Yet another generalization of the Kruskal-Katona theorem Original Research Article

For an n-tuple t = (t1,t2,…,tn) of integers satisfying 1⩽t1⩽t2···⩽tn, T(t)=T denotes the ranked partially ordered set consisting of n-tuples a = (a1,a2,…,an) of integers satisfying tn−ti⩽ai⩽tn, i = 1,2,…,n, partially ordered by defining a to precede c if ai = ci or ci = tn for i = 1,2,…,n. The rank r(a) of a is |{i|ai = tn}|. For 0⩽l⩽n, the set consisting of all elements of rank l is called the lth rank and is denoted Tl. Let b, l and m denote positive integers satisfying b⩽l⩽n and m⩽|Tl|. For a subset A of Tl, Δb A denotes the elements of Tl-b which precede at least one element of A. An algorithm is given for calculating min |Δb A|, where the minimum is taken over all m-element subsets A of Tl. If t1 = t2 = ··· = tn = 1, it reduces to the Kruskal-Katona algorithm.