mthPartition <- function(n, m)
{
    A <- matrix(0, n, n)
    A[1, 1:min(n, 5)] <- 1
    if (n > 1) for (i in 2:n) for (j in 1:n) A[i, j] <- sum(A[i - 1, pmax(j - 1:5, 0)])
 
    for (L in 1:n) {
        if (m <= A[L, n]) break
        ifelse(L < n, m <- m - A[L, n], return(0))
    }
 
    a <- integer(L)
    for (i in 1:L) {
        for (j in 1:5) {
            if (i < L) {
                if (m <= A[L - i, n - j]) break
                m <- m - A[L - i, n - j]
            } else {
                if (j == n) break
            }
        }
 
        a[i] <- j
        n <- n - j
    }
 
    paste(a, collapse = "")
}
 
Q <- list(
    c(2, 1), c(2, 2), c(2, 3),
    c(20, 1), c(20, 2), c(20, 3), c(20, 400096), c(20, 400097),
    c(32, 1), c(32, 2), c(32, 3), c(32, 1000), c(32, 1000000), c(32, 1000000000), c(32, 1333610936), c(32, 1333610937)
)
 
for (q in Q) cat(sprintf("(%d, %d) → %s\n", q[1], q[2], mthPartition(q[1], q[2])))

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