Should I put number combinations like 1111111 onto my lottery ticket?

You should never bet on that kind of sequence. Now, every poster will agree that the odds of any sequence from 000000000 through 999999999 has an equal probability. And if the prize is the same for all winners, it's fine. But, for shared prizes, you will find that you just beat 10 million to 1 odds only to split the pot with dozens of people. To be clear, the odds are the same, no argument. But people's bets will not be 100% random. They will bet your number as well as a pattern of 2's or other single digits. They will bet 1234567. I can't comment whether pi's digits are a common pattern, but the bottom line is to avoid obvious patterns for shared prizes.

When numbers run 1-50 or so, the chance of shared prizes increases when all numbers are below 31, as many people bet dates and stick to 1-31. Not every bettor does this of course, but enough so shared prizes show a skew due to this effect.

Again - odds are the same, but human nature skews the chance of split payout. I hope this answer is clear.

Your intuition is wrong. Compare the two statements

A. The event "the lucky number has all its digits repeated" is much less probable than the event "the lucky number has a few repeated digits"

B. The number 1111111 (which has all its repeated digits) is much less probable than the number 8174249 (which has a few repeated digits).

A is true, B is false.

BTW, this can be related to the "entropy" concept, and the distinction of microstates-vs-macrostates.

Your feeling is incorrect, but there is more to it.

It is in the interest of the lottery organizer for the lottery to be fair (because they have much more to lose in a scandal than they can gain by cheating). Thus it is fairly safe to assume that the lottery combinations are indeed drawn with a uniform distribution, which is to say that all combinations are equally likely. So you are wrong to think that 1111111 is less likely to be drawn than 8174249. Both are equally likely.

Many people are like you, they think some combinations are special, and that these are either more likely or less likely to appear. Your example is 1111111, you find it less likely. Some people find last week's combination to be less likely. Some people think more likely the combination made from those numbers that have occurred most in the past.

My non-scientific explanation of this goes as follows: people's brains automatically look for patterns everywhere. When a pattern is recognized in a thing, the thing gets categorized as "special" and "worthy of attention". This happens with lottery combinations too: any combination that has an obvious pattern, or follows some rule that is easy to describe, will be categorized by our brains as "special". Such special combinations will then be deemed less likely to appear.

In other words, humans are quite bad at dealing with randomness because they cannot help themselves from seeing patterns where there are none.

So, should you play 1111111, or should you avoid 1111111? To answer the question we have to take into account the fact that the prize is shared among all who guessed it. Now, since people are unable to generate random combinations well they tend to play combinations with recognizable patterns: visually or arithmetically pleasing combinations, birth dates, telephone numbers, etc. This seriously skews the combinations that are actually played. For instance, numbers above 31 are less likely to appear, while numbers below 13 are more likely to appear, because people play birth dates.

The upshot of this is that if you play a combination that your brain recognizes as special, and you happen to win, then you will have to share the prize with lots of other people whose brains thought of the same combination. In this sense, even though 1111111 is equally likely as all other combinations, the expected profit is smaller because we know that many other people will play the same combination.

The best strategy to play the lottery is to not play it, because the game is rigged so that your expected profit is negative. However, you may not care about this. For instance, you find pleasure in dreaming about what you would do with the prize, and so you are willing to pay something for it. (This is a perfectly legitimate reason for playing the lottery, I pity those who play because they actually think they can come ahead.)

Anyhow, if you do play the lottery, you should not play obvious combinations, or anything that can be described in one sentence, such as "the birthdays of my pets, increased by 5" (yes, there is going to be someone who has pets born on the same days as you, and who also thinks 5 is his lucky number). By the same reasoning, you should not avoid special combinations because that can be described by "Do not play a combination that has a nice pattern" (many people will use this strategy). The safest procedure is to choose a random combination, and use it no matter what your brain is telling you about its likelyhood. So even if you throw dice and get 1111111, you should use it.

Many lottery organizers will give you the option of choosing a random combination for you. It is in their interest to convince people that they should play randomly chosen combinations, because of the possible fiasco when a pretty combination gets drawn and there are several dozen winners. You should use the organizers random number generator if you believe their programmers are competent enough to get them right. History shows that this is often not the case. For instance, there have been a number of security problems on the web because various components (servers, browsers) used bad random number generators. Just throw dice.