OFFSET
1,2
LINKS
M. Znojil, Non-Hermitian N-state degeneracies: unitary realizations via antisymmetric anharmonicities, arXiv:2010.15014 [quant-ph], 2020.
M. Znojil, Quantum phase transitions mediated by clustered non-Hermitian degeneracies, arXiv:2102.12272 [quant-ph], 2021.
M. Znojil, Quantum phase transitions mediated by clustered non-Hermitian degeneracies, Physical Review E 103 (2021), 032120.
Miloslav Znojil, Bose-Einstein condensation processes with nontrivial geometric multiplicites realized via PT-symmetric and exactly solvable linear-Bose-Hubbard building blocks, arXiv:2108.07110 [quant-ph], 2021.
EXAMPLE
F(n,1) are the sets {0,1}, {0,1,2}, {0,1,2,3}, {0,1,2,3,4}, ...
F(n,2) are the sets {0,2}, {0,2,4}, {0,2,4,6}, {0,2,4,6,8}, ...
F(n,3) are the sets {0,3}, {0,3,6}, {0,3,6,9}, {0,3,6,9,12}, ...
etc., and
G(n,1) are the sets {1}, {1,3}, {1,3,5}, {1,3,5,7}, ...
G(n,2) are the sets {2}, {2,6}, {2,6,10}, {2,6,10,14}, ...
G(n,3) are the sets {3}, {3,9}, {3,9,15}, {3,9,15,21}, ...
etc.
a(2) = 3 because there are three decompositions of F(2,1) = {0,1,2}: the trivial F(2,1), F(1,2) + G(1,1) = {0,2} + {1}, and F(1,1) + G(1,2) = {0,1} + {2}.
The a(3) = 6 decompositions of {0,1,2,3} are:
{{0,1,2,3}},
{{0,1,2}, {3}},
{{0,1}, {2}, {3}},
{{0,2}, {1,3}},
{{0,2}, {1}, {3}},
{{0,3}, {1}, {2}}.
PROG
(PARI)
tiles(S, t, w)={((i, b)->w(b) + sum(i=1, i, if(!bitnegimply(S[i], b), self()(i-1, b-S[i]))))(#S, t)}
Q(j, k, t)={sum(i=0, j-1, 1<<(t+i*k))}
S(n)={concat(concat(vector(n\2+1, k, vector(n\k-1, j, Q(j+2, k, 0)))), concat(vector(n\3+1, k, vector((n\k-1)\2, j, Q(j+1, 2*k, k)))))}
a(n)={tiles(S(n), Q(n+1, 1, 0), b->if(bittest(b, 0), hammingweight(b)-1, 1))} \\ Andrew Howroyd, Oct 29 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Miloslav Znojil, Oct 03 2020
EXTENSIONS
a(10)-a(42) from Andrew Howroyd, Oct 29 2020
STATUS
approved