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Reply-To: Discussion List on the History of Mathematics
From: "Dwayne A. Hickman"
Subject: Closed Form for Stirling Numbers
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
In-Reply-To: <009B599C.001FB580.11@vax.sbu.ac.uk>
Status: OR
There are closed form expressions for Stirling Numbers of the first
and second kind!
While the closed form expression for the second kind is rather nice,
the one for
the first kind "just isn't pretty".
Here goes,
S(n,k) = (-1)^(n-k) * (n-1)! * SUM( from a1=k-1, to n-1, of 1/a1 *
SUM( from a2=k-2, to n-2-a1, of 1/a2 *
SUM( from a3=k-3, to n-3-a2, of 1/a3 * SUM( ...
SUM( from a_(k-1)=1, to n-(k-1)-a_(k-2), of 1/a_(k-1) )...)
For example:
S(n,1) = (-1)^(n-1) * (n-1)!
S(n,2) = (-1)^(n-2) * (n-1)! * SUM( from a1=1, to n-1, of 1/a1 )
S(n,3) = (-1)^(n-3) * (n-1)! * SUM[ from a1=2, to n-1, of 1/a1 *
SUM( from a2=1, to n-2-a1, of 1/a2) ]
It gets long!
Please note that while using the recursive formula to calculate
S(5000,1000) might take about 8 billion calculations, it would take
about 10^15000 calculations using the above closed form expresison
directly. Fortunately, the closed form expression lends itself to
computer programming very well and calculation shortcuts can be
implemented!
Hope this helps some.
Dwayne A. Hickman
Newark, OH USA
From owner-math-history-list@ENTERPRISE.MAA.ORG Tue Aug 12 07:42:17 1997
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Date: Tue, 12 Aug 1997 07:39:51 -0500
Reply-To: Discussion List on the History of Mathematics
From: "Samuel S. Kutler"
Subject: Re: My biography with Stirling Numbers
Comments: To: dhickman1@AOL.COM
Comments: cc: ccstirling@juno.com
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
Status: OR
Dwayne:
I'm writing this primarily to you, but I'm copying the math-history-list
and my sometime student, Catherine Stirling.
Here is how I took notice of Stirling numbers: Catherine Stirling told me
that her father told her that they were descended from a mathematician who
had numbers named after him. In my studies of the calculus, I had learned
about, and I had learned to prove, the Stirling formula for approximating
factorials from Richard Courant's calculus text in the appendix to chapter
VII of volume one. It was my good fortune to be introduced to the calculus
by this wonderful text. It is my understanding that Stirling's formula was
discovered by Abraham De Moivre. Otherwise, I knew nothing about Stirling.
I went to the Dictionary of Scientific Biography to read about James
Stirling. I recommend that article as a starting point for others. They
sent me to a treatise on finite differences by C. Jordan. He has a lengthy
chapter on, and he had great appreciation for, Stirling numbers. He
happened to state that there was no known general formula--by which I
suppose he meant one in closed form, rather than recursively generated--for
obtaining Stirling numbers. His definitions were for signed Stirling
numbers of the first and second kind. Next I put a microfilm copy of
Stirling's original book, in Latin, on our library's reader, but I only
took the trouble to notice that he did have tables of "his" numbers. I also
know, thanks to a reference on the math-history-list that there is an
English translation, but I do not know how to locate one, and I have not
yet checked the catalogue of the Library of Congress. I had already
obtained by that time my copy of THE BOOK OF NUMBERS by Conway & Guy, and I
had seen, and subliminally noticed, references to Stirling numbers. They
advocate calling these numbers Stirling cycle and Stirling set numbers, and
I agree with this as being more informative than first and second kind. I
don't know if they are the first to suggest these designations. I've just
purchased CONCRETE MATHEMATICS by Graham, Knuth, and Patashnik. I
purchased it in fear from Barnes & Noble here in Annapolis. The fear
sprang from the rumor that I had heard that it is out of print. It has
many references to Stirling numbers, and it is still referring to them as
of the first and second kind. I also happen to know that Knuth's three
volume work on algorithms has material on Stirling numbers--certainly in
volume one. I'd like to get it cleared up, if possible, who gave the first
closed form formula for Stirling numbers, and I'd like to check out your
formula that is copied below. All I lack is leisure, but I am not alone in
that.
Best wishes,
Sam Kutler
>There are closed form expressions for Stirling Numbers of the first
>and second kind!
>
>While the closed form expression for the second kind is rather nice,
>the one for
>the first kind "just isn't pretty".
>
>Here goes,
>
>S(n,k) = (-1)^(n-k) * (n-1)! * SUM( from a1=k-1, to n-1, of 1/a1 *
> SUM( from a2=k-2, to n-2-a1, of 1/a2 *
> SUM( from a3=k-3, to n-3-a2, of 1/a3 * SUM( ...
> SUM( from a_(k-1)=1, to n-(k-1)-a_(k-2), of 1/a_(k-1) )...)
>
>
>For example:
>
>S(n,1) = (-1)^(n-1) * (n-1)!
>
>S(n,2) = (-1)^(n-2) * (n-1)! * SUM( from a1=1, to n-1, of 1/a1 )
>
>S(n,3) = (-1)^(n-3) * (n-1)! * SUM[ from a1=2, to n-1, of 1/a1 *
> SUM( from a2=1, to n-2-a1, of 1/a2) ]
>
>It gets long!
>
>Please note that while using the recursive formula to calculate
>S(5000,1000) might take about 8 billion calculations, it would take
>about 10^15000 calculations using the above closed form expresison
>directly. Fortunately, the closed form expression lends itself to
>computer programming very well and calculation shortcuts can be
>implemented!
>
>Hope this helps some.
>
>Dwayne A. Hickman
>Newark, OH USA
From owner-math-history-list@ENTERPRISE.MAA.ORG Wed Aug 13 18:27:24 1997
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Date: Wed, 13 Aug 1997 18:50:00 -0400
Reply-To: Discussion List on the History of Mathematics
From: "Randy K. Schwartz"
Subject: Re: Closed Form for Stirling Numbers
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
Status: OR
On Mon, 11 Aug 1997 at 23:03:24 -0400, Dwayne A. Hickman wrote:
**There are closed form expressions for Stirling Numbers of the first
and second kind!
**While the closed form expression for the second kind is rather nice,
the one for
the first kind "just isn't pretty".
**Here goes,
S(n,k) = (-1)^(n-k) * (n-1)! * SUM( from a1=k-1, to n-1, of 1/a1 *
SUM( from a2=k-2, to n-2-a1, of 1/a2 *
SUM( from a3=k-3, to n-3-a2, of 1/a3 * SUM( ...
SUM( from a_(k-1)=1, to n-(k-1)-a_(k-2), of 1/a_(k-1) )...)
**For example:
S(n,1) = (-1)^(n-1) * (n-1)!
S(n,2) = (-1)^(n-2) * (n-1)! * SUM( from a1=1, to n-1, of 1/a1 )
S(n,3) = (-1)^(n-3) * (n-1)! * SUM[ from a1=2, to n-1, of 1/a1 *
SUM( from a2=1, to n-2-a1, of 1/a2) ]
**It gets long!**
This matter was taken up on this list several months ago. At that time,
on 26 April 1997, Julio Gonzalez Cabillon pointed out that the
** Stirling Cycle Numbers or (unsigned) Stirling numbers of the first kind,
s(m, n) satisfy the following formula:
_____
\ \
\
s(m, n) = ) j_1 . j_2 ... j_(m - n)
/
/____/
j_1, ..., j_(m - n)
where 1 =< j_1 < j_2 < ... < j_(m - n) =< m - 1
[NB: My symbol =< means "not greater than"] **
and Julio went on to provide some examples:
Example 1: s(4, 2) = 11 [m = 4 ; n = 2]
1 =< j_1 < j_2 =< 3
_____
\ \
\
s(4, 2) = ) j_1 . j_2 = (1 x 2) + (1 x 3) + (2 x 3) = 2 + 3 + 6 = 11
/
/____/
j_1, j_2
Example 2: s(5, 2) = 50 [m = 5 ; n = 2]
1 =< j_1 < j_2 < j_3 =< 4
s(4, 2) = (1 x 2 x 3) + (1 x 2 x 4) + (1 x 3 x 4) + (2 x 3 x 4) = 50
These closed-form expressions can easily be modified for signed (as opposed
to unsigned) Stirling numbers of the first kind, and the result seems to be
quite a bit simpler than the one cited by Dwayne.
=====================================================================
Randy K. Schwartz, Chairman email rschwart@schoolcraft.cc.mi.us
Department of Mathematics voice 313/462-4400 extn. 5290
Liberal Arts Building
Schoolcraft College "In the Inn of the world there is room for
18600 Haggerty Road _everyone_. To turn your back on even one
Livonia, MI 48152-2696 person, for whatever reason, is to run
USA the risk of losing the central piece of
fax 313/462-4558 your jigsaw puzzle." - Leo F. Buscaglia
=====================================================================
From <@math.maa.org:math-history-list-owner@maa.org> Tue Jun 10 22:29:07 1997
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Date: Tue, 10 Jun 1997 21:59:54 -0400
From: nyergeg@sbu.ac.uk
To: math-history-list@maa.org
CC: nyergeg@vax.sbu.ac.uk
Message-ID: <009B599C.001FB580.11@vax.sbu.ac.uk>
Subject: Re: Stirling Numbers
Sender: math-history-list-owner@maa.org
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Status: RO
I have only recently joined this list and I noticed an earlier discussion
on a formula for Strirling numbers. Although I am not sure what anyone
means exactly by a 'formula' , nevertheless here is one I quite like, (for
Stirling numbers of the second kind - i.e. Stirling set numbers):
m
_____
\
S(n, m) = (1/m!) (-1)^(m-k) * C(m, k) * k^n
/
____
k = 0
, where 0 <= m <= n and C(m,k) is just the binomial coefficient m choose k.
Using Mathematica syntax, the formula is
S[n_, m_] = (1/m!) * Sum[ (-1)^(m-k) * Binomial[m, k] * k^n, {k,0,m} ]
( The formula can be easily proved by induction)
E.g.
S(7, 4) =
= (1/4!)* [ C(4,0)*0^7 - C(4,1)*1^7 + C(4,2) * 2^7 - C(4,3)*3^7 +
C(4,4) * 4^7 ]
= (1/24) * [ 1*0 - 4*1 + 6*128 - 4*2187 + 1*16384 ] = 350
Best wishes,
Gabor Nyerges
From owner-math-history-list@ENTERPRISE.MAA.ORG Wed Aug 20 00:42 EDT 1997
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Date: Wed, 20 Aug 1997 01:48:37 -0300
Reply-To: Discussion List on the History of Mathematics
From: Julio Gonzalez Cabillon
Subject: Re: My biography with Stirling Numbers
Comments: To: "Samuel S. Kutler"
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
Content-Type: text/plain; charset="us-ascii"
Content-Length: 3240
Status: RO
Dear Sam,
At 07:39 AM 12/08/1997 -0500, you wrote:
>Dwayne:
>
>I'm writing this primarily to you, but I'm copying the math-history-list
>and my sometime student, Catherine Stirling.
>
>Here is how I took notice of Stirling numbers: Catherine Stirling told me
>that her father told her that they were descended from a mathematician who
>had numbers named after him. In my studies of the calculus, I had learned
>about, and I had learned to prove, the Stirling formula for approximating
>factorials from Richard Courant's calculus text in the appendix to chapter
>VII of volume one. It was my good fortune to be introduced to the calculus
>by this wonderful text. It is my understanding that Stirling's formula was
>discovered by Abraham De Moivre. Otherwise, I knew nothing about Stirling.
>I went to the Dictionary of Scientific Biography to read about James
>Stirling. I recommend that article as a starting point for others. They
>sent me to a treatise on finite differences by C. Jordan. He has a lengthy
>chapter on, and he had great appreciation for, Stirling numbers. He
>happened to state that there was no known general formula--by which I
>suppose he meant one in closed form, rather than recursively generated--for
>obtaining Stirling numbers. His definitions were for signed Stirling
>numbers of the first and second kind. Next I put a microfilm copy of
>Stirling's original book, in Latin, on our library's reader, but I only
>took the trouble to notice that he did have tables of "his" numbers. I also
>know, thanks to a reference on the math-history-list that there is an
>English translation, but I do not know how to locate one, and I have not
>yet checked the catalogue of the Library of Congress.
>
Our library has only the Latin edition. The English translation is:
Stirling, James:
"The differential method: or, A treatise concerning
summation and interpolation of infinite series",
translated into English, with the author's approbation,
by Francis Holliday, London: printed for E. Cave, 1749.
>I had already obtained by that time my copy of THE BOOK OF NUMBERS by
>Conway & Guy, and I had seen, and subliminally noticed, references to
>Stirling numbers. They advocate calling these numbers Stirling cycle
>and Stirling set numbers, and I agree with this as being more informative
>than first and second kind.
Early this century, Niels Nielsen (1865-1931) was first to coin the German
terms:
"Stirlingschen Zahlen erster Art" [Stirling numbers of the first kind]
and
"Stirlingschen Zahlen zweiter Art" [Stirling numbers of the second kind]
Nielsen's masterpiece, "Handbuch der Theorie der Gammafunktion"
[B. G. Teubner, Leipzig, 1906], had a great influence, and the terms
progressively found their acceptance.
>I don't know if they are the first to suggest these designations.
Well, I don't think Conway & Guy are the FIRST to suggest the names
*Stirling cycle* and *Stirling (sub)set* numbers. Surely, John might
easily correct my assumption, if otherwise. But it is long before
"The Book of numbers" that I pronounce "n cycle k" and "n subset k"
when reading these formulas aloud.
Best regards,
Julio
From owner-math-history-list@ENTERPRISE.MAA.ORG Wed Aug 20 13:37 EDT 1997
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Reply-To: Discussion List on the History of Mathematics
From: David Fowler
Subject: Re: A notation for Stirling Numbers
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
Content-Type: text/plain; charset="us-ascii"
Content-Length: 872
Status: RO
...
>>obtaining Stirling numbers. His definitions were for signed Stirling
>>numbers of the first and second kind.
...
>Well, I don't think Conway & Guy are the FIRST to suggest the names
>*Stirling cycle* and *Stirling (sub)set* numbers.
...
Do people know the notation for them used in the excellent book (dedicated
to Euler!):
R.L. Graham, D.E. Knuth, & O. Patshnik, Concrete Mathematics, Addison
Wesley, 1989 & often reprinted.
of the first kind: curly brackets n over k
of the second kind: square brackets n over k.
Also Euler numbers: pointed brackets n over k.
All of these are of course generalisations of:
binomial coefficients round brackets n over k.
The book is not clear about whether the proposal is due to them -- "...they
still lack a standard notation. We will write..." (p.234) --, and I don't
know whether it has caught on.
David Fowler
From owner-math-history-list@ENTERPRISE.MAA.ORG Wed Aug 20 17:19 EDT 1997
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Date: Wed, 20 Aug 1997 17:24:43 -0400
Reply-To: Discussion List on the History of Mathematics
From: John Conway
Subject: Re: My biography with Stirling Numbers
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
In-Reply-To: <1.5.4.32.19970820044837.017fec44@adinet.com.uy>
Content-Type: TEXT/PLAIN; charset=US-ASCII
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Status: RO
On Wed, 20 Aug 1997, Julio Gonzalez Cabillon wrote:
>
> >I had already obtained by that time my copy of THE BOOK OF NUMBERS by
> >Conway & Guy, and I had seen, and subliminally noticed, references to
> >Stirling numbers. They advocate calling these numbers Stirling cycle
> >and Stirling set numbers, and I agree with this as being more informative
> >than first and second kind.
>
> Early this century, Niels Nielsen (1865-1931) was first to coin the German
> terms:
>
> "Stirlingschen Zahlen erster Art" [Stirling numbers of the first kind]
>
> and
>
> "Stirlingschen Zahlen zweiter Art" [Stirling numbers of the second kind]
>
> Nielsen's masterpiece, "Handbuch der Theorie der Gammafunktion"
> [B. G. Teubner, Leipzig, 1906], had a great influence, and the terms
> progressively found their acceptance.
>
> >I don't know if they are the first to suggest these designations.
>
> Well, I don't think Conway & Guy are the FIRST to suggest the names
> *Stirling cycle* and *Stirling (sub)set* numbers. Surely, John might
> easily correct my assumption, if otherwise. But it is long before
> "The Book of numbers" that I pronounce "n cycle k" and "n subset k"
> when reading these formulas aloud.
>
> Best regards,
> Julio
>
No, we certainly weren't, since we copied these names from
the book of Graham, Knuth & Patashnik (and I thought we'd said so).
However, I suspect that they WERE the first. Certainly the
set- and cycle- number nomenclature is not very old.
The trouble is that Nielsen put the terms the wrong way round, since
it's the numbers of the second kind that are most often wanted. I have
long felt that in any case the "first and second kind" names were both
colorless and cumbrous, and am very glad indeed to find the new names
meeting with approval.
John Conway
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Reply-To: jgc@adinet.com.uy
From: Julio Gonzalez Cabillon
Subject: Re: A notation for Stirling Numbers
Comments: To: David Fowler
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
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Dear David,
At 06:33 PM 20/08/1997 +0000, you wrote:
>...
>>>obtaining Stirling numbers. His definitions were for signed Stirling
>>>numbers of the first and second kind.
>...
>
>>Well, I don't think Conway & Guy are the FIRST to suggest the names
>>*Stirling cycle* and *Stirling (sub)set* numbers.
>...
>
>Do people know the notation for them used in the excellent book (dedicated
>to Euler!):
>
>R.L. Graham, D.E. Knuth, & O. Patshnik, Concrete Mathematics, Addison
>Wesley, 1989 & often reprinted.
>
>
>of the first kind: curly brackets n over k
>
>of the second kind: square brackets n over k.
>
....
>
>The book is not clear about whether the proposal is due to them -- "...they
>still lack a standard notation. We will write..." (p.234) --, and I don't
>know whether it has caught on.
>
>David Fowler
These notations were suggested in the early sixties by other authors.
Let me check the sources. You may also remember that the "square
brackets
n over k" notation has been used for Gaussian binomial coefficients, and
is still employed. For this q-generalisation of binomial coefficients
I prefer round brackets n over k with a q as an outside subindex on the
right.
My best regards,
Julio
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Date: Wed, 20 Aug 1997 20:56:07 -0300
Reply-To: jgc@adinet.com.uy
From: Julio Gonzalez Cabillon
Subject: Re: My biography with Stirling Numbers
Comments: To: John Conway
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
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On Wed, 20 Aug 1997, John Conway wrote:
>> Well, I don't think Conway & Guy are the FIRST to suggest the names
>> *Stirling cycle* and *Stirling (sub)set* numbers. Surely, John might
>> easily correct my assumption, if otherwise. But it is long before
>> "The Book of numbers" that I pronounce "n cycle k" and "n subset k"
>> when reading these formulas aloud.
>>
>> Best regards,
>> Julio
>>
>
> No, we certainly weren't, since we copied these names from
>the book of Graham, Knuth & Patashnik (and I thought we'd said so).
>However, I suspect that they WERE the first.
They might be. None the less I suspect that the credit should go
to Donald Knuth alone. I don't have their book to hand -- are you
sure that you copied these terms from THAT book? I suspect that
you might have taken this terminology just from Knuth's suggestions,
during private exchanges... I apologise in advance for this assumption,
if otherwise.
>Certainly the set- and cycle- number nomenclature is not very old.
Yes, the nomenclature for the Stirling numbers in the clear-cut way
that you (& Guy) suggest in "The Book of Numbers" is not very old.
Polya and Szego, in their superb book on problems, stated:
"We let S(n, k) stand for the number of different partitions
of a set of n elements into k classes ..."
"We let s(n, k) stand for the number of those permutations of
n elements that are the products of k disjoint cycles."
Next, they went on to mention the unfriendly "first and second kind"
sacred words.
> The trouble is that Nielsen put the terms the wrong way round, since
>it's the numbers of the second kind that are most often wanted.
And from a historical standpoint, it is worth remarking that James
Stirling, in his "Methodus Differentialis" (London, 1730), introduced
his numbers "of the second kind" first (p. 8),
Tabulam priorem.
___________________________________________________________
1 1 1 1 1 1 1 1 1 &c
1 3 7 15 31 63 127 255 &c
1 6 25 90 301 966 3025 &c
1 10 65 350 1701 7770 &c
1 15 140 1050 6951 &c
1 21 266 2646 &c
1 28 461 &c
1 36 &c
1 &c
&c
and then "of the first kind" (p. 11) in second place.
Tabulam posterior.
1
1 1
2 3 1
6 11 6 1
24 50 35 10 1
120 274 225 85 15 1
720 1764 1624 735 175 21 1
5040 13068 13132 6769 1960 322 28 1
40320 109584 105056 67284 22449 4536 546 36 1
____________________________________________________________________
&c &c &c &c &c &c &c &c &c &c
Julio Gonzalez Cabillon
From owner-math-history-list@ENTERPRISE.MAA.ORG Mon Aug 25 12:39 EDT 1997
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Reply-To: Discussion List on the History of Mathematics
From: "Patrick D. F. Ion"
Subject: Re: My biography with Stirling Numbers
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
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Concerning the bracket and brace notations and their cycle and subset
pronunciations, Knuth states, in his note
Knuth, Donald E.; Two notes on notation. Amer. Math. Monthly 99 (1992), no.
5, 403--422. MR 93f:05001
that "I introduced these notations in the first edition of my first book"
(i.e.,
Knuth, Donald E. The art of computer programming. Vol. 1: Fundamental
algorithms. Second printing Addison-Wesley Publishing Co., Reading,
Mass.-London-Don Mills, Ont 1969 xxi+634 pp., MR 44 #3530 ).
He then goes on to further discussion of both the notations for and
properties of the Stirling numbers.
A. E. Fekete, objects strongly to Knuth's notation. The reviewer of his
paper for MR remarks Fekete says that calling the symbol
$\left\{\smallmatrix n\\k\endsmallmatrix\right\}$ a "Stirling subset
number" and verbalizing it as "$n$ subset $k$" would "spell conceptual
disaster". See
Fekete, Antal E.; Apropos "Two notes on notation" [Amer. Math. Monthly 99
(1992), no. 5, 403--422; MR 93f:05001] by D. E. Knuth. Amer. Math. Monthly
101 (1994), no. 8, 771--778. MR 95h:05006
In this connection, since the first part of Knuth's note on notations deals
with it, I wonder if anyone has any ideas about the history of Iversen's
notation. I think I recollect that there was use of that sort of handy
abbreviation before it could readily have derived from Iversen.
Patrick Ion
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Date: Mon, 25 Aug 1997 16:39:13 -0300
Reply-To: jgc@adinet.com.uy
From: Julio Gonzalez Cabillon
Subject: Re: A notation for Stirling Numbers
Comments: To: David Fowler
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
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At 06:56 PM 20/08/1997 -0300, I wrote:
>Dear David,
>
>At 06:33 PM 20/08/1997 +0000, you wrote:
>
>>...
>>>>obtaining Stirling numbers. His definitions were for signed Stirling
>>>>numbers of the first and second kind.
>>...
>>
>>>Well, I don't think Conway & Guy are the FIRST to suggest the names
>>>*Stirling cycle* and *Stirling (sub)set* numbers.
>>...
>>
>>Do people know the notation for them used in the excellent book (dedicated
>>to Euler!):
>>
>>R.L. Graham, D.E. Knuth, & O. Patshnik, Concrete Mathematics, Addison
>>Wesley, 1989 & often reprinted.
>>
>>
>>of the first kind: curly brackets n over k
>>
>>of the second kind: square brackets n over k.
>>
>....
>>
>>The book is not clear about whether the proposal is due to them -- "...they
>>still lack a standard notation. We will write..." (p.234) --, and I don't
>>know whether it has caught on.
>>
>>David Fowler
>
>These notations were suggested in the early sixties by other authors.
>Let me check the sources.
>...
"The great utility of Stirling numbers has become clearer and clearer
with time, and mathematicians have now reached a stage where we can
intelligently choose a notation that will serve us well in the whole
range of applications.
I came into the picture rather late, having never heard of Stirling
numbers until I received my Pd.D. in mathematics. But I soon
encountered
them as I was beginning to analyze the performance of algorithms and
to write the manuscript for my books *The Art of Computer
Programming*.
I quickly realized the truth of Imanul Marx's comment that "these
numbers have similarities with the binomial coefficients (n k);
indeed,
formulas similar to those known for the binomial coefficients are
easily
established" (1). In order to emphasize and to facilitate pattern
recognition when manipulating formulas, Marx recommended using
brackets
symbols [n k] for Stirling numbers of the first kind and brace
symbols
{n k} for Stirling numbers of the second kind. A similar proposal was
being made at about the same time in Italy by Antonio Salmeri (2)."
(3)
Note: I used the symbolisms (n k), [n k] and {n k} instead of the
standard
ones. Email inconveniences!
(1) Marx, Imanuel: "Transformation of series by a variant of Stirling's
numbers", _American Mathematical Monthly_, v. 69, pp. 530-532, June-July
1962.
(2) Salmeri, Antonio: "Introduzione alla teoria dei coefficienti
fattoriali",
Giornale di Matematiche di Battaglini, v. 90 (V. Ser. 10), pp. 44-54,
1962.
(3) Knuth, Donald E.: "Two notes on notation", _American Mathematical
Monthly_,
v. 99, pp. 403-422, May 1992. Cf. page 410.
Julio Gonzalez Cabillon
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Date: Mon, 25 Aug 1997 15:55:00 -0400
Reply-To: Discussion List on the History of Mathematics
From: John Conway
Subject: Re: A notation for Stirling Numbers
Comments: To: Julio Gonzalez Cabillon
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
In-Reply-To: <3401DEDC.6FF8@adinet.com.uy>
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I just noticed that whoever wrote the lines below got Knuth's bracket
notations the wrong way round -
curly brackets are used for the Stirling set (or 2nd kind) numbers
and square brackets for the Stirling cycle (or 1st kind) numbers.
I write in case anyone takes up the reversed version. The mnemonic
is of course that curly brackets are commonly used to define sets.
> >>Do people know the notation for them used in the excellent book (dedicated
> >>to Euler!):
> >>
> >>R.L. Graham, D.E. Knuth, & O. Patshnik, Concrete Mathematics, Addison
> >>Wesley, 1989 & often reprinted.
> >>
> >>
> >>of the first kind: curly brackets n over k
> >>
> >>of the second kind: square brackets n over k.
> >>
> >....
> >>
> >>The book is not clear about whether the proposal is due to them -- "...they
> >>still lack a standard notation. We will write..." (p.234) --, and I don't
> >>know whether it has caught on.
> >>
> >>David Fowler
I think the notation has already caught on among combinatorialists,
particularly those of a computational turn of mind.
John Conway
From owner-math-history-list@ENTERPRISE.MAA.ORG Tue Aug 26 16:25 EDT 1997
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Reply-To: Discussion List on the History of Mathematics
From: "Samuel S. Kutler"
Subject: Nomenclature in Euclid & Knuth
To: MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG
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Friends:
In Book I of Euclid's ELEMENTS there are 23 definitions (horoi--I suppose
it could be translated *limitations* since definition 13 gives
A boundary (horos) is an extremity of anything,
and hence a horos limits the meaning of the term for mathematical
purposes), and none of the definitions is of a parallelogram. Instead
Euclid eases one into the term *parallelogram* thus:
Prop. 33: The straight lines joining equal and parallel straight lines
are themselves equal and parallel.
Prop. 34: In parallelogrammic areas the opposite sides and angles are
equal to one another.
Finally, in
Prop. 35: Parallelograms which are on the same base and in the same
parallels are equal to one another.
Knuth and company in CONCRETE MATHEMATICS--I have the 2nd
edition--similarly ease one into the meaning of the two types of Stirling
Numbers.
On page 257: These [Stirling] numbers come in two flavors,
traditionally called by the no-frills names . . .of the 1st & 2nd kind.
Following Jovan Karamata (1935), they write the 2nd kind in { } and the
first kind in [ ], and they call these symbols
more user-friendly than many other notations that people have tried.
On page 258 & 259 the tables for these numbers are called Stirling's
triangle for subsets & for cycles.
The symbol in curly braces is read n subset k.
The symbol in [ ] is read n cycle k.
Finally, on page 261--if I didn't miss it appearing earlier--they are called
Stirling subset numbers & Stirling cycle numbers.
Conway and Guy shorten the first term to Stirling set numbers.
Best wishes,
Sam