a thing unto itself

letting
${\Bbb N}_1 = \{1\},{\Bbb N}_2 = \{1,2\},$
${\Bbb N}_3 = \{1,2,3\} \ldots$
(i.e., ${\Bbb N}_k = \{i\}_{i=1}^k$… en-sub-kay stands
for the set of positive integers from
one to kay, in other words)…
and immediately turning right around
and redefining {1, 2, 3, … k} as
“N_k” instead, without all the fancy
type…

last monday’s bogart so far
was a ramble on stirling numbers (of the second
and first kind, SN2K & SN1K) and how it was
_introductory_combinatorics_… or should i
just say “op cit.”?). the notations i’ve just
introduced are mine not bogart’s; this will
persist as i imagine.

now i’ve read that section some more and
its pages are littered with my pencil notes.
for instance,
$x^n = \sum_{j=0}^n S(n,j)(x)_j$,
copied out verbatim for the sheer joy
of it, and
* [ (k)_n counts one-to-one F^n’s]
* [ k^n (counts functions)]
* [ S(n,k)k! counts “onto”
functions N_n —->> N_k]
,
written out on p.~50 (where the “falling
factorial” function (k)_n = k!/(k-n)!
is defined) because it seems to me
as clear a summary of what for me
is the heart of the matter as i can
produce at this time: this is what
SN2K’s are *for*, in the context of
some very familiar material (one has
delivered manymany lectures
bearing on “one to one” and “onto”
functions [and even quite a few
devoted to those topics specifically]…
but typically in a context where
“infinitely many” such functions
will be imagined to “exist”… which
makes “counting” the functions sort
of beside the point).

we could go so far as to *define*
stirling numbers of the second kind by
SN2K(n,k) := #{ f | f: N_n —>> N_k}/k!
.
“count the onto functions and divide by
the permutations of the range”… this
of course give the number of *partitions*
of N_n into k “classes” (regardless of
order “within a given class”).
not that i think this would be a good idea.

okay. like i said on monday, a good section
of KPB (2.3 “partitions and stirling numbers”).
i’ve read over all 18 exercises & even set some up.
heck, some could even be considered “done”.
but really i came to ramble about sections 3.1
and 3.2. or so i thought. but *really* i came
to avoid marking up some pages. and i’ve done
about enough of that. for now.