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Triangle read by rows: T(n,k) is the number of edge subsets E of the n-dimensional hypercube graph such that E contains a path between two given nodes at Hamming distance k, 0 <= k <= n.
2

%I #10 Nov 02 2024 11:51:19

%S 1,2,1,16,9,7,4096,2703,2334,2160,4294967296,3425712321,3245350248,

%T 3170502909,3127853061

%N Triangle read by rows: T(n,k) is the number of edge subsets E of the n-dimensional hypercube graph such that E contains a path between two given nodes at Hamming distance k, 0 <= k <= n.

%C 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.

%C 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.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bunkbed_conjecture">Bunkbed conjecture</a>.

%e Triangle begins:

%e 1;

%e 2, 1;

%e 16, 9, 7;

%e 4096, 2703, 2334, 2160;

%e 4294967296, 3425712321, 3245350248, 3170502909, 3127853061;

%e ...

%Y Cf. A061301 (first column), A372705, A373035 (main diagonal).

%K nonn,tabl,more

%O 0,2

%A _Pontus von Brömssen_, May 20 2024