“Trinomial triangle”的意思、由来-开放百科全书

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Trinomial triangle

释义

Properties
Recursion formula
The middle entries
Chess mathematics
Importance in combinatorics
References
Further reading

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

Properties

The -th row corresponds to the coefficients in the polynomial expansion of the expansion of the trinomial raised to the -th power:^[1]

or, symmetrically,

,

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 .

Recursion formula

The trinomial coefficients can be generated using the following recursion formula:^[1]

,

for ,

where for and .

The middle entries

The middle entries of the trinomial triangle {{OEIS|A002426}}

1, 1, 3, 7, 19, 51, 141, 393, 1107, 3139, …

were studied by Euler. The middle entry for the -th row is given by

The corresponding generating function is^[2]

Euler also noted the following exemplum memorabile inductionis fallacis ("notable example of fallacious induction"):

for

where stands for the Fibonacci sequence. For larger , however, this relationship is incorrect. George Andrews explained this fallacy using the general identity^[3]

Chess mathematics

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.^[4] 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
0	1
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
6	1	AABBCC

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,^[4] which can be written in terms of the binomial coefficients as

.

For example,

.

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).

References