Several Problems In Combinatorics Of Posets

This thesis investigates the Mobius function of Dyck path posets and the intersecting property of colored boolean lattices.The first part of this thesis contributes to the computation and applications of the Mobius function of Dyck path posets. The Dyck path poset is ordered by pattern containment. A sequence representation of Dyck paths is presented, which shows that the Dyck path poset is a special case of the generalized subword order defined by Sagan and ratter, but contains the composition poser as a subposet, to which Sagan and Vatter paid more attention. From this it follows the Mobius function of the poset and the Mobius inverse of the rank function, which derives another expression of the rank function. In the end, a kind of Sperner and unimodal subposet is given.The second part of this thesis researches the intersecting property of posers. The authors first introduce a general concept--the colored boolean lattice, which is an extension of the normal boolean lattice, and includes the poset of q-signed sets introduced by Bollobas and Leader (called full colored boolean lattice in the thesis) and the poset of partial permutations introduced by Ku and Leader (called injective colored boolean lattice in the thesis). The authors establish an LYM-type inequality for the injective colored boolean lattice, which deduces the EKR theorem for k-partial permutations given by Ku and Leader immediately, and solve their conjecture on the structure of maximal intersecting families; the thesis gives another example of colored boolean lattice--fixed-point free colored boolean lattice, proves its EKR property, the uniqueness of its maximal intersecting families and an LYM-type inequality of its intersecting antichains; the direct product of colorings (as sets) is also discussed. At last, the authors give a Katona-type proof for intersecting permutations.