Source: wikibot/hall-s-identity
= Hall's identity
{wiki=Hall's_identity}
Hall's identity is a mathematical result related to the theory of partitions and combinatorial identities. Specifically, it provides a relationship involving binomial coefficients, which can be viewed through the lens of combinatorial enumeration. The identity states that for any non-negative integer \\( n \\): \\\[ \\sum_\{k=0\}^\{n\} (-1)^k \\binom\{n\}\{k\} (n-k)^m = (-1)^n \\binom\{m\}\{n\} n!