factorial

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factorial

Maths
the product of all the positive integers from one up to and including a given integer. Factorial zero is assigned the value of one: factorial four is 1 × 2 × 3 × 4. Symbol: n!, where n is the given integer

Factorial

 

The factorial of a given natural number n is the product of all the natural numbers less than or equal to n. The factorial of n is denoted usually by n! Thus,

n! = 1 × 2 × ... × n

For large n an approximate expression for n! is given by Stirling’s formula. The number of permutations of n things taken all at a time is the factorial of n.

factorial

[fak′tȯr·ē·əl]
(mathematics)
The product of all positive integers less than or equal to n; written n !; by convention 0! = 1.

factorial

(mathematics)
The mathematical function that takes a natural number, N, and returns the product of N and all smaller positive integers. This is written

N! = N * (N-1) * (N-2) * ... * 1.

The factorial of zero is one because it is an empty product.

Factorial can be defined recursively as

0! = 1 N! = N * (N-1)! , N > 0

The gamma function is the equivalent for real numbers.

factorial

The number of sequences that can exist with a set of items, derived by multiplying the number of items by the next lowest number until 1 is reached. For example, three items have six sequences (3x2x1=6): 123, 132, 231, 213, 312 and 321. See factor and IFP.
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Thus the factorial function B(3) can only take the values 4 or 6 and therefore we are in one of these two cases.
and D(3) = 8 has the same factorial function as [C.sub.4].
P has the following binomial factorial function B(k) = [2.sup.k-1], where 1 [less than or equal to] k [less than or equal to] n - 1;
n is even and there is a positive integer [alpha] > 1 such that poset P has the binomial factorial function B(k) = [2.sup.k-1] for 1 [less than or equal to] k [less than or equal to] n - 2 and B(n - 1) = [alpha] x [2.sup.n-2] for some positive integer [alpha].
Let P be an Eulerian Sheffer poset of even rank n = 2m + 2 > 4 with the binomial factorial function B(k) = [2.sup.k-1] for 1 [less than or equal to] k [less than or equal to] 2m, and B(2m + 1) = [alpha] x [2.sup.2m], where [alpha] > 1 is a positive integer.
Ehrenborg and Readdy [4] gave a complete classification of the factorial functions of infinite Eulerian binomial posets and infinite Eulerian Sheffer posets.
As we mentioned above, Ehrenborg and Readdy in [4] classify the factorial functions of infinite Eulerian binomial posets and infinite Eulerian Sheffer posets.
The posets P and Q have the same factorial functions and atom functions up to rank 2m.
By considering the factorial functions, Theorem 3.9 implies that the intervals [[??], [a.sub.i]] and [[x.sub.j], [??]] have the same factorial functions as [B.sub.2m] and so they are isomorphic to [B.sub.2m] for
In this section, we give an almost complete classification of the factorial functions and the structure of Eulerian Sheffer posets.
4.1 Characterization of the factorial functions and structure of Eulerian Sheffer posets of rank n [greater than or equal to] 5 for which B (3) = 3!.
have the Sheffer factorial functions D(3) = 4, 6, 8 and 10.
Finite Eulerian posets which are binomial or Sheffer

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