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

Richard Mathar mathar at strw.leidenuniv.nl
Tue Oct 28 21:27:26 CET 2008

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.


> 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

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