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Results tagged with co.combinatorics
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user 51505
Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.
10
votes
Accepted
Number of binary strings with 'at least two consecutives' constraints
We start considering words with no consecutive equal characters at all. These words are called Smirnov words or Carlitz words. (See example III.24 Smirnov words from Analytic Combinatorics by Philipp …
- 516
2
votes
Important formulas in combinatorics
The asymptotic formula of the average number of comparisons used by the Quick Sort algorithm.
\begin{align*}
Q_n=2n(\ln n + \gamma -2)+2\ln n+2\gamma+1+O\left(\frac{1}{n}\right)\tag{1}
\end{align*} …
7
votes
Asymptotics of coefficients of implicitely defined generating function
There are some papers by V. Kruchinin addressing the composition of generating functions
Composita and its properties
Composition of ordinary generating functions
Derivation of Bell Polynomials of t …
- 516
7
votes
Accepted
Is there a simple proof of the following Identity for $\sum_{k=m-1}^l(-1)^{k+m}\frac{k+2}{k+...
We obtain for $l,m\in\mathbb{N}$ with $0\leq m-1 \leq l$:
\begin{align*}
\color{blue}{\sum_{k=m-1}^{l}}&\color{blue}{(-1)^{k+m}\frac{k+2}{k+1}\binom{l}{k}\binom{k+1}{m}}\\
&=\frac{1}{m}\sum_{k=m- …
- 516
1
vote
Proving a particular "Abel type" identiy
Here is a starter. We can simplify the RHS somewhat.
We obtain
\begin{align*}
\frac{1}{2}& \sum_{{n_1+n_2=n-1}\atop{n_1,n_2\geq 0}}\left[ (n_1+1)^{n_1-1} (n_2-2\ell+1)^{n_2} \binom{n_2}{\ell-1}\ …
- 516
3
votes
Accepted
Is there a simple proof of the following binomial Identity (part 2)?
Following the hint @darijgrinberg stated in the comment section with respect to the beauty inside the square brackets we focus on the sum and
we obtain
\begin{align*}
\color{blue}{\sum_{k=m+ …
- 516
3
votes
Combinatorial aspects of continued fractions
Hint: No book recommendation, but in case you are not aware of it you might appreciate the nice survey Combinatorial aspects of continued fractions and applications by Xavier G. Viennot in honor of …
- 516