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50
votes
7answers
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
3answers
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
10answers
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 ...
20
votes
2answers
905 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
1answer
803 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
3answers
3k views

Need help proving that $\sum\limits_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2} \binom{2k+1}{j} \ge 0$

Hello. I have been trying very hard to show that $\sum\limits_{j=0}^{k-1}(-1)^{j+1}(k-j)^{2k-2} \binom{2k+1}{j} \ge 0$ and could not quite get anywhere. This inequality has been verified by computer ...
19
votes
4answers
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
1answer
750 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
1answer
471 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
2answers
593 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 ...
13
votes
1answer
549 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
1answer
882 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 $(M,N)$, ...
13
votes
2answers
218 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 ...
12
votes
6answers
865 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
1answer
280 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
2answers
356 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
2answers
498 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
2answers
769 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
1answer
375 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
5answers
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
1answer
289 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
1answer
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
4answers
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 ...
9
votes
5answers
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
2answers
869 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
2answers
871 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
3answers
591 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
3answers
1k 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
1answer
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
0answers
104 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
0answers
607 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
4answers
659 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
2answers
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
2answers
231 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 $T(z)=z+\frac{T(z)^3}{6}+\frac{T(z^2)T(z)}{2}+\frac{T(z^3)}{3}$
8
votes
1answer
221 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
2answers
313 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
2answers
364 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
1answer
432 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 ...
7
votes
1answer
188 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
1answer
862 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
3answers
464 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
3answers
841 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
2answers
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
1answer
341 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
2answers
749 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
1answer
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
1answer
113 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 $a(n,k)$ ...
6
votes
1answer
518 views

How to extract the diagonal from a bivariate generating function

Let $ F(s,t)= \sum_{i,j} f(i,j) s^i t^j$, which is a bivariate generating funcion of the number $f(i,j)$ for some enumeration problem. Sometimes we know about $F(s,t)$, but what we really need is the ...
6
votes
1answer
1k views

Solving recurrence equation with indexes from negative infinity to positive infinity

In many cases, the recurrence equations that people are solving involves index of only non-negative values. Here I have a recurrence equation which arises from transport of light in an infinite 1D ...
5
votes
2answers
345 views

An interesting calculation of derivative

I was trying to get the probability distribution $p(n)$ from a generating function $G(s)$ like this: $G(s) = e^{a(s-1)^2}=\sum s^np(n)$ I need first to do Maclaurin expansion of the exponential and ...