Talk:Eulerian number

Notation

There doesn't seem to be a standard notation, I've seen E(n,k), A(n,k) and angle brackets used in different sources. So for the sake of consistency I'm picking E(n,k) to be used in the article with a mention of the other notations at the beginning of the article.--RDBury (talk) 18:08, 15 May 2009 (UTC)

After review of some additional sources I'm changing to A(n,k). It's been used in more places than the others and it's more consistent with the A_n(x) notation for the polynomials.--RDBury (talk) 18:56, 15 May 2009 (UTC)

source of the Euler-snippet

Hi I wanted to find the context of the Euler-snippet; wirkstoff, could you please provide the page-number?

Thanks - Gotti 08:04, 31 August 2009 (UTC) —Preceding unsigned comment added by Druseltal2005 (talk • contribs) Found it myself Gotti 20:23, 1 September 2009 (UTC) —Preceding unsigned comment added by Druseltal2005 (talk • contribs)

page 485f. --84.130.170.107 (talk) 22:21, 6 September 2011 (UTC)

Identities

The first of the listed equations in the "Identities"-section was wrong. To be more precise, in the case where n = 0 the left-hand side is a geometric series, yielding ${\frac {x}{1-x}}$ as its limit. But the right-hand side of the equation is zero (we have an empty sum there). I adapted the sum on the right-hand side (the elements are now summed to n not n-1). This corrects the mistake for n = 0 and doesn't change anything for n > 0 because for the index m = n the summand involves a factor A(n,n), which is zero and is this zero itself, contributing nothing to the sum. Now, I'm not an expert in combinatorics and never really worked with Euler numbers. I hope someone, who has experience with these can confirm my correction or look it up in the literature. --TheLaeg (talk) 14:21, 29 March 2010 (UTC)

I adjusted indices of summation in the power-sum identity so it will be valid for n ≥ 0. Zaslav (talk) 20:11, 7 December 2013 (UTC)

Eulerian numbers vs Eulerian polynomials

Premise: the definition and the notation for the Eulerian numbers and polynomials is by no means standard. That said, I see no advantage in the choice of the given definition : "[Eulerian numbers of the first kind] are the coefficients of the Eulerian polynomials:

$A_{n}(x)=\sum _{m=0}^{n}A(n,m)\ x^{n-m}.$"

An obvious instance of standardization wants that if certain numbers C(n,m), for 0≤m≤n, are called the X-numbers, the corresponding X-polynomial of order n is

$\sum _{m=0}^{n}C(n,m)\ x^{m}.$

Allowing inconsistencies between the notation for the coefficients and the notation for polynomials, and adopting free variations such as $\sum _{m=0}^{n}C(n,m)\ x^{n-m}$ or $\sum _{m=0}^{n}C(n+1,m)\ x^{m}$ or $\sum _{m=0}^{n}(-1)^{m}C(n,m)\ x^{m}$ may be of some little local advantage, but turns out into a complete mess as soon as one treats polynomials of several type at one time (suppose we have to write sum down the sum of three special polynomials A_n(x)+B_n(x)+C_n(x), each with its own notation for denoting the coefficients, giving for granted that we are able to keep them all by hart). On the same lines, I don't think that it is an optimal choice having the same letter for the coefficients and the polynomials. In this case, they are too similar, with some risk of confusion. I would go for Knuth solution $\left\langle {n \atop m}\right\rangle ,$ which is also of more practical use. --pma 17:12, 19 November 2010 (UTC)

More on Identities

I've added a third identity on the page, along with how to get it. It relates to the picture posted at the introduction of this page (Euler's Eulerian polynomials). I discovered it a couple of weeks ago and was wondering if anyone has seen it before? I have not, but I am not expert on the subject.

1:26 Dec 07 2010 (PST).

OK but you may as well start from

$\sum _{k=1}^{\infty }k^{n}x^{k}={\frac {\sum _{m=0}^{n}A(n,m)x^{m+1}}{(1-x)^{n+1}}},$

multiply both sides by the coefficient $c_{n}$ of any entire power series $f(x)$, and sum over $n$; of course for $|x|<r$ you can exchange order of summation in the LHS say, provided $f(k)\leq Mr^{-k}$), and obtain

$\sum _{k=1}^{\infty }f(k)x^{k}=\sum _{n=0}^{\infty }c_{n}{\frac {\sum _{m=0}^{n}A(n,m)x^{m+1}}{(1-x)^{n+1}}},$

but what is the point? You can write down a lot of these identities... --pma 00:44, 8 December 2010 (UTC)

Oh I see! I guess this one is kind of special because it's the sum of the actual Eulerian polynomials themselves and it relates with the constant e.

5:32am Dec 08 2010. —Preceding unsigned comment added by 98.164.252.55 (talk) 13:33, 8 December 2010 (UTC)

This article is still lacking of something more relevant: e.g. the recurrence relations for the E. polynomials of first and second kind; it's also lacking of the g.F. of both. So, when this article will be rectified, there is something to add and something to remove! --pma 01:03, 9 December 2010 (UTC)

That's very strange! I remember earlier this year they did in fact have the recursive definition for them... check in the history, it should be there... [Edit: Hm or maybe not!]

Sorry if I wasted your time, please remove the identity if you feel it is useless.

I think this identity is nice but the proof should be removed. Either it is original research, or it is known and can be cited in the literature. I will place the text below here so it is still available if needed. It might be desirable to remove the identity altogether as I don't see a reason to prefer it to other identities, but I won't make that decision. Zaslav (talk) 21:27, 7 December 2013 (UTC)

Another interesting identity is given by the following manipulation:

$e^{k}=\sum _{n=0}^{\infty }{\frac {k^{n}}{n!}}$

$(x^{k})(e^{k})=\sum _{n=0}^{\infty }{\frac {(x^{k})(k^{n})}{n!}}$

$\sum _{k=1}^{\infty }(x^{k})(e^{k})=\sum _{k=1}^{\infty }\sum _{n=0}^{\infty }{\frac {(x^{k})(k^{n})}{n!}}$

For $0\leq x<{\frac {1}{e}}$, the terms on the right side are positive, so we may switch the sum. The terms on the left make a geometric series, and we know that converges. After all of that, we use the above identity to finish the manipulation:

$\sum _{k=1}^{\infty }(xe)^{k}=\sum _{k=1}^{\infty }\sum _{n=0}^{\infty }{\frac {(x^{k})(k^{n})}{n!}}$

${\frac {ex}{1-ex}}=\sum _{n=0}^{\infty }\sum _{k=1}^{\infty }{\frac {(x^{k})(k^{n})}{n!}}=\sum _{n=0}^{\infty }{\frac {\sum _{m=0}^{n}A(n,m)x^{m+1}}{n!(1-x)^{n+1}}}$

Finally, for $0\leq x<{\frac {1}{e}}$ we get

${\frac {e}{1-ex}}=\sum _{n=0}^{\infty }{\frac {\sum _{m=0}^{n}A(n,m)x^{m}}{n!(1-x)^{n+1}}}.$

Notice that the numerator on the right-hand side is the Eulerian polynomial shown at the top of this page.

Analogy to normal distribution & Pascal's triangle?

As n increases, do the values of A(n,k)/n! turn out to approximate a well known function arbitrarily well? Thanks, Rich peterson198.189.194.129 (talk) 02:50, 16 March 2012 (UTC)

Good question, to which I don't know the answer. One observation: you don't have the scaling quite right; we really want to scale vertically by something like ${\frac {\sqrt {n}}{n!}}$ (just based on numerics), which is actually very similar to the Pascal case. (If you just divide by n!, everything goes to 0.) --Joel B. Lewis (talk) 02:49, 17 March 2012 (UTC)

Update: it's normal. See http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2636886#p2636886 --Joel B. Lewis (talk) 19:33, 22 March 2012 (UTC)

thanks!198.189.194.129 (talk) 18:16, 23 March 2012 (UTC)

Discrepancy of the tables of Eulerian numbers regarding $A_{0,0}$

When I use the generating-function for the introduction of the article to create the table of Eulerian numbers, I get that table (indexes (r,c)=(0,0)...(n,m))

[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[1 4 1 0 0 0 0 0 0 0 0 0 0 0 0 0]
[1 11 11 1 0 0 0 0 0 0 0 0 0 0 0 0]

where the element A_{0,0}=1.

When I use the "explicite formula" (from that so-named section) I get

[0 0 0 0 0 0]
[1 0 0 0 0 0]
[1 1 0 0 0 0]
[1 4 1 0 0 0]
[1 11 11 1 0 0]

where A_{0,0}=0. The second version seems to be in use by combinatorists, and even with an prefixed column [1,0,0,0,...], (and called "shifted version", moreover giving it a unit-diagonal), while the first version is also occuring if we use the "polylog"-(or "Li") function rescaled by x and powers of (1-x). Could someone please look at this with the goal to

a) mention it,     
b) make a comment of
         b1) the history of and
         b2) the relevance/rationale in it? 

Note also that the generalization to negative (and even fractional) row-indexes is only possible when the first version is used. — Preceding unsigned comment added by Druseltal2005 (talk • contribs) 07:45, 19 February 2021 (UTC)

Removal of python code

I don't see that having python code in the article adds anything particularly useful; it seems to be original research. I will remove it shortly; but below is a copy for historical reference. 67.198.37.16 (talk) 23:55, 5 January 2024 (UTC)

The following is a Python implementation.

import math  # Python 3.8

def Ank(n, k) -> int:
    """
    Compute A(n, k) using the explicit formula.
    """
    def summand(i):
        return (-1) ** i * math.comb(n + 1, i) * (k + 1 - i) ** n
    return sum(map(summand, range(k + 1)))

def Anks(n) -> list:
    """
    Coefficient list for the n'th polynomial A_n(t).
    """
    return [1] if n == 0 else [Ank(n, k) for k in range(n)]

def eval_polynomial(coeffs, t):
    """
    Polynomial evaluation function.
    """
    return sum(c * t ** k for k, c in enumerate(coeffs))

def An(n, t: float) -> float:
    """
    Polynomial A_n(t).
    """
    return eval_polynomial(Anks(n), t)

# Print the first few polynomials
sup = lambda n: str(n).translate(str.maketrans("0123456789", "⁰¹²³⁴⁵⁶⁷⁸⁹"))
sub = lambda n: str(n).translate(str.maketrans("0123456789", "₀₁₂₃₄₅₆₇₈₉"))

NUM = 8
for n in range(NUM):
    print(f"A{sub(n)}(t) = " + " + ".join(f"{ank} t{sup(k)}" for k, ank in enumerate(Anks(n))))
    # E.g. A₇(t) = 1 t⁰ + 120 t¹ + 1191 t² + 2416 t³ + 1191 t⁴ + 120 t⁵ + 1 t⁶

    # Sanity checks
    assert Ank(n, 1) == 2 ** n - (n + 1)
    assert n == 0 or An(n, 1) == math.factorial(n)  # Cardinality check
    alternating_sum: float = sum((-1)**k * Ank(n, k) / math.comb(n - 1, k) for k in range(n))
    assert n < 2 or abs(alternating_sum) < 1e-13