cp's OEIS Frontend

a(n)/A061301(n) is the probability that the n-dimensional hypercube graph is still connected after each edge has been independently deleted with probability 1/2.

a(n)/A061301(n) is the probability that two given antipodal nodes of the n-dimensional hypercube graph are still connected after each edge has been independently deleted with probability 1/2.

T(n,k)/A061301(n) is the probability that two given nodes at Hamming distance k in the n-dimensional hypercube graph are still connected after each edge has been independently deleted with probability 1/2.

The bunkbed conjecture (the version where all edges, including the posts, have the same probability 1/2 of being retained) holds for the n-dimensional hypercube graph if and only if the (n+1)-st row is nonincreasing.

Proof of formula by Robert Israel: If f is monotone, then for each x in X the set G(x) = {A in 2^X: x in f(A)} is an upset, i.e. if A is in G(x) and A \subset B then B is in G(x). Conversely, if for each x in X the set G(x) is an upset, then f is monotone. And the family {G(x): x in X} determines f, since f(A) = {x: A is in G(x)}. So the cardinality of the set of monotone set-functions is N(|X|)^|X| where N(|X|) is the cardinality of the set of upsets G of 2^X, or equivalently monotone Boolean functions. That is sequence A000372.

This sequence was motivated by a question by Federico Echenique on sci.math.research.

a(n) is also the permanent of the Sylvester Hadamard matrix.

Number of choice functions f:2^A\{{}}->2^A\{{}} where A is an n-element set of variants such that f(X) is a nonempty subset of any nonempty X in 2^A (here 2^A denotes the power set of A).