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How many ways are there to partition a block of 9 into blocks of sizes 1, 2, and 3?

So lets say I have [XXXXXXXXX]. How many ways can I 'fill' it with [X], [XX], and [XXX]

Some ways I could do it is: [X][X][X][X][X][X][X][X][X] (9 1's) [XX][XXX][X][XXX] (2-3-1-3)

Order matters, so one 3 followed by three 2's would be different from 3 2's then 1 3.

Kevin
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  • FYI : A complementary question to this one was asked five years later, this time solved via Generating Functions. Link : https://math.stackexchange.com/questions/3167963/ways-of-constructing-10-unit-high-tower-w-infinite-blocks-1-2-3-units-hig/3168861#3168861 – Martin Hansen Apr 01 '19 at 09:13

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Let $A(n)$ be the number of ways to partition $n$ this way. We have by hand count $A(1)=1,A(2)=2,A(3)=4$ and the recurrence $A(n)=A(n-1)+A(n-2)+A(n-3)$

Ross Millikan
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    To make sense of the recurrence: A block of size $n$ is a block of size $n-3$ with a 3-block attached to the right, or a block of size $n-2$ with a 2-block attached to the right, or a block of size $n-1$ with a $1$-block attached to it.

    The same logic can be used when deducing that the number of ways to tile a $2 \times n$ figure with $2 \times 1$ dominoes yields the fibonacci sequence.

     – MT_ Jan 25 '14 at 22:00