MacMahon parametrization: 3x3 magic squares vs admissible pairs

The map

is a bijection from the $3 \times 3$ magic squares with nonnegative entries and line sum $3e$ onto the admissible parameter pairs

Consequently the two counting functions agree: $M_{3}(3e) = \mathrm{paramCount}(e)$.

Injectivity. MacMahon's centre identity gives $M_{11} = e$; then the diagonal and anti-diagonal identities give $M_{22} = 2e - a$ and $M_{20} = 2e - c$, the row and column identities fill in $M_{01} = 3e - a - c$, $M_{21} = a + c - e$, $M_{10} = e + c - a$, $M_{12} = e + a - c$, and $M_{02} = c$, $M_{00} = a$ by definition. So $(a,c)$ determines $M$ completely.

Surjectivity. Given an admissible pair, the array mkMagic3 e a c has nonnegative entries (that is exactly what admissibility says, together with the implied bounds $a, c \le 2e$) and its three rows, three columns and two diagonals all sum to $3e$; each entry is at most $3e$, so it lies in the search space $\{0,\dots,3e\}$ used by magicCount.

Formalization Note magicCount 3 (3*e) counts arrays with entries in Fin (3*e+1); paramCount e counts the finset paramSet e. The bijection is expressed as an equality of cardinalities.