### Elegia's blog

By Elegia, history, 2 years ago, 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. Comments (15)
 » Auto comment: topic has been updated by Elegia (previous revision, new revision, compare).
 » Is it competitive coding ?
•  » » » After seeing this blog, I went your profile to make sure my guess that you are Chinese!
 » UPD1: simplified some deduction. Thanks!
 » 2 years ago, # | ← Rev. 3 →   How do you calculate the last part with multipoint evaluation? ElegiaAlso, the "tough calculation" part is actually similar to the formula for Eulerian polynomials:$F(x,t) = \displaystyle\sum_{d \ge 0}A_{d}(x)\frac{t^{d}}{d!} = \frac{1-x}{1 - xe^{(1-x)t}},$where $A_{d}(x) = \displaystyle\sum_{k=1}^{d}A(d,k)x^{k}$ is an Eulerian polynomial. Here, $A(d,k)$ is the number of permutations of length $d$ with $k-1$ descents. We can derive a bijection showing that the answer for $k$ is $n! \cdot \sum_{i=1}^{n}\frac{E(i,k)}{i!}$, which is basically the prefix sum of coefficients of $F(x,t)$, and thus we can simplify "prefix sum" to "coefficient of single term" by multiplying $F(x,t)$ with $\frac{1}{1-t}$, which gives the function described in the blog.
•  » » For the $[u^m] f(u)/(1-t\mathrm{e}^u)$ part: \begin{aligned} [u^n] \frac{f(u)}{1-t \mathrm{e}^u} & = \sum_k [u^n] t^k f(u)\mathrm{e}^{ku} \\ &= \sum_k t^k \sum_{0\le j} \frac{k^j}{j!} [u^{n-j}] f(u)\\ &= \sum_k t^k \sum_{0\le j} k^j \frac{f_{n-j}}{j!}\\ &= \sum_k t^k P(k) \end{aligned}For the $[u^m] g(u)/(1-\frac t{1-u})$ part, it might cost $\Theta(n\log^2 n)$ to calculate the coefficient straightly bacause you need to expand the polynomial expressed by the linear combination of $\binom{n+x-1}{n}$ through divide and conquer.Speaking of Eulerian Polynomial: Yes, it's surely Eulerian Polynomial. But what I'm going to show is that there is a way to get the GF with no knowledge of Eulerian Polynomial nor bijection to the permutation. It will be updated a few times later. >_<
•  » » Can you help me to prove this bijection, please.
•  » » » I have done it in my tutorial on Generating Functions here.
 » My implementation: 80415302In your last expression, $(1-t)^{n+2}$ should be $(1-t)^{n+1}$. Or replace the terms after $[u^n]$ with the following: $\frac{-e^u}{(e^uu-e^u+1)(1-te^u)}+\frac{1/(1-u)}{(e^uu-e^u+1)(1-t/(1-u))}$
•  » » corrected, thanks.
•  » » there is a typo in line 347 in you code.
 » stO Elegia! I found that u've been keen on generating function these days XD.
•  » » 2 years already :)
 » 