Say we have a sequence of $n$ positive integers, we can assume they're randomly chosen, let's call it $U_{n}$.
Let $S_{n}$ = sum of $U_{n}$ from $1$ to $n$.
Let $T_{n}$ = sum of $n$ from $1$ to $n$. In other words, $T_{n}$ is the triangle number $T_{n} = \frac{1}{2}n(n+1)$.
Define $F_{n} = U_{n} + n$.
(note that $SF_{n}$, sum of $F_{n}$, is equal to $S_{n} + T_{n}$)
Define two sets $EvenSet$ and $OddSet$ containing $F_{n}$ when it is even and odd respectively.
Let $SEvenF_{n}$ be the sum of numbers in $EvenSet$ and $SOddF_{n}$ be the sum of numbers in $OddSet$.
(note that $SEvenF_{n} + SOddF_{n} = SF_{n} = S_{n} + T_{n}$)
What is the probability of the two pairs ($S_{n}$ and $T_{n}$) and ($SEvenF_{n}$ and $SOddF_{n}$) are the same pairs?
In other words, either "$S_{n} = SEvenF_{n}$ thus $T_{n} = SOddF_{n}$" or "$S_{n} = SOddF_{n}$ thus $T_{n} = SEvenF_{n}$".
How to calculate this probability?
PS: I have a sample data with $n = 114$ in here http://goo.gl/k96FZ
PS2: Is there a way or an algorithm to generate such sequence?
PS3: @ZefChonoles has correctly pointed a concern about Probability Measure. Originally I intended this question to have no restriction on the positive integer set, that is to work on the infinite set of $\mathbb Z^{+}$.
But I believe you are free to impose certain restriction to this problem, as long as you can share insights on approaches in solving this problem.
For example, you can restrict $U_{n}$ and $F_{n}$ to be within $[1, 400]$ for the $n=114$ case. Or you can restrict them to be within $[1, Cn]$ with some constant $C$ for the general case. Or within $[1, Tn]$. Anything that helps, basically. Thanks!