Elegia's blog

By Elegia, history, 7 months ago, In English

We will get the bivariate generating function $$$\widehat F(z, t) = \sum_{n\ge 1}\sum_{k=1}^n A_{n,k}\frac{z^nt^k}{n!}$$$, where $$$A_{n,k}$$$ is the sum of occurrences of $$$k$$$ in all good sequence of length $$$n$$$. Consider the inclusion-exclution principle. For all contribution with $$$\max k = m$$$, we have

$$$ \sum_{j=1}^m \sum_{S \subseteq \{1, 2, \cdots m-1\}} (-1)^{|S|} Q(S, z, t, j) $$$

This means that for all $$$i \in S$$$, we force all recurrences of $$$i$$$ are before $$$i + 1$$$, and the occurrences of $$$j$$$ are counted.

After some tough calculation, we will found out that this equates to $$$\frac{t(\mathrm e^{z(1-t)}-1)}{(1-z) (1-t \mathrm e^{z(1-t)})}$$$. Here are the details:

Details

Now our goal is to calculate

$$$ [z^n]\frac{t(\mathrm e^{z(1-t)}-1)}{(1-z) (1-t \mathrm e^{z(1-t)})} $$$

We consider $$$\left([z^n]\frac{t(\mathrm e^{z(1-t)}-1)}{(1-z) (1-t \mathrm e^{z(1-t)})}\right) + 1 = (1-t)[z^n] \frac 1{(1-z)(1-t\mathrm e^{z(1-t)})}$$$, which looks simpler. We let $$$z = \frac u{1-t}$$$, then

$$$ [z^n]\frac 1{(1-z)(1-t\mathrm e^{z(1-t)})} = (1-t)^n[u^n] \frac1{(1-\frac u{1-t})(1-t\mathrm{e}^u)} $$$

Hence we have

$$$ \begin{aligned} (1-t)[z^n] \frac 1{(1-z)(1-t\mathrm e^{z(1-t)})} &= [u^n]\frac{(1-t)^{n+2}}{(1-\frac{t}{1-u})(1-t\mathrm e^u)(1-u)}\\&= (1-t)^{n+2} [u^n] \left(\frac{-\mathrm e^u}{\left(\mathrm e^u u-\mathrm e^u+1\right) \left(1-t \mathrm e^u\right)}+\frac{\frac{1}{1-u}}{\left(\mathrm e^u u-\mathrm e^u+1\right) (1-\frac{t}{1-u})}\right)\end{aligned} $$$

Noticed that $$$[u^m]\mathrm{e}^{ku} = \frac{k^m}{m!}$$$, this could be calculated straightly through multipoint evaluation with time complexity of $$$\Theta(n\log^2 n)$$$. And $$$[u^m] \frac1{(1-u)^k} = \binom{n+k-1}{n} = \frac{(n+k-1)!}{n!(k-1)!}$$$ so this part could be calculated through a convolution. It will pass if your implementation doesn't have a big constant.

It could also be improved to $$$\Theta(n\log n)$$$ through the Lagrange Inversion formula similar to the original solution, I leave this as an exercise.

UPD1: simplified some deduction.

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