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
had been able to ...
links.jstor.org/sici?sici=0025-570X(200304)76%3A2%3C99%3ACSPATB%3E2.0.CO%3B2-5
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 ...
links.jstor.org/sici?sici=0002-9890(196405)71%3A5%3C498%3ATNOPOA%3E2.0.CO%3B2-R
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
And see also:
The Poetry Handbook: A Guide to Reading Poetry for Pleasure and ... -
Google Books Result
by John Lennard - 2005 - Literary Criticism - 418 pages
For a stanza of five lines the possible number of rhyme-schemes alone
rises ... 3May 1819; N916) 2 The number of rhyme-schemes r(n) for a
stanza of n lines ...
books.google.com/books?isbn=0199265380...
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