Given positive

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

n

and

d

, the

d

-th Macaulay representation of

n

is an expression for

n

as a sum of

binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...

s: :

n = \binom + \binom + \cdots + \binom + \binom.

Here,

c_1,\ldots,c_d

is a uniquely determined, strictly

increasing In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...

of nonnegative integers known as the Macaulay coefficients. For any two positive integers

n_1

and

n_2

n_1

is less than

n_2

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bi ...

the sequence of Macaulay coefficients for

n_1

comes before the sequence of Macaulay coefficients for

n_2

lexicographic order In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of ...

. Macaulay coefficients are also known as the

combinatorial number system In mathematics, and in particular in combinatorics, the combinatorial number system of degree ''k'' (for some positive integer ''k''), also referred to as combinadics, or the Macaulay representation of an integer, is a correspondence between natura ...

References

* * * {{algebra-stub Factorial and binomial topics