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In this paper we study the commutativity property for topological sequence entropy. We prove that if $X$ is a compact metric space and $f,g:X\to X$ are continuous maps then $h_A(f\circ g)=h_A(g\circ f)$ for every increasing sequence $A$ if $X=[0,1]$, and construct a counterexample for the general case. In the interim, we also show that the equality $h_A\left(f\right)=h_A(f{|}_{{\cap }_{n\ge 0}{f}^n\left(X\right)})$ is true if $X=[0,1]$ but does not necessarily hold if $X$ is an arbitrary compact metric space.

Let $W\left(I\right)$ denote the family of continuous maps $f$ from an interval $I=[a,b]$ into itself such that (1) $f\left(a\right)=f\left(b\right)\in \{a,b\}$; (2) they consist of two monotone pieces; and (3) they have periodic points of periods exactly all powers of $2$. The main aim of this paper is to compute explicitly the topological sequence entropy $h_D\left(f\right)$ of any map $f\in W\left(I\right)$ respect to the sequence $D={\left(2^{m-1}\right)}_{m=1}^{\infty }$.