The generating-functions tag has no usage guidance.

**55**

votes

**8**answers

8k views

### What is Lagrange Inversion good for?

I am planning an introductory combinatorics course (mixed grad-undergrad) and am trying to decide whether it is worth budgeting a day for Lagrange inversion. The reason I hesitate is that I know of ...

**44**

votes

**3**answers

5k views

### Hirzebruch's motivation of the Todd class

In Prospects in Mathematics (AM-70), Hirzebruch gives a nice discussion of why the formal power series $f(x) = 1 + b_1 x + b_2 x^2 + \dots$ defining the Todd class must be what it is. In particular, ...

**42**

votes

**10**answers

7k views

### The functional equation $f(f(x))=x+f(x)^2$

I'd like to gather information and references on the following functional equation for power series $$f(f(x))=x+f(x)^2,$$$$f(x)=\sum_{k=1}^\infty c_k x^k$$
(so $c_0=0$ is imposed).
First things that ...

**23**

votes

**3**answers

3k views

### Is the sum $\sum\limits_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2} \binom{2k+1}{j} \ge 0?$

I am trying to prove $\sum\limits_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2} \binom{2k+1}{j} \ge 0$. This inequality has been verified by computer for $k\le40$.
Some clues that might work (kindly provided by ...

**20**

votes

**2**answers

925 views

### Partitions to different parts not exceeding $n$

Consider the polynomial $(1+x)(1+x^2)\dots (1+x^n)=1+x+\dots+x^{n(n+1)/2}$, which enumerates subj. How to prove that it's coefficients increase up to $x^{n(n+1)/4}$ (and hence decrease after this)? Or ...

**20**

votes

**1**answer

810 views

### A strange sum over bipartite graphs

While mucking around with some generating functions related to enumeration of regular bipartite graphs, I stumbled across the following cutie. I wonder if anyone has seen it before, and/or if anyone ...

**19**

votes

**4**answers

1k views

### Are the q-Catalan numbers q-holonomic?

The generating function $f(z)$ of the Catalan numbers which is characterized by $f(z)=1+zf(z)^2$ is D-finite, or holonomic, i.e. it satisfies a linear differential equation with polynomial ...

**19**

votes

**1**answer

783 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 ...

**18**

votes

**1**answer

502 views

### What is the Generating Function for Skew Young Diagrams?

The Problem
This strikes me as a very natural problem which should have been asked (and solved?) already.
For each positive integer k, find a nice expression for the following generating function in ...

**16**

votes

**2**answers

624 views

### Combinatorial meaning of the functional equation for logarithm

If we set $\exp(x)=\sum x^k/k!$, then $\exp(x+y)=\exp(x)\cdot \exp(y)$. In terms of coefficients it means that $(x+y)^n=\sum \frac{n!}{k!(n-k)!} x^ky^{n-k}$, i.e. just binomial expansion.
Now ...

**14**

votes

**2**answers

234 views

### Generating functions for objects with irrational sizes

A problem I'm investigating concerns a combinatorial class in which the 'atoms' have irrational sizes. It seems likely that this is something that has been considered before, but I haven't been able ...

**13**

votes

**3**answers

1k views

### A seemingly simple combinatorial object that must have an easy generating function

One more question related to my earlier "Special" meanders.
I am trying to isolate simplest problems related to it. Here is one.
For a composition (i. e. a tuple of natural numbers) ...

**13**

votes

**1**answer

552 views

### Which sets of lattice points have rational generating functions?

Let $P$ be a subset of $\mathbb N^d$ (or of some normal pointed affine semigroup), and suppose that $f:=\sum_{p\in P}\ t^p\in\mathbb Z[[t_1,\ldots,t_d]]$ is a rational function. What can be said ...

**13**

votes

**1**answer

883 views

### How many pairs (M, N) of sets of size n have M + N = {0, 1, …, n^2-1}?

How many pairs (M, N) of sets of size n have M + N = {0, 1, ..., n^2-1}?
Manfred Schroeder, in Number Theory in Science and Communication, 4th edition, asks (p. 27): find all pairs of sets $(M,N)$, ...

**12**

votes

**6**answers

877 views

### An example of a series that is not differentially algebraic?

Motivated by this question, I remembered a question I was curious about sometime which I am sure has some easy and nice example for it as well, which I just can't think of for some reason. I want an ...

**12**

votes

**1**answer

311 views

### Generating function of the Thue-Morse sequence

Let $T$ be the generating function of the Thue-Morse sequence; thus,
$T(x)=x+x^2+x^4+x^7+\dotsb$. It is known that $T$ satisfies the nice
congruence
$$ (1+x)^3 T^2(x) + (1+x)^2 T(x) + x \equiv 0 ...

**12**

votes

**2**answers

404 views

### Asymptotics of coefficients of implicitely defined generating function

I have two integer sequences $\{a_n\}_{n=0}^\infty$ and $\{b_n\}_{n=0}^\infty$. Explicit formulas for the $a_n$ are known and their asymptotic growth is fully understood. My wish is to also understand ...

**11**

votes

**4**answers

1k views

### Ordinary Generating Function for Bell Numbers

In the OEIS entry for Bell numbers, there appears a generating function
$$\sum_{k=0}^\infty B_k t^k = \sum_{r=0}^\infty \prod_{i=1}^r \frac{t}{1-it}$$
However, I could not locate any proof of ...

**11**

votes

**1**answer

220 views

### Generating function of a sequence is not algebraic

Let we have a sequence $\{a_{n}\}$, such that $\forall n \,\, a_{n}>0$ and $a_{n} \rightarrow\infty, n\rightarrow\infty$. Also let's suppose that we have a subsequence $\{a_{n_{k}}\}$ such that ...

**11**

votes

**2**answers

510 views

### Series defined by a fixed-point functional equation

Description
I my work, I met fixed-point functional equations of a very specific form. Since I am not expert in the domain of functional equations, I have not pushed the study of these very far. Here ...

**11**

votes

**2**answers

773 views

### Positivity of sequences via generating series

There are different ways of showing that a given sequence $a_0,a_1,a_2,\dots$
of integers, say, is nonnegative. For example, one can show that $a_n$ count
something, or express $a_n$ as a (multiple) ...

**11**

votes

**1**answer

377 views

### Number of trivializations of a trivial word in the free group

Let $M$ be the free monoid on $2n$ generators $x_1,X_1,...,x_n,X_n$ and consider the set $T$ of all those elements of $M$ which map to 1 of the free group on $x_1,...,x_n$ under the homomorphism $\pi$ ...

**10**

votes

**5**answers

1k views

### Use of everywhere divergent generating functions

Generating functions are well-known to be much useful in combinatorics. But, maybe just since I am illiteral, all the applications coming in mind deal with power series, which are not just formal, but ...

**10**

votes

**1**answer

292 views

### Laurent polynomials associated to partitions and a $Q$-deformation of $\sigma(d)$

Let $\alpha \vdash d$ be a partition of $d$, i.e. $\alpha = (\alpha_1 \geq \alpha_2 \geq …\geq \alpha_l)$, where $\sum_k \alpha_k = d$. Define a Laurent polynomial in $Q$ as follows:
$$
P_\alpha(Q) = ...

**10**

votes

**1**answer

355 views

### Elephant populations (and Dyck words)

Hello,
I'm relatively new to this forum so apologies if I have tagged my
question incorrectly.
I have been in contact with a wildlife biologist recently concerning
counting elephant populations and ...

**9**

votes

**5**answers

1k views

### Coin flipping and a recurrence relation

How can one solve the following recurrence relation?
$f(n) = 1 + \frac{1}{2^n} \sum_{k = 0}^n {{n}\choose{k}} f(k)$
$f(0) = 0$
As it happens, I can show $f(n) = \Theta(\log n)$ through other means ...

**9**

votes

**2**answers

891 views

### What classes am I missing in the Picard lattice of a Kummer K3 surface?

Constructing the Kummer K3 of an Abelian surface $A$, we have an obvious 22-dimensional collection of classes in $H^2(K3, \mathbb{Z})$ given by the 16 (-2)-curves (which by construction do not ...

**9**

votes

**2**answers

265 views

### Asymptotic growth rate of coefficients of generating function

how to calculate the asymptotic growth rate of coefficients generating function $T(z)$ satisfied this identity
$T(z)=z+\frac{T(z)^3}{6}+\frac{T(z^2)T(z)}{2}+\frac{T(z^3)}{3}$

**9**

votes

**2**answers

877 views

### Can you find linear recurrence relation for dimensions of invariant tensors?

Let $V$ be a finite dimensional highest weight representation of a (semi)-simple Lie algebra. For each $n\ge 0$ take $a_n$ to be the dimension of the space of invariant tensors in $\otimes^n V$.
In ...

**9**

votes

**3**answers

593 views

### Identifying the generating function $ G(a,z) = \sum_{n=0}^{\infty} a^n z^{(n+1)(n+2)/2}. $

I have computed a generating function for a problem involving a particular series, and would like to know if anyone has any references or a categorisation for it? It's
$$
G(a,z) = \sum_{n=0}^{\infty} ...

**9**

votes

**3**answers

2k views

### Solving a general two-term combinatorial recurrence relation

What is known about explicit (not necessarily closed-form) solutions to the recurrence
$$R^n_k= (\alpha n) R^{n-1}_k + (\alpha' n + \beta' k) R^{n-1} _{k-1},$$
with initial condition $R_0^0 = 1$ and ...

**9**

votes

**1**answer

1k views

### Finding recurrence relation for a sequence of polynomials

The sequence
A059710
starts 1,0,1,1,4,10,35,...
This satisfies the polynomial recurrence relation
$$ (n+5)(n+6)a(n)=2(n-1)(2n+5)a(n-1)+(n-1)(19n+18)a(n-2)+14(n-1)(n-2)a(n-3) $$
I have a $q$-analogue ...

**9**

votes

**0**answers

113 views

### types of generating functions (ordinary, exponential, ???) closed under substitution

A nice feature of ordinary and exponential generating functions is that they are closed under substitution: if $F(z)$ and $G(z)$ both have integer coefficients, then $F(G(z))$ also has integer ...

**9**

votes

**0**answers

674 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) = ...

**8**

votes

**4**answers

718 views

### Partitions-Sum of divisors identity

A few years ago I first read about the marvelous Euler identity:
$\sum_{n\in\mathbb{N}}p(n)z^n=\prod_{k\geq1}\frac{1}{1-z^k}$,
where $p(n)$ is the number of partitions of $n$ ($p(0)=1$ by ...

**8**

votes

**2**answers

1k views

### Expressions involving Eulerian numbers of the second kind: trying to show $\sum_{m=0}^{n} (-1)^m(m)m!(2n-m-2)!\left\langle\left\langle n\atop m\right\rangle\right\rangle\neq0$ for even $n$.

Considering the success of a previous question involving Eulerian numbers, I thought I might throw this question into the mix. It comes from some localization computations in GW theory, but in this ...

**8**

votes

**1**answer

523 views

### Another formula for Bell numbers

Here is an observation (thanks to OEIS):
$$\sum_{i=0}^\infty \frac{i^k}{i!}= B_k e,$$ where $B_k$ is the $k$-th Bell number. I might be having reading comprehension issues, but I don't see this ...

**8**

votes

**1**answer

232 views

### Higher moments of information and Renyi entropy

For a given discrete probability distribution, Shannon entropy can be though as an expectation value $\langle - \log p \rangle$ (see also: What is entropy, really?, What is the role of the logarithm ...

**8**

votes

**2**answers

330 views

### Determining the Lambert series for $xq+x^2q^4+x^3q^9+…+x^nq^{n^2}+…$

I am trying to determine the polynomials $P_n(x)$ from
$$
xq+x^2q^4+x^3q^9+...+\ x^nq^{n^2}+...=\sum_{n\geqslant1}\frac{P_n(x)q^n}{1-xq^n};
$$
that is,
$$
\sum_{d|n}x^{\frac ...

**7**

votes

**2**answers

368 views

### What is the number of noncrossing acyclic digraphs?

A noncrossing graph on $n$ vertices is a graph drawn on $n$ points numbered from $1$ to $n$ in counter-clockwise order on a circle such that the edges lie entirely within the circle and do not cross ...

**7**

votes

**1**answer

199 views

### Constructing a generating function using a series with all negative and positive powers of a variable

Trying to count certain combinatorial structures, I arrived at a construction of their generating function through a very inconvenient procedure.
I realize that anybody who will read this has right ...

**7**

votes

**1**answer

920 views

### Generating function for Random Walk Hitting Time, taking the wrong root

In a calculation of the hitting time for a Bernoulli random walk we have to calculate the hitting time $\tau(1)=\inf\{n\ge 0:S_n=1\}$ to reach $+1$ and the generating function has the recursion ...

**6**

votes

**3**answers

374 views

### Tricky two-dimensional recurrence relation

I would like to obtain a closed form for the recurrence relation
$$a_{0,0} = 1,~~~~a_{0,m+1} = 0\\a_{n+1,0} = 2 + \frac 1 2 \cdot(a_{n,0} + a_{n,1})\\a_{n+1,m+1} = \frac 1 2 \cdot (a_{n,m} + ...

**6**

votes

**3**answers

485 views

### A diagonal operation on power series

Given a formal power series $f(x,y)=\sum_{n,m\ge 0}f(n,m)\: x^n y^m$ in two variables $x$ and $y$ over some field of characteristic zero, e.g. the field of complex numbers $\mathbb C$, we define a new ...

**6**

votes

**3**answers

859 views

### What information do the roots of the generating function of the nontrivial zeroes of the Riemann zeta function encode.

Let $a_{m}$ be the imaginary part of the nontrivial roots of the Riemann zeta function $\zeta(s)$. Suppose we have their generating function $u(x)=\sum_{m=1}^{\infty} ...

**6**

votes

**2**answers

1k views

### What are the best known bounds on the number of partitions of $n$ into exactly $k$ distinct parts?

For example, if $n = 10$ and $k = 3$, then the legal partitions are
$$10 = 7 + 2 + 1 = 6 + 3 + 1 = 5 + 4 + 1 = 5 + 3 + 2$$
so the answer is $4$. By choosing $k$ random elements of $\{1,\ldots,2n/k\}$, ...

**6**

votes

**1**answer

343 views

### Generating functions with all non-zero coefficients equal to one

Inspired by this question, I have been wondering if there are any useful generating functions with all non-zero coefficients equal to one. Obviously, the trivial generating function $\frac{1}{1-x}$ ...

**6**

votes

**2**answers

770 views

### Closed form or/and asymptotics of a hypergeometric sum

Dear mathematicians,
I am a computer scientist wandering in the deep sea of combinatorics and asymptotics to pursue a recent interest in average case analysis of algorithms. In doing so, I designed ...

**6**

votes

**1**answer

251 views

### Does the following operation on modular forms yield something modular?

Let $f(z) = \sum_{n=0}^\infty a_nq^n$ be the fourier expansion of a (quasi-)modular form (with $q = e^{2\pi i z}$). Consider the following related functions:
$$f_{m,k}(z) = \sum_{n=0}^\infty a_{mn + ...

**6**

votes

**1**answer

122 views

### Asymptotics of a Bivariate Generating Function

I have the following generating function,
$$G(x,y)=\sum_{n,k \geq 0}a(n,k)x^ny^k = \frac{(y^2-y)x+1}{(y-y^3)x^2-(y+1)x+1}$$
and I am interested in obtaining an asymptotic for the sequence $a(n,k)$ ...