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So I'm trying to come up with a recursive formula $f(n)$ which counts the number of all partitions of $[n]$ into $k$ identical blocks, where the number of elements in each box is more than 1.

What i've tried to do is look at the $n$-th element. If I assume that the $n$-th element is in a box of size $n-2i$, then there's $2i$ element's remaining. Should it follow that there are ${n \choose 2i}$ ways to pick these, and then $f(i)$ valid ways to partition them? Giving $$f(n) = \sum_{i=0}^{n-1} {n\choose 2i}f(i)$$

Does this seem right? If no, any hints you can give me?

MITjanitor
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1 Answers1

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This is a partial answer. First, $f(n)=0$ for $n<2k$ and $$f(2k)=\frac1{k!}\binom{2k}2\binom{2k-2}2\cdots\binom 42\binom 22=\frac{(2k)!}{2^kk!}$$ (you choose $2$ of the $2k$ numbers, then you choose another $2$ of the remaining $2k-2$ numbers, and so on. Finally, you divide by $k!$ since your boxes are identical).

For $n>2k$, suppose that you already put the numbers $1,\dots,n-1$ in the boxes. Then you only can have at most $1$ box with a single element (briefly, a single box), because otherwise the last choice for $n$ would not be enough to avoid single boxes. If none of the box is single, then you can put your number $n$ in any place. If there is a box with a single number, say $j$, then actually you already filled the other $k-1$ boxes with $n-2$ numbers, none of the boxes being single.

The first option contributes $k\cdot f(n-1)$; unfortunately, for the second option I must consider your function $f$, but with $k-1$ instead of $k$. In other words, denoting your function by $f_k$, then the second option contributes $(n-1)\cdot f_{k-1}(n-2)$ (the factor $n-1$ corresponds to the choices for $j$). Consequently the recurrence becomes

$$f_k(n)=kf_k(n-1)+(n-1)f_{k-1}(n-2)\,.$$

I wish to see a solution involving just the function $f_k$,

Matemáticos Chibchas
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