A lucasene is a hexagon chain that is similar to a fibonaccene, an L-fence is a poset the Hasse diagram of which is isomorphic to the directed inner dual graph of the corresponding lucasene. A new class of cubes, which named after matchable Lucas cubes according to the number of its vertices (or elements), are a series of directed or undirected Hasse diagrams of filter lattices of L-fences. The basic properties and several classes of polynomials, e.g. rank generating functions, cube polynomials and degree sequence polynomials, of matchable Lucas cubes are obtained. Some special conclusions on binomial coefficients and Lucas triangle are given.A set P equipped with a binary relation ≤ satisfying reflexivity, antisymmetry and transitivity is said to be a partially ordered set (poset for short). Given any poset P , the dual P * of P is formed by defining x ≤ y to hold in P * if and only if y ≤ x holds in P . A subposet S of P is a chain if any two elements of S are comparable, and denoted by n if |S| = n [3]. Let x ≺ y denote y cover s x in P , i.e. x < y and. The set of all filters of a poset P is denoted by F (P ), and carries the usual anti-inclusion order, forms a finite distributive lattice [3,31] called filter lattice. For a finite lattice L, we denote by0 L (resp.1 L ) the minimum (resp. maximum) element in L.The symmetric difference of two finite sets A and B is defined asis a perfect matching of a graph and C is an M -alternating cycle of the graph, then the symmetric difference of M and edge-set E(C) is another perfect matching of the graph, which is simply denoted by M ⊕ C. Let G be a plane bipartite graph with a perfect matching, and the vertices of G are colored properly black and white such that the two ends of every edge receive different colors. An M -alternating cycle of G is said to be proper, if every edge of the cycle belonging to M goes from white end-vertex to black end-vertex by the clockwise orientation of the cycle; otherwise improper [43]. An inner face of a graph is called a cell if itsboundary is a cycle, and we will say that the cycle is a cell too. For some concepts and notations not explained in the paper, refer to [3,8,31] for poset and lattice, [1,11] for graph theory. Zhang and Zhang [45] extended the concept of Z-transformation graph of a hexagonal system to plane bipartite graphs. Definition 2.1 ([45]) Let G be a plane bipartite graph. The Z-transformation graph Z(G) is defined on M(G): M 1 , M 2 ∈ M(G) are joined by an edge if and only if M 1 ⊕ M 2 is a cell of G. And Z-transformation digraph Z(G) is the orientation of Z(G): an edge M 1 M 2 of Z(G) is oriented from M 1 to M 2 if M 1 ⊕ M 2 form a proper M 1 -alternating (thus improper M 2 -alternating) cell. Let G be a bipartite graph, from Theorem 4.1.1 in [21], we have that G is elementary if and only if G is connected and every edge of G lies in a perfect matching of G. Let G be a plane bipartite graph with a perfect matching, a binary relation ≤ on M(G) is defined as: for M 1 , M 2 ∈ M(G), M 1 ≤ M 2 if and only if Z(G) has a d...

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Umbilical cord blood transplantation (UCBT) and peripheral blood stem cell transplantation (PBSCT) are effective allogeneic treatments for patients with malignant and non-malignant refractory hematological diseases. However, the differences in the immune cell reconstitution and the immune reactions during initial stages post-transplantation are not well established between UCBT and PBSCT. Therefore, in this study, we analyzed the differences in the immune reactions during the early stages (days 7-100 post-transplantation) such as pre-engraftment syndrome (PES), engraftment syndrome (ES), and acute graft-versus-host disease (aGVHD) and the immune cell reconstitution between the UCBT and the PBSCT group of patients. We enrolled a cohort of patients that underwent UCBT or PBSCT and healthy controls (n=25 each) and evaluated their peripheral blood mononuclear cell (PBMC) samples and plasma cytokine (IL-10 and GM-CSF) levels using flow cytometry and ELISA, respectively. Our results showed that the incidences of early immune reactions such as PES, ES, and aGVHD were significantly higher in the UCBT group compared to the PBSCT group. Furthermore, in comparison with the PBSCT group, the UCBT group showed higher proportion and numbers of naïve CD4+ T cells, lower proportion and numbers of Tregs, higher proportion of CD8+ T cells with increased activity, and higher proportion of mature CD56dim CD16+ NK cells during the early stages post-transplantation. Moreover, the plasma levels of GM-CSF were significantly higher in the UCBT group compared to the PBSCT group in the third week after transplantation. Overall, our findings demonstrated significant differences in the post-transplantation immune cell reconstitution between the UCBT and the PBSCT group of patients. These characteristics were associated with significant differences between the UCBT and the PBSCT groups regarding the incidences of immune reactions during the early stages post transplantation.

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The optimal cube factor of a graph, a special kind of component factor, is first introduced. Furthermore, the optimal cube factors of Fibonacci and matchable Lucas cubes are studied; and some results on the Padovan sequence and binomial coefficients are obtained.