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17
votes
1answer
527 views

Monomer-Dimer tatami tilings need better relationships with other math. Summary of results.

A monomer-dimer tiling of a rectangular grid with $r$ rows and $c$ columns satisfies the \emph{tatami} condition if no four tiles meet at any point. (or you can think of it as the removal of a ...
1
vote
1answer
58 views

Matrix Generator for M/M/1 Queue Waiting Time Distribution

I "believe" that generator, $\bf W$, of the waiting time distribution for the M/M/1 queue is given by the following (I'm not sure if this is even correct): ${\bf W} =\left( \begin{array}{ccccc} 0 ...
0
votes
1answer
253 views

Probability generating function zero implies random variable is infinite

Let $V$ be a random variable supported on the nonnegative integers (including $\infty$) and $f(x) = \mathbf E x^V$ be the probability generating function. In our model $V$ is the number of visits to ...
7
votes
0answers
299 views

Generalization of Cauchy's identity

Let $ s_{\lambda} $ be the schur function associated to the partition $ \lambda $. Cauchy's identity (as in Macdonald) states that $$ \sum_{\lambda} s_{\lambda}(X)s_{\lambda}(Y) = ...
5
votes
0answers
262 views

When does a triangle of numbers have a zero row sum?

Suppose we have a triangle of numbers defined by the recurrence relation $$\left| n \atop k \right| = f(n,k) \left| n-1 \atop k \right| +g(n,k) \left| n-1 \atop k-1 \right| + [n=k=0],$$ for some ...
4
votes
0answers
173 views

probabilistic terminology for polynomials with positive coefficients

Given a polynomial $P(x) = p_0 + p_1 x + p_2 x^2 + ... + p_n x^n$ with non-negative coefficients, is there a standard name for (the function of $p_1,...,p_n$ equal to) the variance of an ...
4
votes
0answers
224 views

Generating function related to 2-residues of partitions

Question Express the following power series in two commuting variables $x,y$ as an infinite product, or a short sum of infinite products: $$ ...
3
votes
0answers
113 views

Asymptotics and combinatorics

Wright's expansion of $$ (1-z)^b\exp[A/(1-z)^c], \text{for } A > 0, 0<c<1\tag{1} $$ is, in the words of the late, great Mark Kac "well known to those that know it well". (See, for example, ...
3
votes
0answers
190 views

Generating function for the characteristic function of prime numbers

What do we know about the generating function of $\chi(n)$ (A010051) $$ f(x) = \sum_{n=0}^\infty \chi(n)x^n = \sum_{p\text{ prime}} x^p $$ for $\chi(n)$ the characteristic function of the primes: ...
3
votes
0answers
176 views

closure properties of q-differential equations

I am interested in q-differential equations of the form $p(f(z), f(qz),\dots,f(q^kz))=0$ where $p$ is a polynomial and $k$ an nonnegative integer. I wonder about the closure properties of the class ...
2
votes
0answers
76 views

Lagrangean equations for the generating function of quadrangulations

Let $M(z)$ be the generating function of edge-rooted connected quadrangulations, with $z$ marking the number of edges. I derived the following Lagrangean equations for $M(z)$: $$M(z) = ...
2
votes
0answers
56 views

Rank generating functions on a graded poset and a linear extension of it

Let $A$ be a possibly infinite ensemble and let $\leq$ be a partial order between elements of $A$. We denote $P_{\leq}=(A,\leq)$ the corresponding poset. Furthermore, we suppose that $P$ is graded, ...
2
votes
0answers
219 views

Algebraic Dirichlet series and beyond

I wonder what the "right" notion of "algebraic Dirichlet series" might be. Here I'm thinking of formal Dirichlet series $D(s)=\sum_{n\geq 1} a_n/n^s$, say with $a_n$ being rational numbers. I'm ...
2
votes
0answers
761 views

Which Hadamard Products of Generating Functions Are Known?

The Hadamard product, Schur product, or entrywise product of two generating functions is computed as follows: The Hadamard Product, H(x), given two generating functions f(x) and g(x) where $$ f(x) = ...
1
vote
0answers
106 views

Solving a recurrence (with the form of a convolution) involving binomial coefficients

While dealing with a problem related to intersection of hyperplanes I have come across the following recurrence to obtain the values of $K_{j}$ \begin{array}{cccccccccc} 1 & = & ...
1
vote
0answers
102 views

Is there inverse FFT algorithm for Fourier transform of a integer-valued random variable?

In many applications, it is possible to derive an explicit expression for the Fourier transform of a random variable $X$ $$\varphi (\theta ) = \sum\limits_{n = 0}^\infty {{p_n}} {e^{in\theta }}$$ ...
1
vote
0answers
81 views

Generating series of free PROs

Let \begin{equation} G := \biguplus_{p \geq 0} \: \biguplus_{q \geq 0} G(p, q) \end{equation} be a bigraded set of generators and $\mathcal{F}(G)$ be the free PRO generated by $G$ (see [1] for a net ...
1
vote
0answers
177 views

Generalization of Lagrange inversion with “skewed” formal parameter

I am interested in obtaining an analog of the Lagrange inversion formula, starting from a generalization of the implicit equation. Ordinary Lagrange reversion, as I am familiar with it, starts with ...
1
vote
0answers
186 views

An equation about generating functions and subfactorial

As I promised, I clone the problem from Math.SE to here, in order to find a solution. Suppose $G_n(w)$ is a formal power series (really a probability generating function, see the following ...
1
vote
0answers
295 views

Transfinite Sums Related to a Sequence

Hello, Given a sequence $S$ indexed by the finite ordinals, a limit ordinal $\alpha$, and $k \in \mathbb{N}$, define $S_{\alpha+k}$(the extension of $S$ to $\alpha+k$) to be the sum over the products ...
0
votes
0answers
90 views

Estimating when does a certain binomial sum exceed an upper bound

Given a fixed integer $n > 0$ and $0 \le m \le n$ let us define the numbers $$f_{n,m} = \sum_{i=\lfloor m/2 \rfloor}^m {n-2i \choose n - m -i}{i+1 \choose m - i +1}.$$ For example $f_{n,0} = ...
0
votes
0answers
93 views

Are there generating functions of rational or integral solutions of Diophantine equation that

As we know,there are generating functions for c.e.languages which are some retricted rational or algebraic or transcendental functions dependent on the class of languages like regular ...
0
votes
0answers
87 views

What are the properties of this linear operator?

Suppose $f(x)$ is a function which satisfies the following condition: $$f(x)=\sum_{k=0}^\infty G(2k)\frac{x^{2k}}{(2k)!}$$ Where the generating function $G(x)$ is a "natural" or "discrete-analytic" ...
0
votes
0answers
107 views

Using extended group rings for combinatorial generating functions

In work of mine recently, I have come to investigate generalised recurrence relations. The generalisation I have in mind is where, instead of natural numbers or integers, the recurrence is over some ...
0
votes
0answers
189 views

functional equation, how to solve

Suppose $x_i, y_i \in \mathbb{R}^n$, and $F,G: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$. $$F(x, y) = \frac{x\circ Ay}{x^TAy}$$ $$G(x, y) = \frac{x\circ By}{x^TBy}$$ where $A$ and $B$ are ...
0
votes
0answers
91 views

Theorem Leads for tied-down random walk

scribd.com/doc/87930409/18/Leads-for-tied-down-random-walk [Theorem 3.7, page 38] Could you explain me the last equality in the proof? I mean this: $$\frac{2[\sqrt{1 - s^2t^2} - ...
0
votes
0answers
609 views

Any tips on finding generating functions from recurrence relations involving minimization and maximization?

Any general tips on or examples of finding interesting generating functions from recurrence relations involving minimization and maximization? I'd imagine the case with one term of a minimization or ...