Matrix-chain multiplication

Given a fixed sequence of compatible matrices $A_1,\dots ,A_m$, choose a binary parenthesization of $A_1\cdots A_m$ that minimizes the number of scalar multiplications in ordinary matrix multiplication. The order of the matrices is fixed: parenthesization changes the grouping, not the operand order. Associativity preserves the product while intermediate matrix dimensions change the cost. The possible binary groupings are counted by the Catalan number $C_{m-1}$ for $m\ge 1$.