Re: Bessel Numbers, Borel Numbers.
From: Patrick D. Rockwell (hnhc85a_at_prodigy.net)
Date: 12/03/04
- Next message: Nathan: "Re: Platonism"
- Previous message: Johannes Bergmark: "Circle-line intersection"
- In reply to: G. A. Edgar: "Re: Bessel Numbers, Borel Numbers."
- Next in thread: Patrick D. Rockwell: "Re: Bessel Numbers, Borel Numbers."
- Reply: Patrick D. Rockwell: "Re: Bessel Numbers, Borel Numbers."
- Messages sorted by: [ date ] [ thread ]
Date: Fri, 03 Dec 2004 20:45:18 GMT
"G. A. Edgar" <edgar@math.ohio-state.edu.invalid> wrote in message
news:031220041450524277%edgar@math.ohio-state.edu.invalid...
> In article <QO1sd.38438$Qv5.37455@newssvr33.news.prodigy.com>, Patrick
> D. Rockwell <hnhc85a@prodigy.net> wrote:
>
>> "Peter Webb" <webbfamily-diespamdie@optusnet.com.au> wrote in message
>> news:41b04a77$0$20859$afc38c87@news.optusnet.com.au...
>> >
>> > "Patrick D. Rockwell" <hnhc85a@prodigy.net> wrote in message
>> > news:lhUrd.2612$cy4.460@newssvr31.news.prodigy.com...
>> >> What are Bessel Numbers, and Borel Numbers? How do you generate them?
>> >>
>> >> --
>> >> -------------------------------
>> >> Patrick D. Rockwell
>> >>
>> >
>> > What is Google, and how do you use it?
>> >
>> >
>>
>> I already did! I couldn't find a straightforward answer that I could make
>> sense of. I want to see if someone here can give me a better
>> answer.
>
> My results from Google...
>
> Bessel numbers
>
> http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cg
> i?Anum=A006789
>
> Borel's number
>
> http://www.cs.auckland.ac.nz/CDMTCS/chaitin/summer.html
>
Well, I'm going to study that page on Borel numbers. Thanks. I already
used Google to find that page. One of the ones that I couldn't make much
sense of. The page gives these numbers
1,1,2,5,14,43,143,509,1922,7651,31965,139685,636712,3020203,
14878176,75982829,401654560,2194564531,12377765239,
71980880885,431114329728,2656559925883,16825918195484,
109439943234749,730365368850192
It says that Bessel numbers are the number of nonoverlapping
partitions of an n-set
into equivalence classes.
That much I understand, but it then gives this generating function
G.f. 1/(1-x-x^2/(1-2x-x^2/(1-3x-x^2/...))) (a continued fraction). Could
this generate the above numbers?
I downloaded a PDF file which defines Bessel numbers in the following way.
Let the Bessel number of the second kind B(n, k) be the number of set
partitions of [n] into k
blocks of size one or two, and let the Bessel number of the first kind b(n,
k) be the coefficient of xn?k
in ?y(n)?1(?x) , where y(n)(x) is the nth Bessel polynomial. In this paper,
we show that Bessel numbers
satisfy two properties of Stirling numbers: The two kinds of Bessel numbers
are related by inverse
formulas, and both Bessel numbers of the first kind and the second kind form
log-concave sequences.
By constructing sign-reversing involutions, we prove the inverse formulas.
We review Krattenthaler's
injection for the log-concavity of Bessel numbers of the second kind, and
give a new explicit injection for
the log-concavity of signless Bessel numbers of the first kind.
It then defines b(n,k) = (-1)^(n-k)*(2*n-k-1)!/((2^(n-k)(n-k)!(k-1)!) if
1<=k<=n and 0 if 1<=n<=k
B(n,k)=n!/((2^(n-k)*(n-k)!(2k-n)!) if (n/2)<=k<=n, 0 otherwise.
-- ------------------------------- Patrick D. Rockwell
- Next message: Nathan: "Re: Platonism"
- Previous message: Johannes Bergmark: "Circle-line intersection"
- In reply to: G. A. Edgar: "Re: Bessel Numbers, Borel Numbers."
- Next in thread: Patrick D. Rockwell: "Re: Bessel Numbers, Borel Numbers."
- Reply: Patrick D. Rockwell: "Re: Bessel Numbers, Borel Numbers."
- Messages sorted by: [ date ] [ thread ]
Relevant Pages
|
