On sequential versions of distributional topological complexity

We define a (non-decreasing) sequence $\{\mathsf{dTC}_m(X)\}_{m\ge 2}$ of higher versions of distributional topological complexity ($\mathsf{dTC}$) of a space $X$ introduced by Dranishnikov and Jauhari. This sequence generalizes $\mathsf{dTC}(X)$ in the sense that $\mathsf{dTC}_2(X) = \mathsf{dTC}(X)$, and is a direct analog to the classical sequence $\{\mathsf{TC}_m(X)\}_{m\ge 2}$. We show that like $\mathsf{TC}_m$ and $\mathsf{dTC}$, the sequential versions $\mathsf{dTC}_m$ are also homotopy invariants. Also, $\mathsf{dTC}_m(X)$ relates with the distributional LS-category ($\mathsf{dcat}$) of products of $X$ in the same way as $\mathsf{TC}_m(X)$ relates with the classical LS-category ($\mathsf{cat}$) of products of $X$. On one hand, we show that in general, $\mathsf{dTC}_m$ is a different concept than $\mathsf{TC}_m$ for each $m \ge 2$. On the other hand, by finding sharp cohomological lower bounds to $\mathsf{dTC}_m(X)$, we provide various examples of closed manifolds $X$ for which the sequences $\{\mathsf{TC}_m(X)\}_{m\ge 2}$ and $\{\mathsf{dTC}_m(X)\}_{m\ge 2}$ coincide.