# 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 $E_n$.

19 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
231 views

### odd property of Eulerian numbers

One of the curious features of Pascal's triangle is that each row contains a two power number of odd entries. In fact, the precise number is $2^{b(n)}$ where $b(n)$ denotes the sum of the bits of $n$...
• 2,437
255 views

### Proof for a summation-procedure using the matrix of Eulerian numbers?

I've discussed a procedure for divergent summation using the matrix of Eulerian numbers occasionally in the last years (initially here, and here in MSE and MO but not in that generality and thus(?) ...
71 views

### Connection between Eulerian numbers and number of elements in set of uniform variables greater than the mean?

I was recently investigating the following question: Given $n$ independent $\text{Unif(0, 1)}$ variables $U_1,\ldots,U_n$, let $m$ be the number of elements in $[U_1,\ldots,U_n]$ that are greater than ...
224 views

### Interpretation of Eulerian numbers using the principle of inclusion-exclusion

Eulerian numbers, denoted $e_{n,k}$, are defined as the number of permutations of $[n]$={1,2,...,n} such that there are k "ascents". (For example, the permutation 23541 of [5] would have 3 ascents, ...
109 views

### $\mathbb{Z}$-Polynomials in an Enumeration Identity

I've conjectured the following identity: For $1 \leqslant k \leqslant l \leqslant n$ and $m \in \mathbb{N}$, \begin{align} \sum_{1 \leqslant i_1 < \cdots < i_l \leqslant n} i_{k}^{m} = \sum_{j = ...
• 17.1k
104 views

### About $t$-analogue of the Eulerian polynomials.

A certain way to define the $t$-analogue of the Eulerian polynomials $C_n(x)$ is by $$C_n(x,t)=\sum_{\pi\in S_n}x^{\text{des}(\pi)+1}t^{\text{maj}(\pi)}$$ where $des(\pi)$ is the descents in $\pi$, ...
• 1,536
61 views

### The second-order Eulerian numbers meet the Clausen numbers

The (generalized) Clausen numbers A160014 are defined as $$\operatorname{C}_{n, k} = \prod_{ p\, -\, k\, |\, n} p \quad (p \in \mathbb{P})$$ where $\mathbb{P}$ denotes the primes. The classical ...
166 views

### Permutation statistics in multiple rows

Usually we study the statistics of a permutation written in one row. Is there any result for the statistics of a permutation written in multiple rows? Let me give an example in order to be more clear: ...
• 422
97 views

The Euler numbers $f(n,k)$ are given in Generatingfunctionology as the number of permutations of $[n]$ with exactly $k$ increasing runs (exercise 1.18.c). Its recurrence relation is $$f(n,k) = kf(n-1, ... • 402 2 votes 0 answers 51 views ### Does the sum \sum_{k\ge 0}A(n,k)k^t have a closed form? My friend and I are trying to find a closed form for the sum \sum_{k\ge 0}A(n,k)k^t, in which the notation A(n,k) is for the Eulerian number. (An alternating notation is \left<n\atop k\right&... 2 votes 0 answers 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 ... • 265 1 vote 0 answers 38 views ### Sign alternating for n-th Eulerian polynomial at x=-2 What is the law of sign alternating for the n-th Eulerian polynomial evaluated at x=-2 (A087674)? I have some ideas, but need more (than 100 from related A212846) terms. How and where can I ... • 1,475 1 vote 0 answers 753 views ### Prove Eulerian Number using Combinatorics. For any n ≥ 1 and 1 ≤ k ≤ n, define the “Eulerian number” e(n, k) to be the number of permutations of \{1, 2, . . . , n\} with exactly k −1 descents. So e(n, 1) = e(n, n) = 1, and e(n, k) ... • 434 1 vote 0 answers 117 views ### Formula for q-Analogue of Eulerian Polynomial Define \text{d}(\sigma) and \text{maj}(\sigma) to be the number of descents and the major index of the permutation \sigma, respectively. Define a q-Analogue of the Eulerian Polynomials by A_n(... • 9,205 1 vote 0 answers 58 views ### Somehow "mirroring" the Taylor-expansion of some g(x) In proceeding with an older discussion of a summation-procedure for divergent series using the matrix of Eulerian numbers I came to a rather general formulation but cannot/do-not-know-how-to make sure,... 1 vote 0 answers 152 views ### Is that series-transformation known in the context of divergent summation Background: In the context of divergent summation I'm analyzing the matrix of eulerian numbers for a regular matrix-summation method. Beginning indexes at zero (r for "row", c for "column") the ... 0 votes 0 answers 23 views ### Request bibliographic reference(s) for finite alternating sums with Eulerian numbers I would like to know a bibliographic reference for a math formula. I found this formula on Wikipedia, but no reference is given. It's the 2nd formula (A(n,k) is an Eulerian number):$$\sum_{k=0}^{n-...
• 1
I was reading the proof of the closed-form formula of Eulerian numbers ($A(n,k)$) from Bona's book. I had a doubt in his classification of "removable fences" and how they eliminate the ...
We define the q-analog of the Eulerian polynomial $A_n(x)$ as $A_n(x,q) = \sum_{\sigma \in S_n} x^{d(\sigma)+1}q^{maj(\sigma)}$, where $S_n$ is the set of permutations of $n$ elements, $d(\sigma)$ is ...