A242354 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.
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
Offset: 1
Examples
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
n = 1 with A000081(1) = 1
h(), u(), d(), p() are the 4 four-colored trees of the first and only structure k = 1 (sum is 4 = A136793(1)); for
n = 2 with A000081(2) = 1
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
n = 3 with A000081(3) = 2
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;
References
- 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.
Links
- Martin Paech, Rows n = 1..10, flattened
- E. Kalinowski and W. Gluza, Evaluation of High Order Terms for the Hubbard Model in the Strong-Coupling Limit, arXiv:1106.4938, 2011 (Physical Review B 85, 045105, Jan 2012)
- E. Kalinowski and M. Paech, Table of four-colored Butcher trees B(n,k,m) up to order n = 4.
- M. Paech, E. Kalinowski, W. Apel, G. Gruber, R. Loogen, and E. Jeckelmann, Ground-state energy and beyond: High-accuracy results for the Hubbard model on the Bethe lattice in the strong-coupling limit, DPG Spring Meeting, Berlin, TT 45.91 (2012)
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