On height isotopy classes of embeddings in the plane of a Morse function of a circle

Let (mathrm{SM}_{2n}(S^1,mathbb {R})) be a set of stable Morse functions of an oriented circle such that the number of singular points is (2nin mathbb {N}) and the order of singular values satisfies the particular condition. For an orthogonal projection (pi :mathbb {R}^2rightarrow mathbb {R}), let ({tilde{f}}_0) and ({tilde{f}}_1:S^1rightarrow mathbb {R}^2) be embedding lifts of f. If there is an ambient isotopy (tilde{varphi }_t:mathbb {R}^2rightarrow mathbb {R}^2) ((tin [0,1])) such that ({pi circ tilde{varphi }}_t(y_1,y_2)=y_1) and (tilde{varphi }_1circ {tilde{f}}_0={tilde{f}}_1), we say that ({tilde{f}}_0) and ({tilde{f}}_1) are height isotopic. We define a function (I:mathrm{SM}_{2n}(S^1,mathbb {R})rightarrow mathbb {N}) as follows: I(f) is the number of height isotopy classes of embeddings such that each rotation number is one. In this paper, we determine the maximal value of the function I equals the n-th Baxter number and the minimal value equals (2^{n-1}).