How do I deal with the summations correctly whose upper or lower bounds are symbolic?

Question

I'm new to Wolfram Mathematica and I want to calculate the expression of s (with respect to n) to get the same result as this python code.

s = 0
for i in [1..n]:  # iterate i from 1 to n
    for j in [1..n]:
        for k in [i..j]:  # When i > j, skip it.
            s += 1

For example, when n = 10, it should be $220$ in fact.

I tried this code in Mathematica:

Sum[1, {i, 1, n}, {j, 1, n}, {k, i, j}]

which tells me that the answer is $n^2$, not corresponding to my expectation.

I have found that Mathematica can deal with such summation whose upper bound is less than its lower bound correctly, when the bounds are constants (output $0$ as answer), but when there's a symbolic bound, the result will be something like this:

Sum[1, {i, j, k}]

1 - j + k

When $j > k + 1$, it comes with a negative value. I guess this is the reason for the previous wrong output.

I searched and learnt that I should set GenerateConditions to True for the expression with correct conditions. So I tried this code:

Sum[Sum[1, {k, i, j}, GenerateConditions -> True], {i, 1, n}, {j, 1, n}]

But I got this (copied from MMA online):

ConditionalExpression[Piecewise[{{1, n == 1}}, 0], n <= 1]

It doesn't tell what should be when $n > 1$. I'm really puzzled now, and I want to know how to handle this case properly or what the keywords I could search with?

Answer 1

One way to do this is via Boole:

Sum[Boole[i <= k], {i, 1, n}, {j, 1, n}, {k, i, j}, Assumptions -> n \[Element] Integers]

...which produces: $$\begin{cases}\frac{1}{6} \left(n^3+3 n^2+2 n\right) & n\geq 1 \\0 & \text{True}\end{cases}$$

For $n=10$, this gives 220.

Answer 2

No need for making it so complicated:

Sum[1, {j, 1, n}, {i, 1, j}, {k, i, j}]

1/6 n (1 + n) (2 + n)