A Certain Unscientific Count of Monotone Sequences: Alternative solution¶

Pascal recurrence: the setter's solution¶

alternative.cpp evaluates the coefficient proved in the main editorial. Set \(a=n+m-1\) , \(t=\min(n,m-1)\) , initialize \(C_0=1\) and all other entries to zero. For each row \(i=1,\ldots,a\) , update in descending order \(j=\min(i,t),\ldots,1\) :

\[ C_j\leftarrow C_j+C_{j-1}\pmod p \]

Both values on the right then belong to the previous row, exactly implementing Pascal's identity. Descending order prevents overwriting \(C_{j-1}\) too early. After the row, \(C_j=\binom{i}{j}\) ; consequently the final \(C_t\) is correct. If \(t=0\) , the value remains one.

Time is \(O((n+m)t)\) and auxiliary space \(O(t)\) . The method uses addition only, needs neither a prime modulus nor inverses, and passes the complete tests. Products are preferred here for less computation and storage. Codex (GPT-6) replaced the setter's two-dimensional table with rolling storage while retaining its recurrence.