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%I #12 Jun 19 2014 10:32:22
%S 4,16,64,40,256,160,256,80,1024,640,1024,320,1024,640,544,640,140,
%T 4096,2560,4096,1280,4096,2560,2176,2560,560,4096,2560,4096,1280,4096,
%U 2560,2560,1600,2176,1280,224,16384,10240,16384,5120,16384,10240,8704,10240,2240
%N 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.
%C 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.,
%C n = 1
%C {0} |-> () |-> 10_2
%C n = 2
%C {1} |-> (()) |-> 1100_2
%C n = 3
%C {2} > {1, 1} |-> ((())) > (()()) |-> 111000_2 > 110100_2
%C n = 4
%C {3} > {1, 2} > {1, 1, 1} |-> (((()))) > ((()())) > (()(())) > (()()()) |-> 11110000_2 > 11101000_2 > 11011000_2 > 11010100_2
%C The decimal equivalents of the binary encoded rooted trees in row n are the descending ordered elements of row n in A216648.
%D G. Gruber, Entwicklung einer graphbasierten Methode zur Analyse von Hüpfsequenzen auf Butcherbäumen und deren Implementierung in Haskell, Diploma thesis, Marburg, 2011
%D Eva Kalinowski, Mott-Hubbard-Isolator in hoher Dimension, Dissertation, Marburg: Fachbereich Physik der Philipps-Universität, 2002.
%H Martin Paech, <a href="/A242354/b242354.txt">Rows n = 1..10, flattened</a>
%H E. Kalinowski and W. Gluza, <a href="http://arxiv.org/abs/1106.4938">Evaluation of High Order Terms for the Hubbard Model in the Strong-Coupling Limit</a>, arXiv:1106.4938, 2011 (Physical Review B 85, 045105, Jan 2012)
%H E. Kalinowski and M. Paech, <a href="/A242354/a242354.pdf">Table of four-colored Butcher trees B(n,k,m) up to order n = 4</a>.
%H M. Paech, E. Kalinowski, W. Apel, G. Gruber, R. Loogen, and E. Jeckelmann, <a href="http://www.dpg-verhandlungen.de/year/2012/conference/berlin/part/tt/session/45/contribution/91">Ground-state energy and beyond: High-accuracy results for the Hubbard model on the Bethe lattice in the strong-coupling limit</a>, DPG Spring Meeting, Berlin, TT 45.91 (2012)
%e 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
%e n = 1 with A000081(1) = 1
%e h(), u(), d(), p() are the 4 four-colored trees of the first and only structure k = 1 (sum is 4 = A136793(1)); for
%e n = 2 with A000081(2) = 1
%e h(h()), h(u()), h(d()), h(p()),
%e u(h()), u(u()), u(d()), u(p()),
%e d(h()), d(u()), d(d()), d(p()),
%e 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
%e n = 3 with A000081(3) = 2
%e h(h(h())), h(h(u())), h(h(d())), h(h(p())),
%e h(u(h())), ...
%e ..., p(d(p())),
%e p(p(h())), p(p(u())), p(p(d())), p(p(p())) are the 64 four-colored trees of the structure k = 1 and
%e h(h()h()), h(h()u()), h(h()d()), h(h()p()),
%e h(u()u()), h(u()d()), h(u()p()),
%e h(d()d()), h(d()p()),
%e h(p()p()),
%e ...,
%e p(h()h()), p(h()u()), p(h()d()), p(h()p()),
%e p(u()u()), p(u()d()), p(u()p()),
%e p(d()d()), p(d()p()),
%e p(p()p()) are the 40 four-colored trees of the structure k = 2 (sum is 104 = A136793(3)).
%e Triangle T(n,k) begins:
%e 4;
%e 16;
%e 64, 40;
%e 256, 160, 256, 80;
%e 1024, 640, 1024, 320, 1024, 640, 544, 640, 140;
%Y Row sums give A136793.
%Y Row length is A000081.
%Y Total number of elements up to and including row n is A087803.
%Y Cf. A216648, A242353.
%K nonn,tabf
%O 1,1
%A _Martin Paech_, May 16 2014
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