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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 $E_n$.

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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 $n$...
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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(?) ...
Gottfried Helms's user avatar
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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 ...
Emily Boyajian's user avatar
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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, ...
settheorynoob's user avatar
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$\mathbb{Z}$-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 = ...
user02138's user avatar
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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 $\pi$, ...
Clara's user avatar
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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 ...
Peter Luschny's user avatar
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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: ...
WunderNatur's user avatar
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Understanding the recurrence relation $f(n,k) = kf(n-1, k) + (n-k+1)f(n-1,k-1)$ for the Euler numbers $f(n,k)$

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, ...
Xin Yuan Li's user avatar
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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&...
Renko Usami's user avatar
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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 ...
bronko's user avatar
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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 ...
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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) ...
wtnmath's user avatar
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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(...
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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,...
Gottfried Helms's user avatar
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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 ...
Gottfried Helms's user avatar
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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-...
Roy's user avatar
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Closed form of Eulerian numbers Proof

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 ...
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Identity on the q-analog of the Eulerian polynomial

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 ...
BOlivianoperuano84's user avatar