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

We study the expectation of the discrete total variation statistic $d=\sum_{i=1}^{n-1}|X_{i+1}-X_i|$ 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_{