# How to use Roberto Frucht Formula to find permutations with limited repitition

I asked a question at this Number of arrangements of red, blue, and green balls in which a maximum of three balls of the same color are placed in the five slots. Now the scenario is the same

I have a bag of three balls (look at the link)

• red
• blue
• green

and I have 5 slots.. Every time I pull a ball from the bag I placed it in a slot. I keep randomly pulling balls from the bag and fill five slots..when all five slots are filled, repeat the process

I still want to number of arrangements of red, blue, and green balls in which a maximum of three balls of the same color are placed in the five slots

I found an interesting article by Roberto Frucht (1966), Permutations with limited repetitions that derives a formula to do this.

I want to find the results using this breakthrough formula:

for "the number of $r$-permutations (called variations, r at a time in the older literature) with limited repetition, where each one of the $n$ different things to be permuted may appear at most $s$ times."

Note: In the professor's research he gives an example that shows how many permutations can be computed if 4 items are chosen from 7 items where only a maximum of two items and repeat itself in the arrangement. Note the formula shows that Bell polynomials and Stirling Strings of the second kind play a significant role in deriving the formula.

I am requesting some practical help as to how to use this formula.Specifically, plugging values in and getting the expected results.

• I find this pretty confusing. You "have a bag of three balls", each of distinct color. You don't say whether the balls pulled "randomly" from the bag are used to fill the five slots in a defined order, nor whether the balls are then replaced(?) in the bag, but you somehow "fill five slots," and "when all five slots are filled, repeat the process". To what effect? How are the five slots filled with three balls, and what state remains after each repetition of "the process"? – hardmath Aug 23 '16 at 3:05
• see the link in for the previous question..it has all the details..but for clarification..when someone pull a ball they place it in a slot record the information "red ball slot 1". that person replace the red ball in the bag, pulls another ball and place it in a second slot..they record "blue ball second slot 2" if it is a blue ball....the process goes on untill all 5 slots are filled...now the process starts all over again...now this is random.. – repzero Aug 23 '16 at 3:33
• Now the question is..how many permutation can I have..let say I have 7 balls and I have 4 slots. How many permutations can I have with the condition that a maximum two of the same color occuring/repeated..this is what professor Frucht reasearch was about...since my permutation generator outputs the same number of permutations as shown in a complex example in his research – repzero Aug 23 '16 at 3:35
• I have made a few small changes to your post. Please review to check I did not unintentionally change your meaning. The Question should give a more self-contained statement of a problem, so that a Reader is not required to use those links to understand what you are asking (but it is fine to give them as background references). – hardmath Aug 23 '16 at 5:25
• @hardmath..perfect! – repzero Aug 23 '16 at 10:33