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 related to number of compositions:
“This nightmare occupied some ten pages of manuscript and wound off with a sermon so destructive of all hope to non-Presbyterians that it took the first prize. This composition was considered to be the very finest effort of the evening.... It may be remarked, in passing, that the number of compositions in which the word ‘beauteous’ was over-fondled, and human experience referred to as ‘life’s page,’ was up to the usual average.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)