sw00547math-stirling.nums.and.bicos.at65

09.11.06


Stirling Numbers and Binomial Coefficients


 Yesterday, in the midst of searching for something else,
I found the manilla folder in which I am keeping short
mathematical papers I have written.  Among these is one
wherein I show a way of determining binominal coefficients
which is different from the way in general use.  I wrote it
during the heat of my studying for the GRE.  The date on it
is September 17, 1983.  Alas, my way, while it does work,
is merely of some interest, and is not a viable competitor.
Nonetheless, being pleased with it, I sent it to a most cordial
excellent mathematician in Canada.  In his 83:12:07 letter to
me/ he*        wrote: "Your method for calculating binomial
  coefficients is more complicated than more usual ways of
  doing so, but there is some interest, because you have re-
  discovered the Stirling numbers of the first kind, which
  crop up all over the place in combinatorial analysis."    He
also sent a separate sheet to me on which he uses Pascal's
Triangle to show me how the Stirling numbers of the first kind
operate.  To this he provided further explanatory information.
I had mentioned Pascal's Triangle in my brief paper, but I did
not use it.
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Today I went online to learn more about Stirling numbers.  I
visited two sites.  At site one, after going back & forth several
times from the triangle depiction of the Stirling numbers of the
first kind to the one of the second kind, I suddenly noticed that
the natural number summation sequence is integral to both kinds,
with the important distinctions that each nnss term in the Stirling
numbers of the first kind is preceded by a minus sign, and that
the generated terms therein both alternate from negative to
positive to negative to positive values and increase in value
far more rapidly than do the totally positive terms in the Stirling
numbers of the second kind.  At site two I found what I always
crave: clear examples.  The examples presented there illustrate
combinations.
#
Even though my paper does not "merit publication", I've decided
to publish it.  See entry beneath this one: sw00546math
#
#
Notes:
1. James Stirling was an 18th Century Scottish mathematician.
2. The two triangles showing Stirling numbers of the first kind
    and Stirling numbers of the second kind are located at:
    Stirling number
3. The clear combinations examples are at:
    csuhayward.edu                                               


Brian A. J. Salchert