Let $\Bbb{y} = \big(y_1, y_2, y_3, \dots \big)$ be an infinite sequence of positive real numbers such that following $\Bbb{N} \times \Bbb{N}$ tridiagonal matrix
\begin{equation} T(\Bbb{y}) := \, \begin{pmatrix} 1 & y_1 & 0 & 0 & \\ 1 & 1 & y_2 & 0 & \\ 0 & 1 & 1 & y_3 & \\ 0 & 0 & 1 & 1 & \\ & & & & \ddots \end{pmatrix} \end{equation}
is totally positive in the sense that all of its leading principal minors $[1, \dots,n]$ for $n \geq 1$ are positive; here $[j_1, \dots, j_n]$ denotes the principal minor of $T$ whose row and columns sets are indexed by an (ordered) subset $\{j_1 < \dots < j_n\} \subset \Bbb{N}$. The terminology totally positive is justified since the positivity of $[1, \dots,n]$ for all $n \geq 1$ is in fact equivalent to the positivity of all principal minors of the form $[m,\dots,n]$ with $m \leq n$. I refer the reader to the article of Shih-Wei Yang and Andrei Zelevinsky "Cluster algebras of finite type via coxeter elements and principal minors" as a reference/guide.
For the purposes of this post we'll say that a sequence $\Bbb{y}$ of postive real numbers is totally positive if the corresponding tridiagonal matrix $T(\Bbb{y})$ is totally positive. I should point out that in the current literature a totally positive sequence is usually defined in terms of the total positivity of the corresponding infinite Toeplitz matrix; see for example the work of Lam and Pylyavskyy. One well known method to obtain a totally positive sequence $\Bbb{y} = \big(y_1, y_2, y_3, \dots \big)$ is to choose an auxiliary sequence $\Bbb{t} = \big(t_1, t_2, t_3, \dots \big)$ with $0 < t_n < 1$ for each $n \geq 1$ and then simply set $y_1 = t_1$ and $y_{n+1} = (1 - t_n) \, t_{n+1}$ for all $n \geq 1$. Indeed this recipe gives a parametrization of all totally positive sequences (as I've defined them).
Let us say that a sequence $\Bbb{y}$ is ultra-positive if in addition to being totally positive the quasi-shifted sequence
\begin{equation} \sigma_m(\Bbb{y}) := \, \Big( {y_{m+1} \over {y_m}}, \, y_{m+2}, \, y_{m+3}, \, y_{m+4}, \, \dots \Big) \end{equation}
is totally positive for all $m \geq 1$; equivalently if the infinite tridiagonal matrix
\begin{equation} \displaystyle S_m(\Bbb{y}) := \, \begin{pmatrix} y_m & y_{m+1} & 0 & 0 & \\ 1 & 1 & y_{m+2} & 0 & \\ 0 & 1 & 1 & y_{m+3} & \\ 0 & 0 & 1 & 1 & \\ & & & & \ddots \end{pmatrix} \end{equation}
is totally positive for each $m \geq 1$. In other words the leading principal minors of $S_m(\Bbb{y})$ must be positive for each $m \geq 1$ or more concretely, after using the Laplace expansion, $y_m > y_{m+1}$ and
\begin{equation} y_m[m+2, \dots, n] - y_{m+1}[m+3, \dots,n] > 0 \end{equation}
whenever $m \geq 1$ and $n \geq m+3$. The last inequality can be rewritten as
\begin{equation} \displaystyle {y_{m+1} \over {y_m}} \, < \, \kappa_n(m) := \, {[m+2,\dots,n] \over {[m+3,\dots,n]}} \end{equation}
whenever $m \geq 1$ and $n \geq m+3$. Furthermore, the following short plücker-like relation
\begin{equation} \begin{array}{c} [m, \dots, n] \, [m+1, \dots, n+1] \\ = \\ y_m \cdots y_n \ + \ [m , \dots, n+1] \, [m+1, \dots, n]. \end{array} \end{equation}
together with the fact that $\Bbb{y}$ is totally positive implies that $\kappa_n(m)$ is monotonically decreasing as $n \geq m+3$ increases and therefore
\begin{equation} {y_{m+1} \over {y_m}} \, \leq \, \kappa_\infty(m) := \, \lim_{n \rightarrow \infty} \kappa_n(m) \end{equation}
So we can conclude that $\Bbb{y}$ is ultra-positive if and only if (1) $\Bbb{y}$ is totally positive, (2) the limit $\kappa_\infty(m)$ exists and is positive for all $m \geq 1$, and (3)
\begin{equation} {y_{m+1} \over {y_m}} \, \leq \, \kappa_\infty(m) \end{equation}
for all $m \geq 1$.
An example of an ultra-positive sequence is
\begin{equation} \displaystyle \Bbb{y}_\mathrm{P} := \, \Big({1 \over 2}, \, {1 \over 3}, \, {1 \over 4}, \, \dots \Big) \end{equation}
where $\mathrm{P}$ stands for Plancherel --- the sequence is related to the Plancherel state $\varphi_\mathrm{P}$ of Young-Fibonacci lattice $\Bbb{YF}$ mentioned in this post Differential posets, the Plancherel state $\varphi_\mathrm{P}$, and minimality. In addition any $r$-shift $(y_r, y_{r+1}, y_{r+2}, \dots)$ of an ultra-positive sequence $(y_1, y_2, y_3, \dots)$ is clearly ultra-positive.
Question: How can one produce other examples of ultra-positive sequences and, more generally, how can the space of ultra-positive sequences be parameterized ? More specifically: Is there a variant of the Thoma simplex, perhaps
\begin{equation} \Omega = \left\{ \begin{array}{l} \displaystyle \text{all} \ \ \underline{\omega} = \big(\omega_1 \geq \omega _2 \geq \omega_ 3 \geq \cdots \geq 0 \big) \\ \displaystyle \text{such that} \ \ 1 \geq \omega_1 + \omega_2 + \omega_3 + \cdots \end{array} \right\} \end{equation}
together with a map $\underline{\omega} \mapsto \Bbb{y}_{\underline{\omega}}$ to the space of ultra-positive sequences which is a homeomorphism (in some appropriate topology) ?
thanks, ines.
