Monomial - Number

Number

The number of monomials of degree d in n variables is the number of multicombinations of d elements chosen among the n variables (a variable can be chosen more than once, but order does not matter), which is given by the multiset coefficient . This expression can also be given in the form of a binomial coefficient, as a polynomial expression in d, or using a rising factorial power of d + 1:

\left(\!\!{n\choose d}\!\!\right) = \binom{n+d-1}{d} = \binom{d+(n-1)}{n-1} = \frac{(d+1)\times(d+2)\times\cdots\times(d+n-1)}{1\times2\times\cdots\times(n-1)} = \frac{1}{(n-1)!}(d+1)^{\overline{n-1}}.

The latter forms are particularly useful when one fixes the number of variables and lets the degree vary. From these expressions one sees that for fixed n, the number of monomials of degree d is a polynomial expression in d of degree with leading coefficient .

For example, the number of monomials in three variables of degree d is ; these numbers form the sequence 1, 3, 6, 10, 15, ... of triangular numbers.

Read more about this topic:  Monomial

Famous quotes containing the word number:

    “In a number of other cultures, fathers are not relegated to babysitter status, nor is their ability to be primary nurturers so readily dismissed.... We have evidence that in our own society men can rear and nurture their children competently and that men’s methods, although different from those of women, are imaginative and constructive.”
    —Kyle D. Pruett (20th century)

    “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)

    “After a certain number of years our faces become our biographies. We get to be responsible for our faces.”
    —Cynthia Ozick (b. 1928)