed as differential and integral representations of some elementary functions. This general representation leads to a unique and unified representation of special functions. The integral representation gives the simplified formulations rather than differential representation.

In 1970 George E. Andrews defined the generating functions for P r (n)
1 and P r (n)
2 . In this article these generating functions
are discussed elaborately. This paper shows how to prove the theorem P r (n) P r (n)
3
2 = with a numerical example when n = 9 and
r = 2. In 1966 Andrews defined the terms A¢(n) and B¢(n), but this paper proves the remark A¢(n)= B¢(n) with the help of an
example when n = 10. In 1961 N. Bourbaki defined the term P(n,m) . This paper shows how to prove a Remark in terms of
P(n,m) , where P(n,m) is the number of partitions of the type of enumerated by P r (n)
3 with the further restrictions that
b1 £ 2m .

We analyze properties of the 2-adic valuations of an integer sequence that originates from an explicit evaluation of a quartic integral. We also give a combinatorial interpretation of the valuations of this sequence. Connections with the orbits arising from the Collatz (3x+1) problem are discussed.

We study the enumeration of Hamiltonian cycles on the thin grid cylinder graph $C_m \times P_{n+1}$. We distinguish two types of Hamiltonian cycles, and denote their numbers $h_m^A(n)$ and $h_m^B(n)$. For fixed $m$, both of them satisfy linear homogeneous recurrence relations with constant coefficients, and we derive their generating functions and other related results for $m\leq10$. The computational data we gathered suggests that $h^A_m(n)\sim h^B_m(n)$ when $m$ is even.

We consider an infinite-capacity buffer receiving fluid at a rate depending on the state of an M/M/1 queue. We obtain a new analytic expression for the joint stationary distribution of the buffer level and the state of the M/M/1 queue. This expression is obtained by the use of generating functions which are explicitly inverted. The case of a finite capacity fluid queue is also considered.

ABSTRACT A bargraph is a lattice path in with three allowed steps: the up step , the down step and the horizontal step . It starts at the origin with an up step and terminates as soon as it intersects the -axis again. A down step cannot follow an up step and vice versa. The height of a bargraph is the maximum coordinate attained by the graph. The width is the horizontal distance from the origin till the end. For bargraphs of fixed semi-perimeter we find the generating functions for the total height and the total width and hence find asymptotic estimates for the average height and the average width. Our methodology makes use of a bijection between bargraphs and -avoiding Dyck paths.

— Tumour immune interaction is modelled to evaluate the tumour cell size as a stochastic time dependent model. The life of a tumour cell is assumed to be in hypothetical phases of independently distributed time duration. The analysis uses generating functions to obtain the first few moments of the tumour cell size analyzed. The first few moments are expressed as a function of time and cell proliferation kinetics including the tumour cell escape rate from immune surveillance. Numerical results are obtained and are found to be consistent with the current theory.

We investigate the probability that a random composition (ordered partition) of the positive integer n has no parts occurring exactly j times, where j belongs to a specified finiteforbidden set'A of multiplicities. This probability is also studied in the related case of samples Γ=(Γ1, Γ2,…, Γn) of independent, identically distributed random variables with a geometric distribution. Résumé. Nous examinons la probabilité qu'une composition faite au hasard (une partition ordonnée) du nombre entier positif n n'a pas de partie qui arrivent ...

In this paper we develop a general method to enumerate the congruences of finite summations $\sum_{k=0}^{p-1} \frac{a_k}{m^k} \!\pmod{p}$ and $\sum_{k=0}^{p-1-h} \frac{a_k a_{k+h}}{B^k} \!\pmod{p}$ for the the infinite sequence $\{a_n\}_{n\ge 0}$ with generating functions $(1+x f(x))^\frac{N}{2}$, where $f(x)$ is an integer polynomial and $N$ is an odd integer with $|N|< p$. We also enumerate the congruences of some similar finite summations involving generating functions $\frac{1-\alpha x -\sqrt{1-2(\alpha+\beta)x + Bx^2}}{\beta x}$ and $\frac{1-\alpha x-\sqrt{1-2\alpha x+(\alpha^2-4\beta)x^2}}{2\beta x^2}$.