CF1988F 较草题解

$$\begin{array}{rl}& f_{i,j,k},g_{i,j,k}\to (i\text{ permutation},j\text{ premax or sufmax}，k(a[l]>a[l-1]))\\ & \text{Initialize : }{f}_{1,1,0}=g_{1,1,0}=1\\ & \text{Transfer for f,g}\\ & f_{i,j,k}=f_{i-1,j-1,k-1}+f_{i-1,j,k}\times (k+1)+f_{i-1,j,k-1}\times (i-k-1)\\ & g_{i,j,k}=g_{i-1,j-1,k}+g_{i-1,j,k}\times k+g_{i-1,j,k-1}\times (i-k)\\ & \text{Now you know f, g, a, b and c}\\ & n=\mathtt{\text{forall interger}}\in [1,N],C_{N-1}^{n-1}\times \sum _{i=1,}^n\sum _{j=0\text{ for premax,}}^{i-1}\sum _{k=0\text{ for sufmax,}}^{n-i}\sum _{l=0\text{ for }{f}_k,}^{i-2}\sum _{m=0\text{ for }{g}_k}^{n-i-1}{f}_{i-1,j,l}\times g_{n-i,k,m}\times a_{j+1}\times b_{k+1}\times c_{l+m+[i\ne 1]}\to {\text{ans}}_n\\ & \text{Try to find ans of }\mathcal{O}(N^3)\\ & \text{Define }{F}_{i,k}=\sum _{j=0}^n{f}_{i,j,k}\times a_{j+1},G_{i,k}=\sum _{j=0}^n{g}_{i,j,k}\times b_{j+1}\\ & {\text{ans}}_n=C_{N-1}^{n-1}\times \sum _{i=1,}^n\sum _{l=0\text{ for }{f}_k,}^{i-2}\sum _{m=0\text{ for }{g}_k}^{n-i-1}{F}_{i-1,l}\times G_{n-i,m}\times c_{l+m+[i\ne 1]}\\ & \text{Define }{H}_{i,m}=\sum _{l=0}^{i-1}{F}_{i,l}\times c_{m+l+[i\ne 0]}\\ & {\text{ans}}_n=C_{N-1}^{n-1}\times \sum _{i=1}^n\sum _{m=0\text{ for }{g}_k}^{n-i-1}{G}_{n-i,m}\times H_{i-1,m}\end{array}$$