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A242354
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Number T(n,k) of four-colored rooted trees of order n and structure k; triangle T(n,k), n>=1, 1<=k<=A000081(n), read by rows.
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1
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4, 16, 64, 40, 256, 160, 256, 80, 1024, 640, 1024, 320, 1024, 640, 544, 640, 140, 4096, 2560, 4096, 1280, 4096, 2560, 2176, 2560, 560, 4096, 2560, 4096, 1280, 4096, 2560, 2560, 1600, 2176, 1280, 224, 16384, 10240, 16384, 5120, 16384, 10240, 8704, 10240, 2240
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OFFSET
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1,1
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COMMENTS
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The underlying partitions of n-1 (cf. A000041) for the construction of the trees with n nodes are generated in descending order, the elements within a partition are sorted in ascending order, e.g.,
n = 1
{0} |-> () |-> 10_2
n = 2
{1} |-> (()) |-> 1100_2
n = 3
{2} > {1, 1} |-> ((())) > (()()) |-> 111000_2 > 110100_2
n = 4
{3} > {1, 2} > {1, 1, 1} |-> (((()))) > ((()())) > (()(())) > (()()()) |-> 11110000_2 > 11101000_2 > 11011000_2 > 11010100_2
The decimal equivalents of the binary encoded rooted trees in row n are the descending ordered elements of row n in A216648.
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REFERENCES
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G. Gruber, Entwicklung einer graphbasierten Methode zur Analyse von Hüpfsequenzen auf Butcherbäumen und deren Implementierung in Haskell, Diploma thesis, Marburg, 2011
Eva Kalinowski, Mott-Hubbard-Isolator in hoher Dimension, Dissertation, Marburg: Fachbereich Physik der Philipps-Universität, 2002.
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LINKS
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EXAMPLE
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Let {h, u, d, p} be a set of four colors, corresponding to the four possible "states" of each tree node (lattice site) in the underlying physical problem, namely its occupation with no electron (hole), with one up-spin electron, with one down-spin electron, or with one up-spin and one down-spin electron (pair). (We consider each rooted tree as a cutout of the Bethe lattice in infinite dimensions.) Then for
h(), u(), d(), p() are the 4 four-colored trees of the first and only structure k = 1 (sum is 4 = A136793(1)); for
h(h()), h(u()), h(d()), h(p()),
u(h()), u(u()), u(d()), u(p()),
d(h()), d(u()), d(d()), d(p()),
p(h()), p(u()), p(d()), p(p()) are the 16 four-colored trees of the first and only structure k = 1 (sum is 16 = A136793(2)); for
h(h(h())), h(h(u())), h(h(d())), h(h(p())),
h(u(h())), ...
..., p(d(p())),
p(p(h())), p(p(u())), p(p(d())), p(p(p())) are the 64 four-colored trees of the structure k = 1 and
h(h()h()), h(h()u()), h(h()d()), h(h()p()),
h(u()u()), h(u()d()), h(u()p()),
h(d()d()), h(d()p()),
h(p()p()),
...,
p(h()h()), p(h()u()), p(h()d()), p(h()p()),
p(u()u()), p(u()d()), p(u()p()),
p(d()d()), p(d()p()),
p(p()p()) are the 40 four-colored trees of the structure k = 2 (sum is 104 = A136793(3)).
Triangle T(n,k) begins:
4;
16;
64, 40;
256, 160, 256, 80;
1024, 640, 1024, 320, 1024, 640, 544, 640, 140;
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CROSSREFS
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Total number of elements up to and including row n is A087803.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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