### a thing unto itself

letting

(i.e., … 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

to read about ’em in KPB (kenneth p.~bogart’s

_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,

,

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 *many*many 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.

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