Concrete mathematics : a foundation for computer science

7.1 DOMINO THEORY AND CHANGE 309
(Multiplication isn’t commutative, so we’re on the verge of cheating, by not
distinguishing between left and right division. In our application it doesn’t
matter, because I commutes with everything. But let’s not be picky, unless
our wild ideas lead to paradoxes.)
The next step is to expand this fraction as a power series, using the rule
1
-=
1-z
1 + 2 + z2 + z3 + . . . .
The null tiling I, which is the multiplicative identity for our combinatorial
arithmetic, plays the part of 1, the usual multiplicative identity; and 0 + �
plays z. So we get the expansion
I
I-U-El
= I+I:o+E)+(u+E)2+(u+E)3+~~~
= ~+~:o+e)+(m+m+~+m)
+ (ml+uB+al+rm+Bn+BE+E3l+m3) f... .
This is T, but the tilings are arranged in a different order than we had before.
Every tiling appears exactly once in this sum; for example, CEXE!ll appears
in the expansion of ( 0 + E )‘.
We can get useful information from this infinite sum by compressing it
down, ignoring details that are not of interest. For example, we can imagine
that the patterns become unglued and that the individual dominoes commute
with each other; then a term like IEEIB becomes C1406, because it contains
four verticals and six horizontals. Collecting like terms gives us the series
T =I+O+02-to2+03+2002t04+30202+~4+~~~.
The 20 =2 here represents the two terms of the old expansion, B and ELI, that
have one vertical and two horizontal dominoes; similarly 302 0’ represents the
three terms CB, CH, and Elll. We’re essentially treating II and o as ordinary
(commutative) variables.
We can find a closed form for the coefficients in the commutative version
of T by using the binomial theorem:
I
I- (0 + 02)
= I+(o+o~)+(o+,~)~+(o+~~)~+...
= ~(Ofo2)k
k>O
(7d