[seqfan] game series

[Dmitry Kamenetsky]

Hello Sequence Fans,

Recently I have been looking at the following problem. There is a series of
n games played between two teams. The outcome of each game is either a win
or a loss (no draws).
A team wins the whole series if it wins k=floor(n/2)+1 games or more. Now if
a team reaches the magic number of k wins then the games that follow (if
there are any) are
dead games, because their outcome cannot affect the outcome of the series.
So a natural question arises: out of all the possible 2^n series how many of
them will have
at least one dead game? This forms the sequence 0,0,4,4,20,24,88,116,372,...
This sequence is not in the OEIS and neither is its version for all odd n.

If you could buy tickets only for one of the n games, then which game would
you attend (assuming you had to buy tickets before the series started)?
Personally I would
want to attend the game which decides the series, ie. a game in which one of
the teams reaches k wins. We can find how often the m-th game becomes the
"decider" and then
attend the game with the highest count. It turns out that for even n you
should always attend the last game, while for odd n you should attend either
the last or the second-last
game. I guess these results are not too surprising. The number of times the
last game is the decider gives the sequence A063886=2,2,4,6,12,20,40,70,140.
If we represent a series
as a bit string then the last game will be a decider if the number of 0's
equals the number of 1's (not counting the last bit). This corresponds
precisely to the comment in A063886
by Angel Plaza.

We can also consider the same questions for games with allowed draws.
Comments and ideas are very welcome. Below is the summary of the results.


Sincerely,
Dmitry Kamenetsky

Series Length 1
Number of series with dead games 0
Deciders: 2

Series Length 2
Number of series with dead games 0
Deciders: 0 2

Series Length 3
Number of series with dead games 4
Deciders: 0 4 4

Series Length 4
Number of series with dead games 4
Deciders: 0 0 4 6

Series Length 5
Number of series with dead games 20
Deciders: 0 0 8 12 12

Series Length 6
Number of series with dead games 24
Deciders: 0 0 0 8 16 20

Series Length 7
Number of series with dead games 88
Deciders: 0 0 0 16 32 40 40

Series Length 8
Number of series with dead games 116
Deciders: 0 0 0 0 16 40 60 70

Series Length 9
Number of series with dead games 372
Deciders: 0 0 0 0 32 80 120 140 140

Series Length 10
Number of series with dead games 520
Deciders: 0 0 0 0 0 32 96 168 224 252

Series Length 11
Number of series with dead games 1544
Deciders: 0 0 0 0 0 64 192 336 448 504 504

Series Length 12
Number of series with dead games 2248
Deciders: 0 0 0 0 0 0 64 224 448 672 840 924

Series Length 13
Number of series with dead games 6344
Deciders: 0 0 0 0 0 0 128 448 896 1344 1680 1848 1848

Series Length 14
Number of series with dead games 9520
Deciders: 0 0 0 0 0 0 0 128 512 1152 1920 2640 3168 3432

Series Length 15
Number of series with dead games 25904
Deciders: 0 0 0 0 0 0 0 256 1024 2304 3840 5280 6336 6864 6864

Series Length 16
Number of series with dead games 39796
Deciders: 0 0 0 0 0 0 0 0 256 1152 2880 5280 7920 10296 12012 12870

Series Length 17
Number of series with dead games 105332
Deciders: 0 0 0 0 0 0 0 0 512 2304 5760 10560 15840 20592 24024 25740 25740

Series Length 18
Number of series with dead games 164904
Deciders: 0 0 0 0 0 0 0 0 0 512 2560 7040 14080 22880 32032 40040 45760
48620

Series Length 19
Number of series with dead games 427048
Deciders: 0 0 0 0 0 0 0 0 0 1024 5120 14080 28160 45760 64064 80080 91520
97240 97240

Series Length 20
Number of series with dead games 679064
Deciders: 0 0 0 0 0 0 0 0 0 0 1024 5632 16896 36608 64064 96096 128128
155584 175032 184756