We use an entropy based method to study two graph maximization problems. We upper bound the number of matchings of fixed size in a d-regular graph on N vertices. For 2 N bounded away from 0 and 1, the logarithm of the bound we obtain agrees in its leading term with the logarithm of the number of matchings of size in the graph consisting of N 2d disjoint copies of K d,d . This provides asymptotic evidence for a conjecture of S. Friedland et al. We also obtain an analogous result for independent sets of a fixed size in regular graphs, giving asymptotic evidence for a conjecture of J. Kahn. Our bounds on the number of matchings and independent sets of a fixed size are derived from bounds on the partition function (or generating polynomial) for matchings and independent sets.
The game of plates and olives, introduced by Nicolaescu, begins with an empty table. At each step either an empty plate is put down, an olive is put down on a plate, an olive is removed, an empty plate is removed, or the olives on two plates that both have olives on them are combined on one of the two plates, with the other plate removed. Plates are indistinguishable from one another, as are olives, and there is an inexhaustible supply of each.The game derives from the consideration of Morse functions on the 2-sphere. Specifically, the number of topological equivalence classes of excellent Morse functions on the 2-sphere that have order n (that is, that have 2n + 2 critical points) is the same as the number of ways of returning to an empty table for the first time after exactly 2n + 2 steps. We call this number M n .Nicolaescu gave the lower bound M n ≥ (2n − 1)!! = (2/e) n+o(n) n n and speculated that log M n ∼ n log n. In this note we confirm this speculation, showing that M n ≤ (4/e) n+o(n) n n .
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