Ordered $m$-tuples with fixed number of changes

Question

Given $1\leq k\leq m$, $2\leq d\leq c i\ln i$ and $2\leq i\leq c'\ln(mi\ln i)$ at some $c,c'>0$ how many sequences (lower and upper bounds) are of form $$z_1,\dots,z_m$$ on the condition that

$$0\leq z_1\leq\dots\leq z_m\leq 2^d$$ $$|\{i\in\{1,\dots,m\}: z_i\neq z_{i+1}\}|=k$$

are there?

Is there a standard combinatorics problem associated with such problems?

References would be very helpful.

Answer 1

Apparently, you mean $|\{i\in\{1,\dots,m-1\}: z_i\neq z_{i+1}\}|=k$ as $z_{m+1}$ is undefined. This condition implies that among $z_1, \dots, z_m$ there are $k+1$ district values. Since the number of subsets $\{1,\dots,m-1\}$ of size $k$ equals $\binom{m-1}{k}$, we conclude that the number of sequences of interest equals $$\binom{m-1}{k}\binom{2^d+1}{k+1}.$$