The assumption they make is that for each person you have sex with, that person will have had sex (at that point in time) with (including you) as many unique people as you will ever have sex with.

So for 1 partner, you had sex with one person who had sex with one person (you). 1(your partner) = 1

For 2 partners, you had sex with two people who each had sex with two people. 2 (your partners) + 1 (your first partner's previous partner) + 1 (your second partner's previous partner) = 4.

For 3 partners, you had sex with three people who each had sex with three people. 3 (your partners) + 2 (Partner1's past two partners) + 2 (Partner2's past two partners) + 2 (Partner3's past two partners) = 9;

For 4 partners, you had sex with four people who each had sex with four people. 4 + 3 + 3 + 3 + 3 = 16;

They must have made the miscalculation that since each of your partners had sex with you, that you needed to be removed from the calculation somehow for each of them (even though you weren't exposed to yourself). This reveals the actual trend to be (n + n*(n-1)) = "sexual exposure" which is also n^2.

My guess is that to end up with the trend 2ⁿ -1, they made two mistakes. First they assumed that you have to be removed from the equation for each additional sexual partner, i.e. subtracting (n-1) from each enumeration of exposure. For the first three values, this gives 1, 3, and 7 which matches their pamphlet. The second mistake they made must have been stopping there and assuming that they recognized the pattern as 2^n-1 = "sexual exposure". However, if they had continued with this assumption they would have arrived instead at the trend (n + n(n-1) - (n-1)) = 1 + n(n-1). With this trend the next values would be:

1 + 3 + 3 + 3 + 3 = 1 + 4(3) = 13;

1 + 4 + 4 + 4 + 4 + 4 = 1 + 5(4) = 21;

etc.

So not only are their assumptions ridiculous...

1) It contradicts itself because no one ever has a first time unless they plan to be monogamous forever because the lack of previous sexual partners for anyone's first time is not accounted for at any level.

For example: Suppose you intend to have sex with 3 people in your entire life time, according to the actual trend, your sexual exposure would be 9. However, in order for that to be the case, for each of them, having sex with you would have had to have made their sexual exposure a 9 as well since you were their third partner. But for the first partner, you'd never had sex with anyone before, so his/her sexual exposure is only a 7 despite having had 3 sexual partners. Since 7 doesn't equal 9 the trend can't be accurate for everyone.

2) It contradicts itself by implicitly assuming that each person you have sex with never has sex again and you never have sex with anyone after your final partner, because otherwise there's no way for the trend to account for everyone you have sex with already having the same number of sexual partners as you inevitably end up with.

For example, if someone with 3 partners and an exposure of 9 has sex with another person, the assumption shows that this will raise their sexual exposure to 16, but it can't retroactively add exposure to the people you already had sex with such that they had each ACTUALLY had sex with three people before they slept with you. So each next person someone sleeps with has to have (2n-2) additional sexual partners than you, plus themselves in order to maintain the n² trend. However (2n-2) + 1 for each additional partner is greater than n for all values of n > 1. Since the the trend assumes each person has had sex with as many people as you ever will, this breaks the assumption and shows that trend contradicts itself again.

3) It makes the false assumption that one's partners never have sex with anyone (other than you) that any of your other partners have had sex with (since all past partners are assumed unique). In other words, no two people share more than one partner. This is probably the most reasonable of their simplifying assumptions, but is provably false by utilizing any of the numerous anecdotal stories of "He's with her now!" celebrity culture, porn shoots, swinger testimonials and the like but given the current setting, let's make the assumption that /u/MyLifeSuxNow's tale of the Jenny/Carly/Zack/X permutation makes for a sufficient debunking of this point.

...But their calculations were also the wrong interpretation of their own assumptions in two separate ways in the first place, simply because they made a mistake AND Stopped counting at three before confidently determining the trend was something else.

They'd have had a much closer and more reasonable approximation with the summation of n. That would have at least accounted for the likelihood that older people have had more sex than younger people and that each sexual partner you have looking forward on a timeline would have at least had more chronological opportunity to be keeping up with your current pace and allows everyone to have a first time (though it does assume that each person's first time is with someone else who's having their first time - provably false by anecdote, but not contradictory).

At which point the proper numbers on the graph should be:

1 - 1 2 - 3 3 - 6 4 - 10 5 - 15 6 - 21 7 - 28 8 - 36 9 - 45 10 - 55 11 - 66 12 - 78 13 - 91 14 - 105 15 - 120 16 - 136

TLDR: They are super wrong for lots of reasons in lots of ways, but the reason they came up with 2ⁿ were false assumptions, hasty interpretation, sloppy math, and clearly a categorical resistance to checking their work which extended all the way from whoever wrote the pamphlet through all the people involved in printing it and to the people handing it out.