Unordered-pair partition for ordinal data ($\left(\genfrac{}{}{0ex}{}{n}{2}\right)=C+D+T_X+T_Y+T_{XY}$)

For $n$ observations of two ordered variables, every unordered pair is concordant, discordant, tied only on the first variable, tied only on the second variable, or tied on both. ConsequentlyThe proof is an exhaustive partition: if neither coordinate is tied the two nonzero comparison signs agree or disagree; otherwise exactly one or both coordinates are tied. For contingency-table counts $n_{ij}$ with row totals $n_{i+}$ and column totals $n_{+j}$,Two records tied on both variables belong to each marginal tie total, so subtracting them once from each produces the exclusive $T_X,T_Y$. Using incompatible exclusive and inclusive conventions corrupts Somers' D and Kendall's tau-b even if the untied concordances are correct. The identities also provide a direct arithmetic check on a printed statistical summary.