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

[Olivier Gerard]

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