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A126236
Maximum length of a codeword in Huffman encoding of n symbols, where the k-th symbol has frequency k.
3
1, 2, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11
OFFSET
2,2
LINKS
Cristina Ballantine, George Beck, Mircea Merca, and Bruce Sagan, Elementary symmetric partitions, arXiv:2409.11268 [math.CO], 2024. See p. 20.
M. J. Fisher et al., The birank number of a graph, Congressus Numerant., 204 (2010), 173-180.
Wikipedia, Huffman coding
FORMULA
Conjecture: a(n) = floor(log_2(n)) + floor(log_2(2n/3)). Equivalently, a(n) = a(n-1) + 1 if n has the form 2^k or 3*2^k, a(n) = a(n-1) otherwise. This is true at least for n up to 1000.
EXAMPLE
A Huffman code for n=8 is (1->00000, 2->00001, 3->0001, 4->001, 5->010, 6->011, 7->10, 8->11). The longest codewords have length a(8)=5.
CROSSREFS
Cf. A126014 and A126237. The minimum length of a codeword is in A126235.
Sequence in context: A061716 A362692 A341053 * A198194 A375815 A073047
KEYWORD
nonn,changed
AUTHOR
Dean Hickerson, Dec 21 2006
STATUS
approved