Math on Parth Kohli

More Generating Functions

Mon, 28 Feb 2022 00:00:00 +0000

Today we take a look at the problem ABC241 H.

We are given $N \le 16$ distinct numbers $A_1, \cdots, A_N$ and a sequence describing the amount of supply of each number: a sequence $B_1, \cdots, B_N$ where $B_i$ refers to the supply of the number $A_i$. The score of a combination of $M$ chosen numbers is the product of the numbers. That is, suppose we pick $C_i$ occurences of the $A_i$ where $0 \le C_i \le B_i$ and $\sum C_i = M$, then $\text{score}(C) = \prod_{i = 1}^{N} A_i^{C_i}$. The objective is to find the sum of scores over all possible valid combinations.

Random OR

Sat, 05 Feb 2022 00:00:00 +0000

Today I’ll discuss a probability problem from CodeChef.

The problem is as follows: we have an $N$-bit number $S$ which is initially $0$. At each step, we pick an $N$-bit number $X$ uniformly randomly and set $S := S \mid X$ (the symbol $\mid$ denotes bitwise-or). The process terminates when $S$ becomes $2^N - 1$ (that is, it is impossible to increase it any further because every bit is 1). What is the expected number of steps until termination?

Generating Functions

Mon, 03 Aug 2020 00:00:00 +0000

Let’s look at this math problem from the recent AIsing Programming Contest 2020 on AtCoder.

The problem is to compute the sum of \[\tag{1}(a_2 - a_1)(b_2 - b_1)(c_2 - c_1)(d_2 - d_1)(e_2 - e_1)\] for all tuples such that \[\tag{2} a_1 + a_2 + b_1 + b_2 + \cdots + e_1 + e_2 \le N\] where $$\tag{3} 0 \le a_1 < a_2, ~ 0 \le b_1 < b_2, \cdots,~ 0 \le e_1 < e_2$$The constraint $N \le 10^9$ makes it worse.