"Knitted Sets"

copyright 18 March 1998 Alan KWAN Shiu Ho


In the last article, we compared the number of sequences and triplets in the
deck.  In this article, we will take a look at a concept popular in some
Western versions of mah-jong: "knitted sets".  A knitted set is a 'sequence'
or a 'triplet' that contains, instead of 3 tiles in the same suit, a tile from
each of the 3 suits.  We exclude Honor tiles for the purpose of this
discussion.  Again, some background in elementary Combinatorics of the reader
is assumed.

There are 7 possible numbers for sequences: from 123 to 789.  If we are asking
for knitted sequences with a specific suit order, we have (following the
calculations in the last article)

7 * (4^3)

of them.  Compared with the total number of (regular) sequences in the deck:

s = 3 * 7 * (4^3)

we can see that the former quantity is exactly one-third of the latter.

Now for each number, there are 6 possible suit orders for a knitted sequence.
Thus, if we are not specifying a specific suit order, there are a total of

6 * 7 * (4^3)

knitted sequences in the 136-tile deck, or twice as many as regular sequences.

Now let's look at knitted triplets.  There are 9 numbers for knitted triplets.
Thus, there are simply

9 * (4^3)

knitted triplets in the deck.  That is about 0.43 times the number of regular
sequences, or about 4.24 times the number of regular triplets.

Finally, we put together a ball park figure.  Adding up the numbers and taking
the ratio, we see that there are 

((6 * 7 + 9) * (4^3)) / (s + t) = 2.20...

more than twice as many knitted sets in the deck as regular sets.

The interpretation of the numbers is left to the readers and players.