Consider the symmetric sequence $P_n = \Delta^{n-1}$ of probability measures on finite sets, with coordinatewise $\Sigma_n$-action. There is a natural topological operad structure on $P$ given by multiplication. That is,

Let $p \in \Delta^{n-1}$ be given in coordinates by $p = (p^1,\dots,p^n)$.

For each $i=1,\dots,n$, let $p_i \in \Delta^{n_i-1}$ be given in coordinates by $p_i = (p_i^1,\dots,p_i^{n_i})$.

Then the operad composition is given by

$p \circ (p_1,\dots,p_n) = (p^1 p_1^1, p^1 p_1^2,\dots, p^1 p_1^{n_1}, p_2 p_2^1, \dots, p^n p_n^1, \dots, p^n p_n^{n_n})$.

For example, $(1/3,2/3) \circ ((1),(1/4,1/4,1/2)) = (1/3,1/6,1/6,1/3)$.

(I apologize for the notation -- in particular the tuples on the two sides of the above equation mean different things -- on the left, it's a list of inputs to the $\circ$ operation, while on the right it's a list of coordinates. Another way to write this would be $(p^i)_{i=1}^n \circ ((p_1^j)_{j=1}^{n_1},\dots, (p_n^j)_{j=1}^{n_n}) = (p^i p_i^j)_{1 \leq i \leq n, 1 \leq j \leq n_i}$.)

We can twist this operad by any ~~ convex ~~ increasing homeomorphism $f: [0,1] \to [0,1]$ as follows. Define $f_\ast: \Delta^{n-1} \to \Delta^{n-1}$ by $f_\ast(x_1,\dots,x_n) = (\frac{f(x_1)}{Z},\dots, \frac{f(x_n)}{Z})$ where $Z = f(x_1) + \dots + f(x_n)$; this is a $\Sigma_n$-equivariant homemormophism (~~ invertibility requires $f$ to be convex ~~), and then set

$p \circ^f (p_1,\dots, p_n) = f_\ast^{-1}(f_\ast(p) \circ (f_\ast(p_1),\dots, f_\ast(p_n)))$

**Question:** Is every topological operad structure on the symmetric sequence $P$ of the form $\circ^f$ for some ~~ convex ~~ increasing homeomorphism $f: [0,1] \to [0,1]$?

Guessing the answer is "yes" is motivated by the characterization of quasi-arithmetic means.

**Background:**
The operad $(P,\circ)$ features explicitly in Leinster's operadic characterization of entropy, and twists $(P,\circ^f)$ for certain $f$ are used by Baez, Fritz, and Leinster to characterize certain deformations of entropy. Certainly these operads are implicit in countless mathematical pursuits.

**Explicitly:**
The fixed data is the sequence of topological spaces $(P_0,P_1,\dots)$ and the $\Sigma_n$ action on $P_n$ (this is called a symmetric sequence, though you might also call it a topological species). An operad structure on the symmetric sequence $P$ is a monoid structure for the substitution monoidal product. That is, it consists of continuous operations $\circ: P_n \times P_{n_1} \times \dots \times P_{n_n} \to P_{n_1 + \dots + n_n}$ for each $n,n_1,\dots,n_n \in \mathbb N$ satisfying:

unitality: $1 \circ p = p = p \circ (1, \dots, 1)$ for some $1 \in P_1$

associativity: The two ways of associating $p \circ (p_1,\dots,p_n) \circ (p_{1,1},\dots, p_{n,n_n})$ are equal.

$\Sigma_n$-equivariance: $(\sigma \cdot p) \circ (p_1,\dots, p_n) = \sigma' \cdot (p \circ (p_{\sigma^{-1}1},\dots, p_{\sigma^{-1}n}))$ for each $\sigma \in \Sigma_n$, where $\sigma' \in \Sigma_{n_1 + \dots + n_n}$ is the image of $\sigma$ under the natural inclusion $\Sigma_n \ltimes (\Sigma_{n_1} \times \dots \times \Sigma_{n_n}) \hookrightarrow \Sigma_{n_1+\dots+n_n}$.

The question asks to classify operad structures on the symmetric sequence $P$.