From owner-math-history-list@ENTERPRISE.MAA.ORG Mon Aug 11 23:15:24 1997 Return-Path: Received: from enterprise.maa.org (enterprise.maa.org [206.4.57.253]) by garnet.acns.fsu.edu (8.8.6/8.8.6) with ESMTP id XAA25990 for ; Mon, 11 Aug 1997 23:15:22 -0400 Received: from enterprise (206.4.57.253) by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.115FDF20@enterprise.maa.org>; Mon, 11 Aug 1997 23:19:51 -0400 Received: from ENTERPRISE.MAA.ORG by ENTERPRISE.MAA.ORG (LISTSERV-TCP/IP release 1.8c) with spool id 0834 for MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG; Mon, 11 Aug 1997 23:19:50 -0400 Received: from forum.swarthmore.edu by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.AB5B5840@enterprise.maa.org>; Mon, 11 Aug 1997 23:09:50 -0400 Received: (from http@localhost) by forum.swarthmore.edu (8.8.5/8.8.5/MathForum) id XAA28696 for math-history-list@enterprise.maa.org; Mon, 11 Aug 1997 23:03:24 -0400 (EDT) Epigone-thread: bloiventun X-mailer: epigone Message-ID: Date: Mon, 11 Aug 1997 23:03:24 -0400 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 Return-Path: Received: from enterprise.maa.org (enterprise.maa.org [206.4.57.253]) by garnet.acns.fsu.edu (8.8.6/8.8.6) with ESMTP id HAA11500 for ; Tue, 12 Aug 1997 07:42:15 -0400 Received: from enterprise (206.4.57.253) by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.D6DDB8D0@enterprise.maa.org>; Tue, 12 Aug 1997 7:46:27 -0400 Received: from ENTERPRISE.MAA.ORG by ENTERPRISE.MAA.ORG (LISTSERV-TCP/IP release 1.8c) with spool id 0869 for MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG; Tue, 12 Aug 1997 07:46:26 -0400 Received: from whorfin.sjca.edu by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.D6699A40@enterprise.maa.org>; Tue, 12 Aug 1997 7:46:26 -0400 Received: from [199.89.180.243] (slip-4.sjca.edu [199.89.180.243]) by whorfin.sjca.edu (8.8.5/8.8.5) with SMTP id HAA07736; Tue, 12 Aug 1997 07:39:50 -0400 (EDT) Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Message-ID: 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 Return-Path: Received: from enterprise.maa.org (enterprise.maa.org [206.4.57.253]) by garnet.acns.fsu.edu (8.8.6/8.8.6) with ESMTP id SAA49564 for ; Wed, 13 Aug 1997 18:27:22 -0400 Received: from enterprise (206.4.57.253) by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.2A0B7780@enterprise.maa.org>; Wed, 13 Aug 1997 18:31:50 -0400 Received: from ENTERPRISE.MAA.ORG by ENTERPRISE.MAA.ORG (LISTSERV-TCP/IP release 1.8c) with spool id 0309 for MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG; Wed, 13 Aug 1997 18:31:49 -0400 Received: from schoolcraft.cc.mi.us (198.108.104.2) by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.299D7370@enterprise.maa.org>; Wed, 13 Aug 1997 18:31:49 -0400 Received: by firewall.schoolcraft.cc.mi.us id <28690-1>; Wed, 13 Aug 1997 18:27:07 -0400 Encoding: 92 TEXT X-Mailer: Microsoft Mail V3.0 Message-ID: <97Aug13.182707edt.28690-1@firewall.schoolcraft.cc.mi.us> 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 Received: from math.maa.org (math.maa.org [206.4.57.254]) by garnet.acns.fsu.edu (8.8.3/8.7.3) with SMTP id WAA73258 for ; Tue, 10 Jun 1997 22:29:06 -0400 Received: by math.maa.org id <90648>; Tue, 10 Jun 1997 22:02:14 -0400 Received: from CSDAlpha2.sbu.ac.uk ([136.148.1.111]) by math.maa.org with SMTP id <90627>; Tue, 10 Jun 1997 22:02:08 -0400 Received: from ice.sbu.ac.uk (minda.sbu.ac.uk [136.148.1.11]) by CSDAlpha2.sbu.ac.uk (8.8.5/8.7.3) with SMTP id DAA26052 for ; Wed, 11 Jun 1997 03:02:08 +0100 (BST) Received: from HYDRA (BIG::SYSTEM) by ICE (MX V4.2 AXP) with SMTP (DECnet); Wed, 11 Jun 1997 02:58:18 BST Received: by vax.sbu.ac.uk (MX V4.2 VAX) id 11; Wed, 11 Jun 1997 02:59:55 BST 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 Precedence: bulk Comments: send subscribe/unsubscribe messages to majordomo@maa.org 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 Received: from enterprise.maa.org (enterprise.maa.org [206.4.57.253]) by polaris.net (8.8.6/8.7.6) with ESMTP id AAA10281 for ; Wed, 20 Aug 1997 00:42:42 -0400 (EDT) Received: from enterprise (206.4.57.253) by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.B9A819C0@enterprise.maa.org>; Wed, 20 Aug 1997 0:55:17 -0400 Received: from ENTERPRISE.MAA.ORG by ENTERPRISE.MAA.ORG (LISTSERV-TCP/IP release 1.8c) with spool id 0642 for MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG; Wed, 20 Aug 1997 00:55:05 -0400 Received: from mail.adinet.com.uy by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.B20747E0@enterprise.maa.org>; Wed, 20 Aug 1997 0:55:04 -0400 Received: from jgc (r127-42.adinet.com.uy [207.3.127.42]) by mail.adinet.com.uy with SMTP (8.7.1/8.7.1) id BAA13579; Wed, 20 Aug 1997 01:52:12 -0300 (SAT) X-Sender: jgc@adinet.com.uy X-Mailer: Windows Eudora Light Version 1.5.4 (32) Mime-Version: 1.0 Message-ID: <1.5.4.32.19970820044837.017fec44@adinet.com.uy> 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 Received: from enterprise.maa.org (enterprise.maa.org [206.4.57.253]) by polaris.net (8.8.6/8.7.6) with ESMTP id NAA02013 for ; Wed, 20 Aug 1997 13:37:55 -0400 (EDT) Received: from enterprise (206.4.57.253) by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.F56B5B40@enterprise.maa.org>; Wed, 20 Aug 1997 13:50:02 -0400 Received: from ENTERPRISE.MAA.ORG by ENTERPRISE.MAA.ORG (LISTSERV-TCP/IP release 1.8c) with spool id 0153 for MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG; Wed, 20 Aug 1997 13:49:58 -0400 Received: from ribble.maths.warwick.ac.uk by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.8BF22000@enterprise.maa.org>; Wed, 20 Aug 1997 13:39:56 -0400 Received: from [137.205.97.92] by ribble.maths.warwick.ac.uk; Wed, 20 Aug 1997 18:33:02 +0100 (BST) X-Sender: dhf@ribble.maths.warwick.ac.uk Mime-Version: 1.0 Message-ID: Date: Wed, 20 Aug 1997 18:33:04 +0000 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 Received: from enterprise.maa.org (enterprise.maa.org [206.4.57.253]) by polaris.net (8.8.6/8.7.6) with ESMTP id RAA10536 for ; Wed, 20 Aug 1997 17:19:11 -0400 (EDT) Received: from enterprise (206.4.57.253) by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.EC413930@enterprise.maa.org>; Wed, 20 Aug 1997 17:31:42 -0400 Received: from ENTERPRISE.MAA.ORG by ENTERPRISE.MAA.ORG (LISTSERV-TCP/IP release 1.8c) with spool id 0764 for MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG; Wed, 20 Aug 1997 17:31:37 -0400 Received: from and.Princeton.EDU by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.E8893A40@enterprise.maa.org>; Wed, 20 Aug 1997 17:31:35 -0400 Received: from coffee.Princeton.EDU (conway@coffee.Princeton.EDU [128.112.16.102]) by and.Princeton.EDU (8.8.5/8.8.3) with ESMTP id RAA12959 for ; Wed, 20 Aug 1997 17:24:44 -0400 (EDT) Received: (conway@localhost) by coffee.Princeton.EDU (8.7.5/8.6.12) id RAA04144; Wed, 20 Aug 1997 17:24:43 -0400 (EDT) MIME-Version: 1.0 Message-ID: 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 Content-Length: 1830 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 From owner-math-history-list@ENTERPRISE.MAA.ORG Wed Aug 20 17:51 EDT 1997 Received: from enterprise.maa.org (enterprise.maa.org [206.4.57.253]) by polaris.net (8.8.6/8.7.6) with ESMTP id RAA14439 for ; Wed, 20 Aug 1997 17:51:14 -0400 (EDT) Received: from enterprise (206.4.57.253) by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.63449CD0@enterprise.maa.org>; Wed, 20 Aug 1997 18:03:39 -0400 Received: from ENTERPRISE.MAA.ORG by ENTERPRISE.MAA.ORG (LISTSERV-TCP/IP release 1.8c) with spool id 0797 for MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG; Wed, 20 Aug 1997 18:03:35 -0400 Received: from mail.adinet.com.uy by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.5C4D5160@enterprise.maa.org>; Wed, 20 Aug 1997 18:03:27 -0400 Received: from jgc (r44-73.adinet.com.uy [206.99.44.73]) by mail.adinet.com.uy with SMTP (8.7.1/8.7.1) id TAA07770; Wed, 20 Aug 1997 19:00:33 -0300 (SAT) X-Mailer: Mozilla 3.01Gold (Win95; I) MIME-Version: 1.0 Content-Transfer-Encoding: 7bit Message-ID: <33FB6824.6429@adinet.com.uy> Date: Wed, 20 Aug 1997 18:56:52 -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 Content-Type: text/plain; charset=us-ascii Content-Length: 1222 Status: RO 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 From owner-math-history-list@ENTERPRISE.MAA.ORG Wed Aug 20 19:50 EDT 1997 Received: from enterprise.maa.org (enterprise.maa.org [206.4.57.253]) by polaris.net (8.8.6/8.7.6) with ESMTP id TAA28261 for ; Wed, 20 Aug 1997 19:50:39 -0400 (EDT) Received: from enterprise (206.4.57.253) by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.16D634B0@enterprise.maa.org>; Wed, 20 Aug 1997 20:03:12 -0400 Received: from ENTERPRISE.MAA.ORG by ENTERPRISE.MAA.ORG (LISTSERV-TCP/IP release 1.8c) with spool id 0849 for MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG; Wed, 20 Aug 1997 20:03:08 -0400 Received: from mail.adinet.com.uy by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.112C02B0@enterprise.maa.org>; Wed, 20 Aug 1997 20:03:03 -0400 Received: from jgc (r44-110.adinet.com.uy [206.99.44.110]) by mail.adinet.com.uy with SMTP (8.7.1/8.7.1) id UAA15407; Wed, 20 Aug 1997 20:59:47 -0300 (SAT) X-Mailer: Mozilla 3.01Gold (Win95; I) MIME-Version: 1.0 Content-Transfer-Encoding: 7bit Message-ID: <33FB8417.3098@adinet.com.uy> 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 Content-Type: text/plain; charset=us-ascii Content-Length: 3265 Status: RO 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 Received: from enterprise.maa.org (enterprise.maa.org [206.4.57.253]) by polaris.net (8.8.6/8.7.6) with ESMTP id MAA01698 for ; Mon, 25 Aug 1997 12:39:38 -0400 (EDT) Received: from enterprise (206.4.57.253) by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.BEA51E40@enterprise.maa.org>; Mon, 25 Aug 1997 12:52:28 -0400 Received: from ENTERPRISE.MAA.ORG by ENTERPRISE.MAA.ORG (LISTSERV-TCP/IP release 1.8c) with spool id 1354 for MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG; Mon, 25 Aug 1997 12:52:28 -0400 Received: from math.ams.org by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.BE340CF0@enterprise.maa.org>; Mon, 25 Aug 1997 12:52:28 -0400 Received: from axp14.ams.org by math.ams.org via smtpd (for enterprise.maa.org [206.4.57.253]) with SMTP; 25 Aug 1997 16:45:46 UT Received: from mr3.mr.ams.org by AXP14.AMS.ORG (PMDF V5.1-8 #1) with SMTP id <01IMUVAR7JYO0007JV@AXP14.AMS.ORG> for MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG; Mon, 25 Aug 1997 12:45:45 EST Received: from mr4.mr.ams.org by mr3.mr.ams.org; (5.65v3.2/1.1.8.2/30Oct96-1013AM) id AA21719; Mon, 25 Aug 1997 12:45:43 -0400 Received: from pion.mr.ams.org by mr4.mr.ams.org; (5.65v3.2/1.1.8.2/28Sep94-0231PM) id AA09525; Mon, 25 Aug 1997 12:45:42 -0400 X-Sender: ion@mr4.mr.ams.org MIME-version: 1.0 Message-ID: Date: Mon, 25 Aug 1997 12:48:15 -0400 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 In-Reply-To: <33FB8417.3098@adinet.com.uy> Content-Type: text/plain; charset="us-ascii" Content-Length: 1378 Status: RO 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 From owner-math-history-list@ENTERPRISE.MAA.ORG Mon Aug 25 15:34 EDT 1997 Received: from enterprise.maa.org (enterprise.maa.org [206.4.57.253]) by polaris.net (8.8.6/8.7.6) with ESMTP id PAA00349 for ; Mon, 25 Aug 1997 15:33:54 -0400 (EDT) Received: from enterprise (206.4.57.253) by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.0A369740@enterprise.maa.org>; Mon, 25 Aug 1997 15:46:23 -0400 Received: from ENTERPRISE.MAA.ORG by ENTERPRISE.MAA.ORG (LISTSERV-TCP/IP release 1.8c) with spool id 1435 for MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG; Mon, 25 Aug 1997 15:46:23 -0400 Received: from mail.adinet.com.uy by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.09221A00@enterprise.maa.org>; Mon, 25 Aug 1997 15:46:21 -0400 Received: from jgc (r127-167.adinet.com.uy [207.3.127.167]) by mail.adinet.com.uy with SMTP (8.7.1/8.7.1) id QAA01886; Mon, 25 Aug 1997 16:43:22 -0300 (SAT) X-Mailer: Mozilla 3.01Gold (Win95; I) MIME-Version: 1.0 Content-Transfer-Encoding: 7bit Message-ID: <3401DEDC.6FF8@adinet.com.uy> 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 Content-Type: text/plain; charset=us-ascii Content-Length: 2675 Status: RO 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 From owner-math-history-list@ENTERPRISE.MAA.ORG Mon Aug 25 15:49 EDT 1997 Received: from enterprise.maa.org (enterprise.maa.org [206.4.57.253]) by polaris.net (8.8.6/8.7.6) with ESMTP id PAA02635 for ; Mon, 25 Aug 1997 15:49:22 -0400 (EDT) Received: from enterprise (206.4.57.253) by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.337C02F0@enterprise.maa.org>; Mon, 25 Aug 1997 16:01:51 -0400 Received: from ENTERPRISE.MAA.ORG by ENTERPRISE.MAA.ORG (LISTSERV-TCP/IP release 1.8c) with spool id 1446 for MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG; Mon, 25 Aug 1997 16:01:51 -0400 Received: from and.Princeton.EDU by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.324BE760@enterprise.maa.org>; Mon, 25 Aug 1997 16:01:49 -0400 Received: from tea.Princeton.EDU (conway@tea.Princeton.EDU [128.112.16.7]) by and.Princeton.EDU (8.8.5/8.8.3) with ESMTP id PAA21931; Mon, 25 Aug 1997 15:55:01 -0400 (EDT) Received: (conway@localhost) by tea.Princeton.EDU (8.7.5/8.6.12) id PAA07493; Mon, 25 Aug 1997 15:55:00 -0400 (EDT) MIME-Version: 1.0 Message-ID: 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> Content-Type: TEXT/PLAIN; charset=US-ASCII Content-Length: 1092 Status: RO 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 Received: from enterprise.maa.org (enterprise.maa.org [206.4.57.253]) by polaris.net (8.8.6/8.7.6) with ESMTP id QAA06681 for ; Tue, 26 Aug 1997 16:25:19 -0400 (EDT) Received: from enterprise (206.4.57.253) by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.72378950@enterprise.maa.org>; Tue, 26 Aug 1997 16:38:13 -0400 Received: from ENTERPRISE.MAA.ORG by ENTERPRISE.MAA.ORG (LISTSERV-TCP/IP release 1.8c) with spool id 1973 for MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG; Tue, 26 Aug 1997 16:38:12 -0400 Received: from whorfin.sjca.edu by enterprise.maa.org (LSMTP for Windows NT v1.1a) with SMTP id <0.7195A400@enterprise.maa.org>; Tue, 26 Aug 1997 16:38:12 -0400 Received: from [199.89.180.241] (slip-2.sjca.edu [199.89.180.241]) by whorfin.sjca.edu (8.8.5/8.8.5) with SMTP id QAA22013 for ; Tue, 26 Aug 1997 16:31:28 -0400 (EDT) X-Sender: reltuk@mailhost.sjca.edu (Unverified) Mime-Version: 1.0 Message-ID: Date: Tue, 26 Aug 1997 16:31:43 -0500 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 Content-Type: text/plain; charset="us-ascii" Content-Length: 1679 Status: RO 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