Combinatorics

IMPORTANT

**Conny Hansson has pointed out a potential flaw in the numbers below. This chart correctly counts the number of ways to make a scoring pattern, but it does not account for arranging the tiles. This means the total number of regular hands is larger than the combinations of tiles that would make those hands.

To illustrate: Imagine three chows of 123 dots, plus a pung of 7 dots, plus a pair of 9 dots. Those same tiles could make pungs of 1, 2, 3, and 7 of dots with a pair of 9's. This chart counts those hands separately. Whether it should or not is another matter.

This table can be used to compare the relative frequency of obtaining each of the scoring elements in a random collection of tiles that happen to be a regular hand.

It shows the number of possible ways to create each of the scoring patterns. That is to say the number of combinations of tiles in all possible regular hands. No regard is given to game play: no chow, pung, or kong claims are considered. In practice, this means that patterns with sequences will have a higher result than triplets (as there are more tiles available), and that multiple honor sets will be easier to obtain than similar numbered sets (as players discard single honors more frequently).

Nothing that requires a wait (such as Nine Gates, Edge Wait, etc.) are possible to compute with the same methods, nor is any computation involving having concealed versus exposed sets.

Note that Regular Hands means hands that have four sets of three tiles and a pair. The table also has irregular hands in it (such as Seven Pairs), but they are not included in the total count for regular hands. Similarly, hands with irregular elements (such as anything knitted) is also not counted in that total.

Table Key

Column Name Meaning
A World Series Of Mahjong Overview (aka Zung Jung) v3.2 Fan Number
B Zung Jung v3.2 Score
C Chinese Official Overview (aka Mahjong Competition Rules) Fan Number
D MCR Score
E World Mahjong Federation Overview (aka WMPA) Fan Number
F WMFed Score
G EMA Riichi Yaku - Concealed(Exposed) - * Special

Results

Scoring Pattern Combinations A B C D E F G
Regular Hands 11,369,205,075,492
No Honors 7,154,156,054,616 76 1
Four Chows 6,715,895,624,442 1.1 5
Four Chows No Value Honors 5,339,360,797,812 *1(0)
Mixed Double Chow 4,973,054,207,424 70 1
Four Chows No Honors 4,788,746,867,160 63 2 15 1
One Pung 3,809,325,606,336
Short Straight 3,537,856,860,870 71 1
One Voided Suit 2,633,031,836,760 75 1
Pung of Terminal or Wind 1,828,674,922,116 73 1
Value Pair 1,786,219,355,376
Seven Pairs 1,569,298,171,584 10.2 30 19 24
Seven Unique Pairs 1,505,948,184,576 24 2 2
Two Suits Only 1,158,898,125,528
Two Terminal Chows 978,440,223,168 72 1
All Simples 903,760,799,976 1.3 5 68 2 13 1 1
Mixed Shifted Chows 823,042,473,984 51 6
Value Honor 820,346,243,136 3.1 10
Two Pungs 774,405,495,000
Dragon Pung 564,711,862,608 59 2 6,7,8 1 1
Two Identical Chows 448,675,374,720 5.1.1 10 69 1 14 1 1(0)
Mixed Triple Chow 357,580,366,530 6.1 35 48 8 19 2 2(1)
Two Similar Chows Twice 267,760,375,104
Knitted Straight 237,892,534,272 35 12
Seat Wind 208,720,839,984 61 2 4 1 1
Prevalent Wind 208,720,839,984 60 2 5 1 1
Knitted Straight, Set and Pair 204,069,666,816 35 12
Mixed Straight 203,069,423,616 39 8
Lesser Honors and Knitted Tiles 135,291,469,824 34 12
Straight 116,688,273,408 7.1 40 28 16 18 2 2(1)
Tile Hog 108,496,998,180 64 2
Six Honors and Knitted Tiles 101,468,602,368 34 12
Small Two Winds 92,739,602,304
All Types 72,593,123,328 52 6 17 2
Three Pungs 67,441,343,634
Three Pure Shifted Chows 58,253,661,696 30 16
Greater Honors and Knitted Tiles 57,982,058,496 20 24
Two Consecutive Pungs 42,449,544,000
Double Pung 41,281,971,264 65 2
Five Honors and Knitted Straight 33,822,867,456 34+35 24
All Fives 29,819,607,684 31 16
Outside Hand 28,400,308,860 8.1.1 40 55 4 16 2 2(1)
Half Flush 21,438,061,428 2.1.1 40 50 6 26 3 3(2)
Big Two Winds 13,352,952,960
Big Two Dragons 6,986,887,488 54 6
Terminal in Each Set 6,368,641,140 8.1.2 50 28 3 2
Upper Four 2,938,382,904 36,37 12
All Pungs 2,137,006,080 4.1 30 49 6 23 2 2
Full Flush 1,446,713,304 2.1.2 80 22 24 31 6 6(5)
Two Identical Chows Twice 1,399,133,304 5.1.2 60 27 3 3(0)
Thirteen Orphans 1,308,622,848 10.1 160 7 88 40 15 *L
Small Three Similar Pungs 1,253,507,136 6.2.1 30
Small Three Winds 1,055,296,512 3.3.1 30
Reversible Tiles 931,561,760 40 8
Mixed Shifted Pungs 459,835,392 42 8
Four Pure Shifted Chows 434,221,056 16 32
Three-Suited Terminal Chows 301,989,888 29 16
Three Identical Chows 279,401,280 5.1.3 120 23 24
Three Pure Shifted Pungs 278,299,266 7.2.1 100 24 24
Small Three Dragons 265,510,656 3.2.1 40 10 64 30 4 2
Three Similar Pungs 103,947,264 6.2.2 120 32 16 20 2 2
Upper Tiles 63,943,416 25,26,27 24
Big Three Winds 62,115,840 3.3.2 120 38 12
Big Three Dragons 16,137,216 3.2.2 130 2 88 33 12 L
All Terminals and Honors 9,838,080 8.1.3 100 18 32 29 4 2
Two Similar Pungs Twice 7,741,440
All Even 6,082,560 21 24
Seven Shifted Pairs 2,519,424 6 88
Small Four Winds 2,248,704 3.3.3 320 9 64 32 12 L
Pure Terminal Chows 839,808 13 64
Four Consecutive Pungs 829,440 7.2.2 200 15 48
All Honors 161,280 3.4 320 11 64 37 12 L
All Green 155,316 3 88 L
Big Four Winds 46,080 3.3.4 400 1 88 41 20 2L
All Terminals 46,080 8.1.4 400 8 64 39 15 L
Four Identical Chows 3,906 5.1.4 480 14 48

Methodology

For the regular hands (four sets and a pair), the first set steps through all of the pungs, then all of the chows. Each of the other three sets steps from the previous set to the end of that list (so as to avoid counting permutations). The pair then steps through each of the 34 tile types. (So the first combination examined is four pungs of one dot, and a pair of the same, which would require 14 identical tiles!) Any combinations that require more than four of any one tile are skipped. (So the first combination counted has pungs of 1 dot, 2 dots, 3 dots, 4 dots, and a pair of 5 dots.)

A count of ways to combine tiles to make that hand is then counted. (Continuing the example, there are 4 ways to make each of the dot pungs, and 6 ways to make the pair, or 4 * 4 * 4 * 4 * 6 or 1536 ways to make that hand.) Then each of the patterns is tested to see if the hand conforms, testing the more-restrictive patterns first, and the count is added to the pattern's total if the test passes. (The example counts towards Four Consecutive Pungs, but not Three Consecutive Pungs.) Further implications are not removed. (Our example hand is counted in All Pungs.)

For seven pairs hands, each of the seven pairs is stepped from the previous set through the remaining tiles.

Some of the results are computed directly such as Thirteen Orphans is (4^12 * 6 * 13). Greater Honors and Knitted Tiles as well as Lesser Honors and Knitted Tiles are computed similarly. Someone please check my math!

The list of exclusions:

  • Upper Four excludes Upper Tiles (That is, a hand that scores for Upper Tiles is NOT counted toward Upper Four)
  • One Pung excludes two or more; Two Pungs excludes three or more; Three Pungs excludes four pungs
  • Two Terminal Chows excludes Three Suited- and Pure-Terminal Chows
  • Outside Hand excludes Pure Lesser Terminals, Terminals and Honors, All Terminals
  • Pure Lesser Terminals excludes All Terminals
  • All Terminals excludes All Terminals and Honors
  • Half- and Full-Flush exclude All Honors
  • Half Flush excludes Full Flush
  • Big Four Winds excludes all smaller Winds; etc.
  • Big Three Dragons excludes all smaller Dragons; etc.
  • Mixed Double Chow excludes Two Similar Sequences Twice
  • Two Identical Sequences excludes three or more identical sequences; etc.
  • Short Straight excludes Straight
  • Three Pure Shifted Chows excludes Four Pure Shifted Chows
  • Double Pung excludes Double Pung Twice
  • Two Consecutive Pungs excludes three or more; etc.
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page revision: 44, last edited: 01 Oct 2011 01:00
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