For $n\in \mathbb{N}$, consider non-negative integer-valued functions $f$ on $\{1,2,\dots ,n\}$ satisfying $f(i)\ge f(j)$ for $i>j$ and ${\sum }_{i=1}^n(i+f(i))=2023$. Choose $n$ such that ${\sum }_{i=1}^{n}f(i)$ is the least. How many such functions exist in that case?

Step 1: Expand the summation constraint:

Step 2: We want to minimize the sum of $f(i)$, which is given by:

Since $f(i)\ge 0$, the sum of $f(i)$ must be non-negative. To minimize this sum, we must choose the largest integer $n$ such that $\frac{n(n+1)}{2}\le 2023$.

Step 3: We solve for $n$:

• For $n=63$: $\frac{63\times 64}{2}=2016\le 2023$.

• For $n=64$: $\frac{64\times 65}{2}=2080>2023$.

Thus, the optimal choice is $n=63$.

Step 4: The sum of $f(i)$ for $n=63$ is:

Step 5: Since $f(i)$ is a non-decreasing, non-negative integer-valued function, the sequence $f(1)\le f(2)\le \cdots \le f(63)$ with sum equal to $7$ represents a partition of the integer 7 into at most 63 parts.

Step 6: Since the number of parts (63) is much larger than the sum (7), this is simply the partition function $p(7)$. The number of partitions of 7 is exactly $15$. Thus, exactly $15$ such functions exist.