Looking at formula (4) at the top of page 3 of the paper makes it clear that Gessel uses the usual convention, in which summation over $i_1+\cdots+i_j = n$ is a summation over all non-negative integers $i_1,\ldots,i_j$ which sum to $n$, rather than all partitions. In the case of (4) we also have a lower bound on $i_1,\ldots,i_j$. If he wanted to sum over all partitions he would have added the condition $i_1\geq\cdots\geq i_j$, so in (4), instead of $i,j,k \geq 2$ he would have had $i\geq j\geq k\geq 2$.

Reading further down the paper, the definition you quote, formula (5), appears on the top of page (4), followed by the description "where the sum is over positive integers $i_1,\ldots,i_j$". Again, it is not mentioned that $i_1,\ldots,i_j$ are ordered.

The meaning of a definition involving multiple sums with Bernoulli numbers
In my septic system, I have a sump pump that pumps the liquid into the drainage field. We have a tank alarm in this tank with the pump, but someone (not sure if it was me or someone else) forgot to turn back on the alarm one time. So now that 1000 gallon tank is over flowing. The pump circuit has been blowing for me at the fuse box. It seems to blow the circuit based on getting a bunch of rain. I'll do a separate question for that.

My question is this. I can hear the pump run for a bit (I haven't kept track of the time) then stop. After a while it will start again. Is there a safety or something that would cause the pump to not run until there is no water left? This happens no matter if the float is hooked up or I am just plugged in straight. Or could it be the drainage field is full of water and needs to seep out so there is room for more?

I know the tank isn't empty cause shortly after the pump stops, water starts seeping out of the lid again. (and the alarm light is still on).
