[seqfan] A005002 inconsistent with wikipedia entry about Stirling numbers of the second kind?

[Richard Mathar]

It ought be possible to show by s.o. with some flexibility in
converting e.g.f's to binomial sums to demonstrate that
(apart from offset) A005000 and A006505 are the same -- this is
just proposed by comparing all the numbers.

Richard

> From seqfan-owner at ext.jussieu.fr  Tue Jul 15 09:32:58 2008
> Date: Tue, 15 Jul 2008 09:31:47 +0200
> From: "Olivier Gerard" <olivier.gerard at gmail.com>
> To: "Jonathan Post" <jvospost3 at gmail.com>
> Subject: Re: A005002 inconsistent with wikipedia entry about Stirling numbers of the second kind?
> Cc: SeqFan <seqfan at ext.jussieu.fr>
> 
> Dear Jonathan
> 
> On Tue, Jul 15, 2008 at 08:15, Jonathan Post <jvospost3 at gmail.com> wrote:
> > Does anyone have, in hardcopy, the  J. Riordan 1979 reference cited?
> > There seems to be an inconsistency between a wikipedia entry about
> > Stirling numbers of the second kind, allegedly to count rhyme schemes,
> > and A005002. Embarassing to me, as a much-published poet to be
> > confused here.
> >
> > Both differ from:
> > http://acm.uva.es/archive/nuevoportal/data/problem.php?p=2871
> >
> 
> The difference is due to the fact that A005002 (as well as A005000 and A005003)
> counts a particular kind of rhyming schemes with additional
> constraints whereas the reference
> to the fact that Stirling numbers count rhyming schemes is more
> accurately covered by  A000296  which is the total number of rhyming schemes
> (and a convolution on Stirling numbers of the second kind  /  Bell numbers)
> and also in the entry for Bell numbers where there is a slightly more
> detailed explanation:
> 
> " Number of distinct rhyme schemes for a poem of n lines: a rhyme
> scheme is a string of letters (eg, 'abba') such that the leftmost
> letter is always 'a' and no letter may be greater than one more than
> the greatest letter to its left. Thus 'aac' is not valid since 'c' is
> more than one greater than 'a'. For example, a(3)=5 because there are
> 5 rhyme schemes. aaa, aab, aba, abb, abc. - Bill Blewett
> (BillBle(AT)microsoft.com), Mar 23 2004 "
> 
> A005001 sums the Bell numbers (which are the sums of the Stirling
> numbers of the second kind),
> starting from n=0
> 
> You are right that there should be a detailled example and definition
> of the kind of rhyming scheme
> counted by   A005000 - A005003  so that we can extend the sequences by a formula
> to arbitrary n.
> 
> 
> Olivier