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Questions tagged [analytic-combinatorics]

Use for questions related to counting combinatorial objects.

Analytic combinatorics is a technique for counting combinatorial objects. Analytic combinatorics uses the internal structure of the objects to derive formulas for their generating functions and then complex analysis techniques to get asymptotics.

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Generalizing the growth of sums of two squares

Consider the set $S$ of numbers which are the sum of two (integer) squares, and define $S(n)$ as the number of members of $S$ in $\{1,2,\ldots,n\}.$ It is well-known that $$ S(n) \sim \frac{Kn}{\sqrt{\log n}} $$ with a constant $K$ (0.7642...,…
Charles
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Prove that $\sum\limits_{j=k}^n\,(-1)^{j-k}\,\binom{j}{k}\,\binom{2n-j}{j}\,2^{2(n-j)}=\binom{2n+1}{2k+1}$.

In an attempt to answer this thread, I discovered an identity involving binomial coefficients. However, I am not able to find a proof. All tricks are welcome. Let $n$ and $k$ be nonnegative integers with $k\leq n$. Prove that…
Batominovski
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Generating Functions and Linear Diophantine Inequalities

The following exercise is from Analytic Combinatorics by Philippe Flajolet and Robert Sedgewick, page 46. A $k$-composition of $n$ is an ordered $k$-tuple of non-negative integers whose sum is $n$. Consider the class $\mathcal{F}$ of compositions…
A.E
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Multivariate Faà di Bruno's formula

I'm attempting to implement a computer algebra function using the combinatoric version of Faà di Bruno's formula presented by Michael Hardy in Combinatorics of Partial Derivatives that "collapses" partitions to account for multiple variables. The…
Sophia Gold
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The number of lattice points in d-dimensional ball

The following paper states that the number of lattice points in a $d$-dimensional ball of radius $R$ is $V_d R^d + O(R^\alpha)$ where $\alpha = d - 2$ and $V_d$ is the volume of the unit $d$-dimensional ball. The paper cites a German book Einführung…
Guy
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Combinatorial interpretation of $\frac{1}{1-e^{x+1}}$ or $\frac{1}{1-e^{x-1}}$

After like 100 hours searching info to unravel a mystery about expressions on $e$ like $\frac{e^{6} A_{5}\left(e^{-1}\right)}{e^{6}\left(1-e^{-1}\right)^{6}} \approx 5 !$ talked by me in here, user Jair Taylor redirected me to this generating…
Alejandro Quinche
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Combinatorial interpretation of rational function on e

Over the last few weeks I have become obsessed with expressions like $$ \frac{e+4 e^{2}+e^{3}}{(1-e)^{4}}, $$ $$ \frac{e+26 e^{2}+66 e^{3}+26 e^{4}+e^{5}}{(1-e)^{6}}, $$ or $$ \frac{e+120 e^{2}+1191 e^{3}+2416 e^{4}+1191 e^{5}+120…
Alejandro Quinche
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Deriving exponential generating function for central trinomial coefficients

A recent question (link) asked for a derivation of the (ordinary) generating function for the central trinomial coefficients $\{T_n\}$. But the OEIS page (A002426) also lists an exponential generating function $$\sum_{n=0}^\infty T_n…
Semiclassical
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Prove that $\sum\limits_{j=0}^k\,j\,\binom{n}{j}\,\binom{n-j}{2k-2j}\,2^{2k-2j}=n\binom{2n-2}{2k-2}$

I encountered this in my homework. I derived two ways to solve the problem and the answer which I have tested using programming, seem to be the same, but I am not sure how to prove this equation. Let $n$ and $k$ be nonnegative integers with $k\leq…
Yilin L.
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Flajolet & Sedgewick: How to compute the variance of the number of cycles in a random permutation?

I am reading the book Analytic Combinatorics 4ed by Sedgewick and Flajolet. On page 160 at Example III.4 the authors derive the variance of the number of cycles in a random permutation. I can follow the authors up to the part where the…
3nondatur
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Are there good bounds on the sum of inverse degrees for a graph?

I'm looking for any literature on the sum $\sum\limits_{v\in V} \frac{1}{\deg(v)}$, but when I look for inverted degree sum, not much comes up about this specifically. I figure that it is either old and common knowledge, or nobody has looked at…
Robert Short
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Number of ways to represent $100$ as a sum of $1, 5, 10, 25$

In how many ways can you represent $100$ as a sum using only these numbers: $1, 5, 10, 25$ if the order does not matter? What if the order did matter? My solution: Let the power of $x$ denote our current sum. We need to take any number of ones,…
Aemilius
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Generating functions for tail length and rho-length

I am trying to obtain generating functions for tail length and rho length of a random point in a random mapping. Let $\phi:\{1,2,\ldots,n\}\to \{1,2,\ldots,n\}$ be a random function. Consider the directed graph whose nodes are the elements…
Sam
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XOR operation inner-product proof

Consider the binary power set $\underline{K}$ which contains binary strings of length $N$. Example: Let $N=3$, we then have $$\underline{K} = \big\{\{000\},\{100\},\{010\},\{001\},\{110\},\{101\},\{011\},\{111\} \big\}.$$ Now consider two vector…
John Doe
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Bounding integrals using asymptotic expansions of the integrand

Im following the book "Analytic Combinatorics" by Flajolet and Sedgwick. I'm having trouble understanding the last part of the proof of the theorem regarding the standard function scale. In particular im confused on how to bound the right part of…
h3h325
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