Solving $$f(x, k) = f(x, k-1) + f(x-1, k-1) + \dots + f(1, k-1)$$ in terms of $x$

Question

I'm having trouble determining the complexity of an algorithm. Let's say the number of operations of my algorithm is described by $$f(x, k) = f(x, k-1) + f(x-1, k-1) + f(x-2, k-1) + \dots + f(1, k-1)$$ where $f(x, 1) = x$

How can I describe the growth of $f(x)$ in terms of

1. $x$
2. $k$
3. $x$ when $k=x-1$

For 1), I believe $f(x, k)$ would be exponential in $x$ as $k$ can be taken to be a constant which would reduce the above equation to being similar to

$$f(x) = f(x-1) + f(x-2) + \dots + f(1) $$

Similarly, for 2), I believe $f(x, k)$ would be exponential in $k$ as I would need to take $x$ as a constant which would lead to the below equation

$$f(k) = xf(k-1) $$

Need help with 3) and also validating my thinking for 1) and 2)

Answer 1

The answers to all three questions become easier once we proved the following formula for all positive integer $k$ and $x$. $$ f(x,k) = \frac{x(x+1)(x+2)\cdots(x+k-1)}{k!}$$ Proof: The formula is easily seen to be true for $k=1$ and all positive integer $x$.

Assume the formula is true for some positive integer $k$ and all positive integer $x$. Then, $$\begin{align} &f(x, k+1)\\ &=f(x, k) + f(x-1, k) + f(x-2, k) + \dots + f(1, k)\\ &=\frac1{k!}(x(x+1)(x+2)\cdots(x+k-1) + (x-1)x(x+1)(x+2)\cdots(x+k-2)\\ &\quad\quad+\cdots+1\cdot2\cdots k\\ &=\frac1{k!}\left(\frac{x(x+1)(x+2)\cdots(x+k)-(x-1)x(x+1)\cdots(x+k-1)}{k+1}\right.\\ &\quad\quad+\frac{(x-1)x(x+1)\cdots(x+k-1)-(x-2)(x-1)x\cdots(x+k-2)}{k+1}\\ &\quad\quad+\cdots\\ &\quad\quad+\frac{2\cdot3\cdots(k+1)-1\cdot2\cdots k}{k+1}\\ &\quad\quad+\left.\frac{1\cdot2\cdots k}{k+1}\right)\\ &=\frac1{k!}\frac{x(x+1)(x+2)\cdots(x+k)}{k+1}\\ &=\frac{x(x+1)(x+2)\cdots(x+(k+1)-1)}{(k+1)!}\end{align}$$

By mathematical induction on $k$, the formula is true for all positive integer $k$ and $x$.

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Here are the answers to your three cases.

1) For fixed $k$ and $x\to\infty$, $$f(x,k)=\frac{x^k}{k!}\frac xx\frac{x+1}x\cdots\frac{x+k-1}{x}\sim\frac{x^k}{k!}$$ That is, $f(x,k)=\Omega(x^k)$.

2) For fixed $x$ and $k\to\infty$, by Stirling's approximation, $$\begin{align}f(x,k)&=\frac{(x-1+k)!}{(x-1)!k!}\\ &\sim\frac1{(x-1)!}\frac{\sqrt{2\pi(x-1+k)}\left(\frac{x-1+k}{e}\right)^{x-1+k}}{\sqrt{2\pi k}\left(\frac{k}{e}\right)^{k}}\\ &=\frac1{(x-1)!}\sqrt\frac{x-1+k}k (x-1+k)^{x-1}\left(\frac{x-1+k}{k}\right)^k\frac{1}{e^{x-1}}\\ &=\frac1{(x-1)!}\sqrt\frac{x-1+k}k\left(\frac{x-1+k}{k}\right)^{x-1}k^{x-1}\left(1+\frac{1}{\frac k{x-1}}\right)^{\frac k{x-1}(x-1)}\frac{1}{e^{x-1}}\\ &\sim\frac1{(x-1)!}k^{x-1}e^{x-1}\frac{1}{e^{x-1}}\\ &=\frac{k^{x-1}}{(x-1)!}\end{align}$$ That is, $f(x,k)=\Omega(k^{x-1})$

3) Let $k=x-1$ go to infinity, again by Stirling's approximation, $$f(x,k) = \frac{(2k)!}{k!k!}\sim \frac{\sqrt{2\pi2k}\left(\frac{2k}{e}\right)^{2k}}{\sqrt{2\pi k}\left(\frac{k}{e}\right)^{k}\sqrt{2\pi k}\left(\frac{k}{e}\right)^{k}}=\frac{2^{2k}}{\sqrt{\pi k}}$$ That is, $f(k+1,k)=\Omega(k^{-1/2}2^{2k})$.

Answer 2

You can write your recursion as \begin{equation} \begin{split} f(x,k) &= \sum\limits_{i_1=1}^x f(i_1,k-1)\\ &= \sum\limits_{i_1=1}^x \sum\limits_{i_2=1}^{i_1} f(i_2,k-2)\\ &= \vdots \\ &= \sum\limits_{i_1=1}^x \sum\limits_{i_2=1}^{i_1} \ldots \sum\limits_{i_{k-1}=1}^{i_{k-2}} f(i_{k-1},1)\\ &= \sum\limits_{i_1=1}^x \sum\limits_{i_2=1}^{i_1} \ldots \sum\limits_{i_{k-1}=1}^{i_{k-2}} i_{k-1}\\ \end{split} \end{equation} We know that from Faulhaber's formula \begin{equation} \sum\limits_{i_{k-1}=1}^{i_{k-2}} i_{k-1} = \frac{1}{2}O(i_{k-2}^2) \end{equation} So \begin{equation} f(x,k) = \sum\limits_{i_1=1}^x \sum\limits_{i_2=1}^{i_1} \ldots \sum\limits_{i_{k-2}=1}^{i_{k-3}} \frac{1}{2}O(i_{k-2}^2) \end{equation} Also \begin{equation} \sum\limits_{i_{k-2}=1}^{i_{k-3}} O(i_{k-2}^2) = \frac{1}{3}O(i_{k-3}^3) \end{equation} If you keep doing this, you'd get \begin{equation} f(x,k) = \frac{1}{2 \times 3 \times 4 \times \ldots \times (k-1) } \sum\limits_{i_1=1}^x O(i_1^{k-1}) = \frac{1}{2 \times 3 \times 4 \times \ldots \times (k-1) \times k} O(x^k) = \frac{1}{k!} O(x^k) \end{equation}

Answer 3

For $k \geq 1$, define $$ T_k(z) = \sum_{n=0}^\infty f(n,k) z^n, $$ where we define $f(0,k) = 0$ (which agrees with the definition for $k=1$).

Since $f(n,1) = n$ for all $n$, we get $$ T_1(z) = \sum_{n=0}^\infty nz^n = z\sum_{n=0}^\infty nz^{n-1} = z \frac{d}{dz} \sum_{n=0}^\infty z^n = z \frac{d}{dz} \frac{1}{1-z} = \frac{z}{(1-z)^2}. $$ For $k > 1$, we have $$ T_k(z) = \sum_{n=0}^\infty f(n,k)z^n = \sum_{n=0}^\infty \sum_{m=0}^n f(m,k-1) z^n = \sum_{m=0}^\infty f(m,k-1) \sum_{n=m}^\infty z^n = \\ \sum_{m=0}^\infty f(m,k-1) z^m \sum_{n=0}^\infty z^n = \sum_{m=0}^\infty f(m,k-1) \frac{z^m}{1-z} = \frac{T_{k-1}(z)}{1-z}. $$ A short induction shows that $$ T_k(z) = \frac{z}{(1-z)^{k+1}}. $$ You can now consider the bivariate generating function $$ S(z,w) = \sum_{n=0}^\infty \sum_{k=1}^\infty f(n,k) z^n w^k = \sum_{k=1}^\infty w^k T_k(z) = z \sum_{k=1}^\infty w^k (1-z)^{-k-1} = \\ \frac{z}{1-z} \sum_{k=1}^\infty \left(\frac{w}{1-z}\right)^k = \frac{zw}{(1-z)^2}\frac{1}{1-w/(1-z)} = \frac{zw}{(1-z)(1-z-w)}. $$ From here you can in principle deduce all asymptotic properties. It is also easy to extract the exact formula, since the coefficients of $\frac{1}{(1-z)^k}$ are known.