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Questions related to combinatorics and discrete mathematical structures

37 votes

How can one byte hold 256 possibilities?

I just did a quick animation to convince you. Yellow is one, blue is zero.
A.Schulz's user avatar
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7 votes
Accepted

Simplify complexity of n multichoose k

Edit: This answer is for $k<n$. Without bounding $k$ in terms of $n$ the expression is unbounded. If $k=n-1$ then your expression becomes $O\left ({{2(n-1)}\choose{n-1}}\right)$. Notice that by Stir …
A.Schulz's user avatar
  • 12.3k
2 votes

Number of possible min heaps

The second level can contain other numbers than 2 and 3. See for example 1 / \ 2 5 / \ 3 7 / \ 4 6 It is also unclear if you count isomorphic trees. See this …
A.Schulz's user avatar
  • 12.3k
27 votes
Accepted

How many different max-heaps exist for a list of n integers?

I did not find a closed form, but according to this entry in the Online Encyclopedia of Integer Sequences the sequence starts with 1, 1, 1, 2, 3, 8, 20, 80, 210, 896, 3360, 19200, 79200, 506880, 274 …
A.Schulz's user avatar
  • 12.3k
3 votes

What is the probability of friendship conditioned on the number of mutual friends

Okay, this is my approach to this problem. I think the key concept here is the clustering coefficient $c$. The value of $c$ is denotes how likely it is that two nodes are connected by an edge, if the …
A.Schulz's user avatar
  • 12.3k