In combinatorial mathematics, the Lubell–Yamamoto–Meshalkin inequality, more commonly known as the LYM inequality, is an inequality on the sizes of sets in a Sperner family, proved by Bollobás (1965), Lubell (1966), Meshalkin (1963), and Yamamoto (1954).
To include the initials of all four discoverers, it is sometimes referred to as the YBLM inequality.
Then Lubell (1966) proves the Lubell–Yamamoto–Meshalkin inequality by a double counting argument in which he counts the permutations of U in two different ways.
First, by counting all permutations of U identified with {1, …, n } directly, one finds that there are n!
But secondly, one can generate a permutation (i.e., an order) of the elements of U by selecting a set S in A and choosing a map that sends {1, … , |S | } to S. If |S | = k, the set S is associated in this way with k!(n − k)!