The trinomial triangle is a variation of Pascal's triangle. The difference between the two is that an entry in the trinomial triangle is the sum of the three (rather than the two in Pascal's triangle) entries above it:
The -th entry of the -th row is denoted by
Rows are counted starting from 0. The entries of the -th row are indexed starting with from the left, and the middle entry has index 0. The symmetry of the entries of a row about the middle entry is expressed by the relationship
hence the alternative name trinomial coefficients because of their relationship to the multinomial coefficients:
Furthermore, the diagonals have interesting properties, such as their relationship to the triangular numbers.
The sum of the elements of -th row is .
- for ,
where for and .
The middle entries
- 1, 1, 3, 7, 19, 51, 141, 393, 1107, 3139, …
were studied by Euler. The middle entry for the -th row is given by
Euler also noted the following exemplum memorabile inductionis fallacis ("notable example of fallacious induction"):
The triangle corresponds to the number of possible paths that can be taken by the king in a game of chess. The entry in a cell represents the number of different paths (using a minimum number of moves) the king can take to reach the cell.
Importance in combinatorics
The coefficient of in the polynomial expansion of specifies the number of different ways of randomly drawing cards from two sets of identical playing cards. For example, in such a card game with two sets of the three cards A, B, C, the choices look like this:
|Number of selected cards||Number of options||Options|
|1||3||A, B, C|
|2||6||AA, AB, AC, BB, BC, CC|
|3||7||AAB, AAC, ABB, ABC, ACC, BBC, BCC|
|4||6||AABB, AABC, AACC, ABBC, ABCC, BBCC|
|5||3||AABBC, AABCC, ABBCC|
In particular, this results in as the number of different hands in a game of Doppelkopf.
Alternatively, it is also possible to arrive at this number by considering the number of ways of choosing pairs of identical cards from the two sets, which is . The remaining cards can then be chosen in ways, which can be written in terms of the binomial coefficients as
The example above corresponds to the three ways of selecting two cards without pairs of identical cards (AB, AC, BC) and the three ways of selecting a pair of identical cards (AA, BB, CC).
- Weisstein, Eric W., "Trinominal Coefficient", MathWorld.
- Weisstein, Eric W., "Central Trinomial Coefficient", MathWorld.
- George Andrews, Three Aspects for Partitions. Séminaire Lotharingien de Combinatoire, B25f (1990) Online copy
- Andreas Stiller: Pärchenmathematik. Trinomiale und Doppelkopf. ("Pair mathematics. Trinomials and the game of Doppelkopf"). c't Heft 10/2005, p. 181ff