What is of still here? What is of Q? What is of Stirling? Is it closed?

Oh, and doing the same calculation in Excel for C=24, S=24 (taking the values for T(24,F) from rows 277-300 of the "flattened" table) gives A_av = ~8.641905

I don't believe a closed form solution to your problem is possible, but I believe I can make it computationally rather easy, unless you're looking at values of C and/or S running into the thousands.

Look at the problem this way: after placing C cards randomly into S slots, let the number of slots containing at least one card be F ("Full" slots). Thus, F cards must be touching the Felt in that configuration, and the number of cards not touching the felt is A=C-F (A for "Angelic" cards, for the feet of angels never touch the ground). Find the average value of F, and the average value of A is then trivial:

A_av=C-F_av

Let p(C,S,F) refer to the probability that, after dealing C cards into S slots, exactly F slots have one or more cards in them. The number of ways C distinct cards may be dealt into exactly F out of S slots is equal to the number of ways C distinct cards may be dealt into a given F slots with the restriction that each slot must have at least one card, times the number of ways F slots may be chosen out of a total of S slots. The denominator is simply the total number of ways C cards may be dealt into S slots at all, which as you note in your post is simply S^C. Thus we have:

p(C,S,F)=[T(C,F) * (^S_F)]/S^C

T(C,F) is the number of ways C cards may be dealt into F non-empty slots (C≥F, of course). It is given as sequence A019538 at the Online Encyclopedia of Integer Sequences. Unfortunately there is no closed-form expression for T(C,F), but a recursive formula is given there. Also note the entry includes a link to a "flattened" list which gives values up to T(100,100).

(^S_F) is the usual binomial coefficient, S!/[F!(S-F)!]

As an aside, this may be of interest but is not necessary to this solution:

T(C,F)=F!*{^C_F}

where {^C_F} is the Stirling number of the second kind, and much more on those can be found by Googling, including an explicit statement that there is no closed-form expression for them.

Thus,

   min(C,S)
Fav = Σ   F*p(C,S,F)
     F=1


Also, you can check your work by showing that

min(C,S)
   Σ   p(C,S,F) = 1
  F=1


or, equivalently, that the sum of the numerators is is equal to S^C

Here are some examples which illustrate how the calculation proceeds. (I haven't shown the computation of T(C,F) for these, just took them from the table). I've given three examples, one with C>S, one with C=S, one with C<S.

Note that the maximum value of F is the lesser of C or S.


Example 1: C=7 S=5

F T(7,F) (5F) T(7,F)*(5F) F*T(7,F)*(5F)
- ------ ---- ----------- -------------
1      1    5           5             5
2    126   10        1260          2520
3   1806   10       18060         54180
4   8400    5       42000        168000
5  16800    1       16800         84000
                    -----        ------
Sums                78125        308705

Check: 78125=57

Fav = 308705/78125 = ~3.951
Aav = 7 - 308705/78125 = 238170/78125 = ~3.049



Example 2: C=6, S=6

F T(6,F) (6F) T(6,F)*(6F) F*T(6,F)*(6F)
- ------ ---- ----------- -------------
1      1    6           6             6
2     62   15         930          1860
3    540   20       10800         32400
4   1560   15       23400         93600
5   1800    6       10800         54000
6    720    1         720          4320
                    -----        ------
Sums                46656        186186

Check: 46656=66

Fav = 186186/46656 = ~3.991
Aav = 6 - 186186/46656 = 93750/46656 = ~2.009



Example 3: C=4, S=9

F T(4,F) (9F) T(4,F)*(9F) F*T(4,F)*(9F)
- ------ ---- ----------- -------------
1      1    9           9             9
2     14   36         504          1008
3     36   84        3024          9072
4     24  126        3024         12096
                     ----         -----
Sums                 6561         22185

Check: 6561=94

Fav = 22185/6561 = ~3.381
Aav = 4 - 22185/6561 = 4059/6561 = ~0.619


I don't know if this helps but for C=S, the number of cards on top of other cards tends to C/e as C goes to infinity.

"The hydrogen atom..." is missing only Q.
"A wedge..." is missing J, Q, and Z.
"Not that there..." is missing J, X, and Z.

If a spammer ever manages to construct a proper pangram, the results could be dire.

"Tale lara croft tomb, cradle nagiaa bronx, pack beyond. Saints, few recognized catholic church! Lille addition, passengers vehicles freight. Land km, sq mi region. Demand april may executive addict dies marathon, game sirius!"

Oh, so close to a pangram. Missing only j. I suggest:

"Tale lara croft tomb, cradle nagiaa bronx, pack beyond. Saints, few recognized catholic church! Lille addition, passengers vehicles freight. Land km, sq mi region. Demand april may executive addict dies jogging, game sirius!"

hullo. Teucer has revived your curvature of the earth nomic:

http://einos-nomic.blogspot.com/

Thanks much for the message, Emily! As I said in the entry (and I meant it!), I was rather horrified to find myself thinking that way; I posted it not so much because I think it's a correct reaction as to document and wonder about the fact that it was the reaction that I had.

On the specific point:

1) Not at all! Wall Street is a bunch of private firms, not run or funded by American taxpayers anymore than it's run or funded by Venezuelan or Welsh taxpayers.

2) It's true that not bailing out the current failing institutions would likely have led to quite a bit of pain, banks closing, etc. However, now that Congress HAS bailed them out, they will take even MORE risks (since, after all, if anything goes wrong someone will step in to save them), and the NEXT meltdown will lead to even more pain. Sacrificing long-term health for the short-term illusion of health: bad idea!

3) I know, isn't it awful? :) I think it would be ideal if countries could freely trade with one another, but ALSO arrange things so that if one country has a problem like this, other countries don't find themselves so entangled. Probably impossible, but hey...

Edited by author 10-04-2008 07:57 PM
 Hello! I have been a long time reader of your log (since the good ol' sims 2 days) and what kept me reading after the depature of The Sims was your very unique take on world news and the fact that you are actually quite funny.
However I must post a protest about the comment you made on 30th of September regarding the "bailout" of Wall Street:

"Ha ha ha ha ha ha bite me!

Run your own fucking country, assholes.

If your economy depends upon the US government stealing a trillion dollars from its citizens, then as the lolcats say, U R DOIN IT WRONG.

[rude sounds]" <--that comment in paticular

I am from Wales, in Europe, and we don't actually run our own country (Gordon Brown does it for us) but I digress.

The points that this rambling message is trying make are:

1. Many people such as yourself are complaining that the goverment will need money from the US taxpayer to help Wall Street. Isn't Wall Street actually run and funded by American taxpayers? Admittedly, not every American is a shareholder, but I'm sure a fair number are.

2. While I do agree with your point that Wall Street would recover from the recession eventually, many small banks which are closing now will have no way of recovering. Our towns local bank has been shut down because the economy simply isn't stable enough to justify owning the bank. If America had "bailed out" Wall Street, many banks would have been saved from closure.

3. "Run your own fucking country, assholes."- Globalisation (sp?) means that no country can truly have its own economy any more. If a similar crash had happened in Britain months ago, then America would have felt the reprucussions, albeit on a smaller scale. And at the end of the day, America is HUUUGE (pop. 305 mil.) especially compared to Wales (pop. 3 mil.) so what chance do we stand really?

Sorry for cluttering up your message board with my overly long post. Apologies also for my spelling and grammer (English isn't my first language). An please let me assure you that this is in no way a personal attack on yourself. I just like debating :D