What is the $i$th number $T_! (n,k,i)$ such that $T_! (n,k,i)$ is the sum of $n \in \mathbb{N}$ distinct positive integer factorials in $k \in \mathbb{N}$ distinct ways (ignoring ordering, parentheses, etc.) for a given $(n,k) \in \mathbb{N}^2:2 \leq n,k$, for the first few $i \in \mathbb{N}$ (where $T_!$ increases strictly monotonically with respect to $i$)? In other words, for a fixed $i$, $T_! (n,k,i) = \sum_{j=1}^n (t_{j, 1}!) = ... = \sum_{j=1}^n (t_{j, k}!)$ where $t_{j_0,k_0}! \not= t_{j_1,k_0}! \forall j_0 \not= j_1; t_{j,k} \in \mathbb{N} \forall (j,k) \in \mathbb{N}^2$ and with the other aforementioned conditions. (Just pick a small $n$ and $k$ (each) that is a prime and list the terms in this sequence for the given triplet for all positive integer $i$'s less than or equal to some small odd prime (your choice).)
How about for basically the same set up but with the similar but distinct equality, for fixed $i$, $U_! (n,k,i) = \sum_{j=1}^n (t_{j, 1}*(t_{j, 1}!)) = ... = \sum_{j=1}^n (t_{j, k}*(t_{j, k}!))$?
What about for the same set-ups but with the added restriction that the $t$'s mentioned must be prime (in the typical ring $\mathbb{Z}$), (where the taxicab-like functions are labelled $P_!$ and $V_!$ respectively)?
How did you find these sequences?
The basic sums of distinct factorial numbers are:
There are no equal sums of distinct factorial numbers which don't boil down to one of those two.
Proof: consider the numbers in the factorial number system. The factorials are $0_{10}!=10_!$, $1_{10}!=10_!$, $2_{10}!=100_!$, $3_{10}!=1000_!$, $\ldots$. Because $n > 2 \implies \sum_{i=0}^{n-1} i! < n!$, in a sum of distinct factorials the only possible carry is $0_{10}! + 1_{10}! = 10_! + 10_! = 100_!$, so a sum of distinct factorials represented in factorial base contains only digits $0$ and $1$ with the exception of the $2!$s place, which may be a $2$; and in any number satisfying that restriction on the digits, the part above the $2!$s place can be uniquely decomposed into a sum of distinct factorials.