# Questions tagged [eulerian-numbers]

For questions about the Eulerian numbers $A_{n,k}$, defined as the number of permutations in the symmetric group $S_n$ having $k$ descents. Not to be confused with Euler’s number $e$ or the Euler numbers $E_n$.

64 questions
Filter by
Sorted by
Tagged with
369 views

23 views

1 vote
103 views

1 vote
129 views

### Power series of $x/(1-ae^{-x})$.

I am looking for a power series expansion for the function $x(1-ae^{-x})^{-1}$ (perhaps for $0<a<1$). Using the Bernoulli numbers, I can write \begin{align*} \frac{x}{1-ae^{-x}} &= \frac{x}{...
364 views

### Recurrence formula for the Eulerian and derangement polynomial

For the Eulerian polynomial $$A_n(x)=\sum_{\pi\in S_n}x^{\mathrm{des}(\pi)}$$ it is well known, that we have the nice recurrence formula $$A_n(x)=\sum_{k=0}^{n-1}\binom{n}{k}A_k(x)(x-1)^{n-1-k}.$$ I ...
214 views

### The relation of the Bernoulli numbers to the Catalan numbers

The Bernoulli numbers $B_n$ are the backbone of calculus, and according to B. Mazur, they "act as a unifying force, holding together seemingly disparate fields of mathematics." The Catalan ...
97 views

194 views

### Simon Newcomb's problem

I am looking for an answer to the following problem. Let $S$ be the multiset $\{1^{d_1},2^{d_2},\dots,m^{d_m}\}$ $A_{S,k}$ is the number of permutations of $S$ with $k-1$ descents and no descent at ...
60 views

### Is my use of derivatives correct in this infinite sum?

I was inspired by a question that seeked to find a generating function for the sum of powers. We set $s(n,p)=\sum_{k=0}^n k^p$, now we are looking for $G(x,p)= \sum_{n=0}^{\infty}s(n,p)x^n$. ...
1 vote
The problem phrases as follows: Let $\sigma = (\sigma_1, . . . , \sigma_n)$ be a permutation. We say that element $i$ is progressive if $\sigma_i > i$. We write $pro(\sigma)$ for the number of ...