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

Jonathan Post jvospost3 at gmail.com
Tue Jul 15 08:15:04 CEST 2008

```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

Subject: unsubmitted COMMENT FROM Jonathan Vos Post RE A005002

%I A005002
%S A005002 1, 4, 13, 41, 134, 471, 1819, 7778
%N A005002 Number of rhyme schemes.
%C A005002 wikipedia: "The Stirling numbers of the second kind can
represent the total number of rhyme schemes for a poem of n lines.
S(n,k) gives the number of possible rhyming schemes for n lines using
k unique rhyming syllables."
%H A005002 <a href="http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind#Rhyming_Schemes">Rhyming
Schemes, in Wikipedia entry for Stirling numbers of the second
kind</a>.
%e A005002 a(3) = 4 because for a poem of 3 lines, there is 1 rhyme
scheme using just 1 rhyme (aaa), 3 rhyme schemes using two rhymes
(aab, aba, abb), and one rhyme scheme using 3 rhymes (abc); we exclude
the last as only technically a rhyme scheme but with no rhymes, and
have 1 + 3 = 4.
a(4) = 13 because for a poem of 4 lines, there is 1 rhyme scheme using
just 1 rhyme (aaaa); 7 rhyme schemes using two rhymes (aaab, aaba,
abaa, baaa, aabb, abab, abba), noting that abbb is equivalent to baaa,
and that bbaa is equivalent to aabb; 6 schemes using 3 rhymes (aabc,
abac, abca, abbc, abcb, abcc); 1 using no rhymes (abcd) which we
exclude, giving 1 + 7 + 6 = 14, but -- whoops -- that's not 13 as
desired.
%Y A005002 Cf. A008277.
%O A005002 1,2
%K A005002 ,nonn,
%A A005002 Jonathan Vos Post (jvospost3 at gmail.com), Jul 15 2008

These in turn suggest that the wikiedpia entry comes from:

JSTOR: C. S. Peirce and the Bell Numbers
S(n, k) is the number of rhyme schemes for a stanza of n lines with k
rhyming .... The number of rhyme schemes we could have obtained if we

or

JSTOR: The Number of Partitions of a Set
A great many problems of enumeration can be interpreted as counting
the number of partitions of a finite set; for example, the number of
rhyme schemes for n ...

The former may refer to:
the number of rhyme schemes, or logical relatives (C. S. Pierce, Amer.
J. Math. vol. 3 (1880) p. 48) with m plexes of nuance

The Poetry Handbook: A Guide to Reading Poetry for Pleasure and ... -