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In cryptography, permutations are used to rearrange the elements within a set, such as the letters in an alphabet. In the context of a monoalphabetic cryptosystem, a cryptographic permutation is the unique mapping of each plaintext letter to a different ciphertext letter.

Imagine a simple alphabet 'A' to 'Z', rearranged into a random order. This rearranged alphabet represents the key to encrypt and decrypt messages. The security of a monoalphabetic cipher directly correlates to the number of possible permutations. The number of permutations of a set of n elements is denoted as n!, where '!' represents the factorial operation. Hence, for the 26 letters of the English alphabet, there are 26! possible permutations, creating a vast number of potential keys. Although this number may seem large, other factors, discussed in later sections, affect the actual security of these cryptosystems.

The factorial of a number, expressed as n!, is the product of all positive integers from 1 up to n. It's an important mathematical concept for understanding permutations in cryptography, as well as in probability and combinatorics. The factorial of 26 is calculated as:
\[ 26! = 26 \times 25 \times 24 \times \ldots \times 3 \times 2 \times 1 \]
This represents the total number of ways to arrange 26 unique items, or in our case, letters. Factorial values grow astronomically with even small increases in n, leading to a very large number of unique mappings for monoalphabetic cryptosystems. However, despite the seemingly immense number of possible permutations, factorials alone do not guarantee security, as we'll explore in relation to frequency analysis and computational power.

Frequency analysis is a technique used to break ciphers by analyzing the frequency at which certain letters or patterns occur in the ciphertext. In any language, certain letters and combinations occur with predictable regularity. For instance, in English, 'E' is the most commonly used letter.

Attackers use this predictability by counting the frequency of each letter in a ciphertext and matching it with the known frequency of letters in the language. This is particularly effective against monoalphabetic ciphers because if a letter maps consistently to a single ciphertext letter, patterns can be detected more easily. This vulnerability is a significant weakness of monoalphabetic cryptosystems. Consequently, the security of these systems is compromised, and modern ciphers have moved towards more complex systems that are less susceptible to frequency analysis.

Cryptography security refers to the strength and resilience of a cryptosystem against unauthorized decryption. While the factorial calculation of permutations can suggest a high level of security for monoalphabetic cryptosystems due to their large number of potential keys, the reality is quite different. Modern cryptography standards require ciphers to withstand various types of attacks, including brute force and statistical analysis.

Even though a monoalphabetic cipher can have 26! possible configurations, modern computational power makes brute-force attacks feasible, where an attacker systematically checks all possible keys. Moreover, the susceptibility of these systems to frequency analysis further undermines their security. For these reasons, monoalphabetic cryptosystems are considered insecure by contemporary standards, and more advanced techniques such as polyalphabetic ciphers and public-key cryptography are used to safeguard sensitive information.