Related papers: A new Rational Generating Function for the Frobeni…
We study variants of the \emph{Frobenius coin-exchange problem}: given $n$ positive relatively prime parameters, what is the largest integer that cannot be represented as a nonnegative integral linear combination of the given integers? This…
The Frobenius Coin Problem is a classic question in mathematics: given coins of specified denominations, what is the largest amount that cannot be formed using only those coins? This brief work covers a variation of such question, posing a…
Expanding on recent results of another an algorithm is presented that provides solution to the Frobenius Coin Problem in worst case O(n^2) in the magnitude of the largest denomination.
The classical Frobenius problem is to compute the largest number g not representable as a non-negative integer linear combination of non-negative integers x_1, x_2, ..., x_k, where gcd(x_1, x_2, ..., x_k) = 1. In this paper we consider…
The Frobenius coin problem in three variables, for three positive relatively prime integers $a_1< a_2< a_3$ asks to find the largest number not representable as $a_1x_1+a_2x_2+a_3x_3$ with non-negative integer coefficients $x_1$, $x_2$ and…
This paper presents a new methodology to count the number of numerical semigroups of given genus or Frobenius number. We apply generating function tools to the bounded polyhedron that classifies the semigroups with given genus (or Frobenius…
Using the notion of the composita, we obtain a method of solving iterative functional equations of the form $A^{2^n}(x)=F(x)$, where $F(x)=\sum_{n>0} f(n)x^n$, $f(1)\neq 0$. We prove that if $F(x)=\sum_{n>0} f(n)x^n$ has integer…
In this paper we study the coefficients of the powers of an ordinary generating function and their properties. A new class of functions based on compositions of an integer $n$ is introduced and is termed composita. We present theorems about…
Since their introduction by Andrews, generalized Frobenius partitions have interested a number of authors, many of whom have worked out explicit formulas for their generating functions in specific cases. This has uncovered interesting…
The powers of generating functions and its properties are analyzed. A new class of functions is introduced, based on the application of compositions of an integer $n$, called composita. The methods for obtaining reciprocal and reverse…
The Coin Change problem, also known as the Change-Making problem, is a well-studied combinatorial optimization problem, which involves minimizing the number of coins needed to make a specific change amount using a given set of coin…
We introduce a generating function associated to the homogeneous generators of a graded algebra that measures how far is this algebra from being finitely generated. For the case of some algebras of Frobenius endomorphisms we describe this…
For a simple connected graph $G$, the $Q$-generating function of the numbers $N_k$ of semi-edge walks of length $k$ in $G$ is defined by $W_Q(t)=\sum\nolimits_{k = 0}^\infty {N_k t^k }$. This paper reveals that the $Q$-generating function…
In this paper we present a generating function approach to two counting problems in elementary quantum mechanics. The first is to find the total ways of distributing identical particles among different states. The second is to find the…
The change-making problem consists of representing a certain amount of money with the least possible number of coins, from a given, pre-established set of denominations. The greedy algorithm works by choosing the coins of largest possible…
We compute generating functions for the sum of the real-valued character degrees of the finite general linear and unitary groups, through symmetric function computations. For the finite general linear group, we get a new combinatorial proof…
A solution is proposed for the problem of composition of ordinary generating functions. A new class of functions that provides a composition of ordinary generating functions is introduced; main theorems are presented; compositae are written…
Endowing the set of functional graphs (FGs) with the sum (disjoint union of graphs) and product (standard direct product on graphs) operations induces on FGs a structure of a commutative semiring R. The operations on R can be naturally…
This paper presents a new method to solve functional equations of multivariate generating functions, such as $$F(r,s)=e(r,s)+xf(r,s)F(1,1)+xg(r,s)F(qr,1)+xh(r,s)F(qr,qs),$$ giving a formula for $F(r,s)$ in terms of a sum over finite…
We examine two different ways of encoding a counting function, as a rational generating function and explicitly as a function (defined piecewise using the greatest integer function). We prove that, if the degree and number of input…