Martin Paech - cp's OEIS frontend

User: Martin Paech

Martin Paech has authored 3 sequences.

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

Martin Paech, May 16 2014

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.

			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;

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.

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)

Row sums give A136793.
Row length is A000081.
Total number of elements up to and including row n is A087803.
Cf. A216648, A242353.

A242353 Number T(n,k) of two-colored rooted trees of order n and structure k; triangle T(n,k), n>=1, 1<=k<=A000081(n), read by rows.

Original entry on oeis.org

2, 4, 8, 6, 16, 12, 16, 8, 32, 24, 32, 16, 32, 24, 20, 24, 10, 64, 48, 64, 32, 64, 48, 40, 48, 20, 64, 48, 64, 32, 64, 48, 48, 36, 40, 32, 12, 128, 96, 128, 64, 128, 96, 80, 96, 40, 128, 96, 128, 64, 128, 96, 96, 72, 80, 64, 24, 128, 96, 128, 64, 128, 96, 80

Offset: 1

Martin Paech, May 11 2014

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.

			Let {u, d} be a set of two colors, corresponding each with the up-spin and down-spin electrons in the underlying physical problem. (We consider each rooted tree as a cutout of the Bethe lattice in infinite dimensions.) Then for
n = 1 with A000081(1) = 1
  u(), d() are the 2 two-colored trees of the first and only structure k = 1 (sum is 2 = A038055(1)); for
n = 2 with A000081(2) = 1
  u(u()), u(d()), d(u()), d(d()) are the 4 two-colored trees of the first and only structure k = 1 (sum is 4 = A038055(2)); for
n = 3 with A000081(3) = 2
  u(u(u())), u(u(d())), u(d(u())), u(d(d())), d(u(u())), d(u(d())), d(d(u())), d(d(d())) are the 8 two-colored trees of the structure k = 1 and
  u(u()u()), u(u()d()), u(d()d()), d(u()u()), d(u()d()), d(d()d()) are the 6 two-colored trees of the structure k = 2 (sum is 14 = A038055(3)).
Triangle T(n,k) begins:
2;
4;
8,   6;
16, 12, 16,  8;
32, 24, 32, 16, 32, 24, 20, 24, 10;

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.

Martin Paech, Rows n = 1..14, 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 two-colored Butcher trees B(n,k,m) up to order n = 5.
M. Paech, A sonification of this sequence, created with MUSICALGORITHMS, using simple 'division operation' instead of modulo scaling (3047 elements, 240 bpm).
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)

Row sums give A038055.
Row length is A000081.
Total number of elements up to and including row n is A087803.
Cf. A216648.

A240605 Total number of distinct sequences for the number of double occupancy in the underlying Fermion problem (see comment), i.e., the number of distinct hopping sequences (cf. A198761, A225823) in four-colored rooted trees with n nodes, starting and ending with the same coloring in two colors (cf. A198760, corresponding to zero double-occupancy).

Original entry on oeis.org

1, 2, 10, 59, 397, 2878, 21266, 162732, 1253128, 9839212, 77644825, 620377508, 4981522538, 40351448045, 328421827064, 2690586461296, 22139293490054, 183106636176023, 1520309861062921, 12675106437486945, 106033283581264574, 890035798660219755

Offset: 2

Martin Paech, Apr 09 2014

nonn

The sequences of double-occupancy are generated by the operators T_{+U}, T_{-U}, and T_{0} defined in eq. (8) in Phys. Rev. B 85, 045105 (2012), see below.
Also the number of "island altitude-profiles" of length 2n-1, see examples, which satisfy the following requirements:
(1) Every profile starts and ends at sea-level (zero double-occupancies).
(2) The height increases and decreases with every step at most one unit.
(3) The maximum height does not go beyond floor(n/2).
(4) The minimum height does not fall below sea-level.
(5) Sea-level could only be reached after an even number of steps.
(6) For n even, no plateaus exist at maximum height (= n/2).
(7) For n even, two peaks at maximum height have an even distance.

			n = 2
  0 1 0  |->  T_{+U} T_{-U}  |->  /\
n = 3
                                                     __
  0 1 1 1 0  |->  T_{+U} T_{ 0} T_{ 0} T_{-U}  |->  /  \
  0 1 0 1 0  |->  T_{+U} T_{-U} T_{+U} T_{-U}  |->  /\/\
n = 4
                                                                       ____
  0 1 1 1 1 1 0  |->  T_{+U} T_{ 0} T_{ 0} T_{ 0} T_{ 0} T_{-U}  |->  /    \
                                                                       __/\
  0 1 1 1 2 1 0  |->  T_{+U} T_{ 0} T_{ 0} T_{+U} T_{-U} T_{-U}  |->  /    \
                                                                       __
  0 1 1 1 0 1 0  |->  T_{+U} T_{ 0} T_{ 0} T_{-U} T_{+U} T_{-U}  |->  /  \/\
                                                                       _/\_
  0 1 1 2 1 1 0  |->  T_{+U} T_{ 0} T_{+U} T_{-U} T_{ 0} T_{-U}  |->  /    \
                                                                       /\__
  0 1 2 1 1 1 0  |->  T_{+U} T_{+U} T_{-U} T_{ 0} T_{ 0} T_{-U}  |->  /    \
                                                                       /\/\
  0 1 2 1 2 1 0  |->  T_{+U} T_{+U} T_{-U} T_{+U} T_{-U} T_{-U}  |->  /    \
                                                                       /\
  0 1 2 1 0 1 0  |->  T_{+U} T_{+U} T_{-U} T_{-U} T_{+U} T_{-U}  |->  /  \/\
                                                                         __
  0 1 0 1 1 1 0  |->  T_{+U} T_{-U} T_{+U} T_{ 0} T_{ 0} T_{-U}  |->  /\/  \
                                                                         /\
  0 1 0 1 2 1 0  |->  T_{+U} T_{-U} T_{+U} T_{+U} T_{-U} T_{-U}  |->  /\/  \
  0 1 0 1 0 1 0  |->  T_{+U} T_{-U} T_{+U} T_{-U} T_{+U} T_{-U}  |->  /\/\/\

Alois P. Heinz, Table of n, a(n) for n = 2..400
E. Kalinowski and W. Gluza, Evaluation of High Order Terms for the Hubbard Model in the Strong-Coupling Limit, arXiv:1106.4938 [cond-mat.str-el], 2011 (Physical Review B 85, 045105, Jan 2012)
E. Kalinowski and M. Paech, Table of island altitude-profiles I(n,k) up to order n = 6.
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)

Cf. A198760, A198761, A225823.

b:= proc(x, y, m, v, d) option remember; `if`(y>x or y<0 or
       y>m or v and y=m and d=1 or y=0 and irem(x, 2)=1, 0,
      `if`(x=0, 1, `if`(v and y=m or y=0, 0, b(x-1, y, m, v,
      `if`(d=2, 2, 1-d)))+ `if`(y=0 or y=1 and irem(x, 2)=0, 0,
       b(x-1, y-1, m, v, `if`(d=2, `if`(v and y=m, 1, 2), 1-d)))+
       b(x-1, y+1, m, v, `if`(d=2, 2, 1-d))))
    end:
a:= n-> b(2*n-2, 0, iquo(n, 2, 'r'), r=0, 2):
seq(a(n), n=2..30);  # Alois P. Heinz, May 09 2014

b[x_, y_, m_, v_, d_] := b[x, y, m, v, d] = If[y>x || y<0 || y>m || v && y == m && d==1 || y==0 && Mod[x, 2]==1, 0, If[x==0, 1, If[v && y==m || y==0, 0, b[x-1, y, m, v, If[d==2, 2, 1-d]]] + If[y==0 || y==1 && Mod[x, 2]==0, 0, b[x-1, y-1, m, v, If[d==2, If[v && y==m, 1, 2], 1-d]]] + b[x-1, y+1, m, v, If[d==2, 2, 1-d]]]]; a[n_] := b[2*n-2, 0, Quotient[n, 2], Mod[ n, 2]==0, 2]; Table[a[n], {n, 2, 30}] (* Jean-François Alcover, Feb 24 2016, after Alois P. Heinz *)

Terms a(16) and a(17) are calculated on a HP Integrity Superdome 2-16s by courtesy of Hewlett-Packard Development Company, L.P., by Martin Paech, May 08 2014 (The used algorithm generates explicitly all distinct sequences of double-occupancy, i.e. all valid "island altitude-profiles", and counts them.)

a(18)-a(23) from Alois P. Heinz, May 08 2014

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User: Martin Paech

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.

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A242353 Number T(n,k) of two-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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