Abstract
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 $M_n \geq (2n-1)!! = (2/e)^{n+o(n)}n^n$ and speculated that $\log M_n \sim n\log n$. In this note we confirm this speculation, showing that $M_n \leq (4/e)^{n+o(n)}n^n$.
Highlights
The basic aim of Morse theory is to gain knowledge of the topology of a manifold by studying smooth functions on it
If f has m critical points x1, . . . , xm ordered such that f (x1) < · · · < f, a slicing of f is an increasing sequence a0, . . . , am so that a0 < f (x1) < a1 < · · · < am−1 < f < am
Motivated by Hilbert’s 16th problem calling for a study of the topology of real algebraic varieties, Arnold [1] raised the broad question of the possible structures of excellent Morse functions on various manifolds, and in particular on Sn, and the specific enumerative question of how the number of possible structures grows as a function of the number of critical points
Summary
The basic aim of Morse theory is to gain knowledge of the topology of a manifold by studying smooth functions on it. Let f and g be excellent Morse functions on X each with m critical points, and let a0, . On the sphere S2 an excellent Morse function has 2n + 2 critical points for some n 0, of which exactly n are saddle points (with the rest being either local minima or local maxima). Motivated by a question of Arnold [1], in [5] Nicolaescu considered the question of the number of topological equivalence classes of excellent Morse functions on the 2-sphere S2 with n saddle points. He later speculated [6] that the lower bound is essentially log Tn2 ∼ n log n (2).
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