איש בחטאו ימות

Here we have two identical sayings about the direct influence of a father's actions. However, it is obvious that the Tanach does not like this idea. We see this in many shades:

Amatziah's father, Yoash, was brutally mudered in his prime. However, his son killed only the killers, and not their sons, in accordance with the Torah's view (as expressed in Devarim, Yechezkel, and Yirmiyahu.) This is in fact the climax and relaxing of the theme in the perakim leading up:

In one of the most confusing stories of Melachim Bet, the king of Moav, during battle, tries to break to the king of Edom, the ally of Israel, and fails. But then the king of Moav sacrifices either his son, or the king of Edom's son, and a wrath comes on Israel. I would suggest that the wrath is the king of Edom, and the son is his. The Israelites turn back because they fail to protect this right of all chidren - which is not to die for the father, or directly because of the father. Next...

Elisha has attached himself to the woman from Shunem, and then suddenly, her child is pained in the head while working in the field, and then dies in his mother's hand. Elisha cures the child, because of his ties with the Shunemite. Although this is definitely a positive action and Hashem surely willed it, the action itself runs foul of the general philosophy. We continue...

The famine is rampant, as is, evidently, cannabilism. Most famines are brought because of the iniquities of the people. So here are the sinful people, killing their kids as a result of their sins. This is also a flagrant violation of our moral rule.

Yehu ben Yehoshafat ben Nimshi sent letters to Achav's 70 descendants assuring them they could keep their father's house. It was a letter of peace, despite what Yehu thought of Achav. However, he wrote also to the people in the Shomron, where these 70 lived, and they slaughtered them. Again, we have an obvious perversion of the rule.

So we see that in Melachim Bet, we have these situations all the time. Amatziah, a good king, finally did something about it. Yechezkel and Yirmiyahu also sought to address this.

Here we have the end of the sad story of Korach. Every man and everything belonging to them or belonging to Korach gets swallowed up.

So then how do many songs of the Benei Korach survive if everything went overboard?

And here we have the answer. They simply did not die. And this is because they acted righteously, and so, here is another case of the Torah breaking the mold of אבות לא יומתו על בנים. But let us go the other way....

Now what is happening here? Didn't we just say the sins of the father count for nothing on the sons?

Well, we have the operative word: לשונאי, which tells us that if you act like your sinful father, you will carry his same iniquity and will accordingly be punished.