# 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
831 views

• 1,503
188 views

• 170k
144 views

### A recurrence of the second-order Eulerian polynomials

Recently, some of the remarkable properties of second-order Eulerian numbers $\left\langle\!\!\left\langle n\atop k\right\rangle\!\!\right\rangle$ A340556 have been proved on MSE [ a , b , c ] ...
363 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 ...
• 293
160 views

• 7,243
166 views

1k views

### Half of the binomial theorem

The binomial theorem states that the generating function $\sum_{k=0}^n {n \choose k} x^k$ is equal to $(1+x)^n$ for any $n$. For a given $n$, let $$B(x)=\sum_{k=0}^n {2n+1\choose k} x^k.$$ That is, ...
• 415