2

Notations/Definitions: A composition of a natural number $n \in \mathbb{N}$ (I use $0 \notin \mathbb{N}$) is a sequence of natural numbers $n_1,n_2, ..., n_d \in \mathbb{N}$ such that $n_1+...+n_d = n$. (repetions are allowed, order matters) Denote the number of compositions of a natural number $n \in \mathbb{N}$ consisting only of 1,2 by $k_n$. Denote the number of compositions of $n$, that only consist of terms from $\{2,3,...\}$ by $\tilde{k}_n$. Let $K(z) := \sum_{i = 0}^{\infty} k_n \cdot z^n$ and $\tilde{K}(z) := \sum_{i = 0}^{\infty} \tilde{k}_n \cdot z^n$.

What I'm trying to prove: The number of compositions of $n \in \mathbb{N}$ consisting of parts $\leq 2$ equals the number of compositions of (n+2) consisting of parts $\geq 2$. Or in my notation: $k_n = \tilde{k}_{n+2}$.

Example: 4 has five compositions consisting of parts $\leq 2$ (1+1+1+1, 1+1+2, 1+2+1, 2+1+1, 2+2) and 6 has five compositions of parts $\geq 2$ (2+2+2, 3+3, 6, 4+2, 2+4).

Proof: I'm trying to prove the statement above by using generating functions. As the compositions of $n$ only consist of numbers from $\{1,2\}$ the generating function of $k_n$ is given by $$ K(z) = \frac{1}{1-GF(\{1,2\})} = \frac{1}{1-(z+z^2)}. $$ As the compositions of $n$ only consist of numbers from $\{2,3,...\}$ the generating function of $\tilde{k}_n$ is given by $$ \tilde{K}(z) = \frac{1}{1-GF(\{2,3,...\})} = \frac{1}{1-(z^2+z^3+...)} = \frac{1}{\frac{-z^2-z+1}{1-z}}. $$ My idea to achieve my desired result ($\tilde{k}_{n+2} = k_n$) was to show $$\sum_{i = 0}^{\infty} \tilde{k}_{n+2} \cdot z^n = \frac{\tilde{K}(z) - \tilde{k}_0 - \tilde{k}_1 \cdot z}{z^2} \overset{!}{=} K(z)$$ but the last equality doesn't seem to hold. (I think $\tilde{k}_0 = \tilde{k}_1 = 0$, is this correct?)

One more thing I realized is that $K(z) = \frac{1}{1-z} \cdot \tilde{K}(z)$ yields $$ k_n = \sum_{i = 0}^n \tilde{k}_i,$$ which I think shouldn't be correct.

Because of the last two points I think at least one of the generating functions is not correct, but I cannot find a mistake. Thank you in advance for any help!

lioness99a
  • 5,019
liouville
  • 63
  • 2
    Someone asked this same problem here. Your mistake is with $\tilde k_0$; you should have $\tilde k_0=1$, as there is the empty composition. – Mike Earnest Mar 22 '19 at 16:57
  • Thank you for your help. I haven't seen the other post, should I delete mine? If I use $\tilde{k}_0 = 1$ and $\tilde{k}_1 = 0$ i get the result I wanted to show. The last thing I don't fully understand is: Why does the empty composition matter for $\tilde{k}_0$ but not for $\tilde{k}_1$? – liouville Mar 29 '19 at 13:40
  • The empty composition has zero parts, so it contributes to $\tilde k_0$. There are zero compositions with $1$ part summing to $1$; any composition with one part must sum to at least $2$. Therefore, $\tilde k_1=0$. – Mike Earnest Mar 29 '19 at 15:12

0 Answers0