How many partitions of 100 into 10 parts are there with all parts having DIFFERENT sizes?
I tried brute force, but ill take to long, and i can't find a Simple way to get it.
I was thinking in summation with (i) ... But i can't reach the answer.
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If the parts are $p_1<p_2<\dots<p_10$, then the differences $d_1=p_1$, $d_k = p_k-p_{k-1}$ are positive integers satisfying $10d_1+9d_2+8d_3+\cdots+d_{10}=100$; equivalently, if $c_k=d_k-1$, then the $c_k$ are nonnegative integers satisfying $10c_1+9c_2+8c_3+\cdots+c_{10}=45$. Counting such solutions should be pretty easy by looping over the variables. – Greg Martin Apr 29 '21 at 11:27
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So what i want os p(100,10)? – puctw Apr 29 '21 at 11:37
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How i do a summation of i=1 to k? Of p(10,i)??? Is that correct? – puctw Apr 29 '21 at 11:39
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Can we use inclusion-exclusion principle ? – Apr 29 '21 at 11:49
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You can try with that – puctw Apr 29 '21 at 11:50
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Like I said yesterday, $p$ as a function in this context is the name for the function where we don't care about the part sizes being distinct. You want to use the letter Q, not the letter P here... you want $q(100,10)$ in order to emphasize the part sizes being distinct. – JMoravitz Apr 29 '21 at 14:55
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Like I said yesterday, this is a duplicate. Like I said yesterday, you can find more information on mathworld.wolfram, scrolling down the page to where it talks about $q(n,k)$. See also this question – JMoravitz Apr 29 '21 at 14:57
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Does this answer your question? Partition an integer $n$ into exactly $k$ distinct parts – JMoravitz Apr 29 '21 at 14:57
1 Answers
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Sequence https://oeis.org/A008289 contains a table of the number of ways to split an integer n into k parts of different size. It gives a recursive formula which produces 33401 in row 100 and column 10 of the table:
1
1 1
1 1 0
1 2 0 0
1 2 1 0 0
1 3 1 0 0 0
1 3 2 0 0 0 0
1 4 3 0 0 0 0 0
1 4 4 1 0 0 0 0 0
1 5 5 1 0 0 0 0 0 0
1 5 7 2 0 0 0 0 0 0 0
1 6 8 3 0 0 0 0 0 0 0 0
1 6 10 5 0 0 0 0 0 0 0 0 0
1 7 12 6 1 0 0 0 0 0 0 0 0 0
1 7 14 9 1 0 0 0 0 0 0 0 0 0 0
1 8 16 11 2 0 0 0 0 0 0 0 0 0 0 0
1 8 19 15 3 0 0 0 0 0 0 0 0 0 0 0 0
1 9 21 18 5 0 0 0 0 0 0 0 0 0 0 0 0 0
1 9 24 23 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0
R. J. Mathar
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