Discrete total variation under the multinomial, Part~1: expectation under the uniform null

We study the expectation of the discrete total variation statistic $d={\sum }_{i=1}^{n-1}\mathrm{\mid }{X}_{i+1}-X_i\mathrm{\mid }$ of counts $X_1,\dots ,X_n$ over $n$ ordered bins with sample size $N$, under the uniform multinomial null. In the notation used throughout the series, a reference distribution $\mathbf{r}$ defines $d_{\mathbf{r}}(x)=\mathrm{\parallel }x-N\mathbf{r}{\mathrm{\parallel }}_V$; here $\mathbf{r}=\mathbf{u}$ is uniform and we write simply $d$. We obtain an exact single-sum formula, a closed form in central trinomial coefficients valid for every $n\ge 2$, with one-term evaluations at $n=2,4$ and a complete D-finite characterization for $n=3$. The per-edge expectation $W_N=E\mathrm{\mid }{X}_2-X_1\mathrm{\mid }$ satisfies an exact three-term recurrence, from which we derive its complete $1\mathrm{/}N$ expansion with the $n$-independent constant $E(d^{\circ }\mathrm{/}\sqrt{Nn})\to 2\mathrm{/}\sqrt{\pi }$ for the cyclic statistic $d^{\circ }=d_{\mathbf{u}}^{\circ }$, independent of the number of bins; the complementary sparse regime (fixed $N$, $n\to \mathrm{\infty }$) is treated in a separate paper. Conditioning on a two-bin total reduces each edge to a symmetric random walk; as the subsequent parts of the series will show, this method carries the analysis to non-uniform reference laws.