Number of Compositions
Conventionally the empty composition is counted as the sole composition of 0, and there are no compositions of negative integers. There are 2n−1 compositions of n ≥ 1; here is a proof:
Placing either a plus sign or a comma in each of the n − 1 boxes of the array
produces a unique composition of n. Conversely, every composition of n determines an assignment of pluses and commas. Since there are n − 1 binary choices, the result follows. The same argument shows that the number of compositions of n into exactly k parts is given by the binomial coefficient . Note that by summing over all possible number of parts we recover 2n−1 as the total number of compositions of n:
For weak compositions, the number is, since each k-composition of n + k corresponds to a weak one of n by the rule → .
Read more about this topic: Composition (number Theory)
Famous quotes containing the words number of and/or number:
“I can’t quite define my aversion to asking questions of strangers. From snatches of family battles which I have heard drifting up from railway stations and street corners, I gather that there are a great many men who share my dislike for it, as well as an equal number of women who ... believe it to be the solution to most of this world’s problems.”
—Robert Benchley (1889–1945)
“As equality increases, so does the number of people struggling for predominance.”
—Mason Cooley (b. 1927)