If P is an upper semilattice whose Hasse diagram is a tree and whose cartesian powers are Macaulay, it is shown that Hasse diagram of P is actually a spider in which all the legs ...
We show that any k-regular bipartite graph with 2n vertices has at least ((k−1)k−1 kk−2 )n perfect matchings (1-factors). Equivalently, this is a lower bound on the permanent...