Applications of the symmetric chain decomposition of the lattice of divisors (Q1337572)

The de Bruijn-Tengbergen-Kruyswijk symmetric chain decomposition of the lattice \(D_ N\) of divisors of a number \(N\) was given an elegant `parenthesization' description by \textit{C. Greene} and \textit{D. J. Kleitman} [J. Comb. Theory, Ser. A 20, 80-88 (1976; Zbl 0361.05015)]. This is now used to give neat proofs of three results: (i) the Clements- Griggs result concerning the strict unimodality of the rank numbers of \(D_ N\); (2) obtaining a minimum cover by intervals of the union of consecutive rank sets in \(D_ N\); (3) extending \textit{D. J. Kleitman's} general solution [Adv. Math. 5, 155-157 (1970; Zbl 0195.407)] of the Littlewood-Offord problem, on sums of vectors in a unit ball, to the case where limited repetitions are permitted.